Discontinuity, Nonlinearity, and Complexity

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Volume 5 Issue 4 December 2016

ISSN 2164‐6376 (print) ISSN 2164‐6414 (online) 

An Interdisciplinary Journal of

Discontinuity, Nonlinearity, and Complexity

Discontinuity, Nonlinearity, and Complexity Editors Valentin Afraimovich San Luis Potosi University, IICO-UASLP, Av.Karakorum 1470 Lomas 4a Seccion, San Luis Potosi, SLP 78210, Mexico Fax: +52 444 825 0198 Email: [email protected]

Lev Ostrovsky Zel Technologies/NOAA ESRL, Boulder CO 80305, USA Fax: +1 303 497 5862 Email: [email protected]

Xavier Leoncini Centre de Physique Théorique, Aix-Marseille Université, CPT Campus de Luminy, Case 907 13288 Marseille Cedex 9, France Fax: +33 4 91 26 95 53 Email: [email protected]

Dimitry Volchenkov The Center of Excellence Cognitive Interaction Technology Universität Bielefeld, Mathematische Physik Universitätsstraße 25 D-33615 Bielefeld, Germany Fax: +49 521 106 6455 Email: [email protected]

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Dumitru Baleanu Department of Mathematics and Computer Sciences Cankaya University, Balgat 06530 Ankara, Turkey Email: [email protected]

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Ravi P. Agarwal Department of Mathematics Texas A&M University – Kingsville, Kingsville, TX 78363-8202 USA Email: [email protected]

Editorial Board Vadim S. Anishchenko Department of Physics Saratov State University Astrakhanskaya 83, 410026, Saratov, Russia Fax: (845-2)-51-4549 Email: [email protected]

Continued on back materials

An Interdisciplinary Journal of Discontinuity, Nonlinearity, and Complexity Volume 5, Issue 4, December 2016

Editors Valentin Afraimovich Xavier Leoncini Lev Ostrovsky Dimitry Volchenkov

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Discontinuity, Nonlinearity, and Complexity 5(4) (2016) 341–353

Discontinuity, Nonlinearity, and Complexity https://lhscientificpublishing.com/Journals/DNC-Default.aspx

Further Results on the Stability of Neural Network for Solving Variational Inequalities Mi Zhou1†, Xiaolan Liu2,3‡ 1 School

of Science and Technology, Sanya College, Sanya, Hainan 572022, China of Science, Sichuan University of Science and Engineering, Zigong, Sichuan 643000, China 3 Sichuan Province University Key Laboratory of Bridge Non-destruction Detecting and Engineering Computing, Zigong, Sichuan 643000, China 2 College

Submission Info Communicated by A.C.J. Luo Received 1 November 2015 Accepted 17 December 2015 Available online 1 January 2017 Keywords Variational inequalities Neural network Positive semi-definite Continuously differentiable Exponential stability

Abstract This paper analyzes and proves the global Lyapunov stability of the neural network proposed by Yashtini and Malek when the mapping is continuously differentiable and the Jacobian matrix of the mapping is positive semi-definite. Furthermore, the neural network is shown to be exponentially stable under stronger conditions. In particular, the stability results can be applied to the stability analysis of variational inequalities with linear constraints and bounded constraints. Some examples show that the proposed neural network can be used to solve the various nonlinear optimization problems. The new results improve the existing ones in the literature.

©2016 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction We consider the following variational inequality: to find x∗ ∈ K such that F(x∗ ), y − x∗  ≥ 0,

for all x ∈ K.

(1)

where x = (x1 , . . . , xn )T ∈ Rn , F is a continuous mapping from Rn into itself, and K is a nonempty closed convex subset in Rn and ·, · denotes the usual inner product in Rn . We denote the variational inequality † Mi Zhou was

supported by Natural Science Foundation of Hainan Province (Grant No.114014), Scientific Research Fund of Hainan Province Education Department (Grant No.Hnjg2016ZD-20). Xiao-lan Liu was partially supported by National Natural Science Foundation of China (Grant No.61573010), Opening Project of Sichuan Province University Key Laboratory of Bridge Non-destruction Detecting and Engineering Computing(No.2015QZJ01),Artificial Intelligence of Key Laboratory of Sichuan Province(No.2015RZJ01), Scientific Research Fund of Sichuan Provincial Education Department(No.14ZB0208 No.16ZA0256), Scientific Research Fund of Sichuan University of Science and Engineering (No.2014RC01 No.2014RC03). ‡ Corresponding author. Email address: [email protected] ISSN 2164 − 6376, eISSN 2164 − 6414/$-see front materials © 2016 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/DNC.2016.12.001

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problem (1) by V I(F, K). V I(F, K) includes nonlinear complementarity problems(K = Rn+ ), systems of nonlinear equations(K = Rn ) as special cases. In many engineering and scientific fields including traffic equilibrium and network economics problems, K often has the following structure: K = {x ∈ Rn |Ax − b ≥ 0, Bx = c, x ≥ 0}, where

A ∈ Rm×n , rank(A) = m, B ∈ Rr×n , rank(B) = r, 0 ≤ m, r ≤ n, b ∈ Rm , b ∈ Rr .

By attaching lagrange multiplier y ∈ Rm to nonlinear constraint Ax − b ≥ 0, and lagrange multiplier z ∈ Rr to linear constraint Bx = c, according to the Karush-Kuhn-Tucker condition(refer to [1]), we obtain an equivalent form of the problem (1): to find y∗ ∈ Rm and z∗ ∈ Rr , where y∗ ≥ 0 such that u∗ = (x∗ , y∗ , z∗ )T is the solution of the following problem: (2) G(u∗ ), u − u∗  ≥ 0, for all u ∈ K0 . where the set K0 = {u = (x, y, z)|x ≥ 0, y ≥ 0, z

is free in sign} and ⎞ ⎛ F(x) − AT y − BT z ⎠. Ax − b G(u) = ⎝ Bx − c

We denote the problem (2) by V I(G, K0 ). As we are all known, x∗ is the solution of V I(F, K) if and only if = (x∗ , y∗ , z∗ ) is the solution of V I(G, K0 ). For solving V I(F, K), many researchers proposed the dynamical systems and studied the stability of it. For details, the readers can refer to [2], [3] and the references therein. The dynamical systems are generated by the equation:

u∗

dx = PK (x − α F(x)) − x. dt where α is a positive constant. Recently, neural networks for optimization problems have achieved many significant results. Among then, Kennedy and Chua [4] proposed a neural network which employs both the gradient method and penalty function method for solving nonlinear problems. Their energy function can be viewed as an ”inexact” penalty function, and thus the true optimizer can only be obtained when the penalty parameter is infinite. Xia and Wang [5] proposed a neural network for the problem (1). However, their model needs to estimate the Lipschitz constant and its structure is quite complex. It is well known that it is hard to estimate this constant in practice. By overcoming this shortfall, few primal and dual neural networks with two layers and one-layer structure were suggested in [7] and [8], [9] and the references therein. Some significant work has been done in recent years, see [6] and the references therein. For instance, in order to solve problem V I(G, K0 ), Yashtini and Malek [10] proposed the following neural network: ⎛ ⎞ (x − F(x) + AT y + BT z)+ − x du ⎠. = H(u) = ⎝ (3) (y − Ax + b)+ − y dt −Bx + c It is easy to see that H(u) = PK0 (u − G(u)) − u, where PK0 (·) : Rn+m+r → K0 is a projection operator defined by PK0 (u) = argminv∈K0 u − v . Three assumptions for the stability in the sense of Lyapunov and globally convergence of the dynamical systems (3) were developed as follows ( [10]): (A1) The mapping F is once differentiable on an open set including K. (A2) The mapping F is monotone on K. (A3) ∇F is positive definite on Rn+ = {x ∈ Rn |x ≥ 0}. Although the above three assumptions are weaker than the ones which is given by Xia(2004) [11], they are still strong. We does not need the assumption (A2) to ascertain the stability of dynamical system (3) in the sense

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of Lyapunov, it can be removed. Meanwhile, the assumption (A3) can be replaced by positive semi-definiteness of the mapping F. Example 1. Consider the problem of finding the optimal solution of the following constrained optimization problem: min f (x) = 0.4x1 + x1 2 + x2 2 + 0.5x3 2 + 0.5x4 2 +

x1 3 30

s.t. − 0.5x1 − x2 + x4 ≥ −0.5 x1 + 0.5x2 − x3 = 0.4 x≥0 The problem has only one solution x∗ = (0.2792, 0.2416, −0.0000, 0)T . x∗ is also the solution of the corresponding variational inequality where F(x) = ∇ f (x) and K = {x ∈ Rn | − 0.5x1 − x2 + x4 ≥ −0.5, x1 + 0.5x2 − x3 = 0.4, x ≥ 0}, A = (−0.5, −1, 0, 1), b = −0.5, B = (1, 0.5, −1, 0), C = −0.4. Therefore, ⎞ ⎛ 0.4 + 2x1 + 0.2x21 ⎟ ⎜ 2x2 ⎟. F(x) = ⎜ ⎠ ⎝ x3 x4 and



2 + 0.2x1 ⎜ 0 ∇F(x) = ⎜ ⎝ 0 0

⎞ 000 2 0 0⎟ ⎟. 0 1 0⎠ 001

Note that ∇F(x) is positive semi-definite on R4+ . The condition (A3) can not be used to ascertain the stability of dynamical systems (3) for solving the problem, because ∇F(x) is only positive semi-definite, but not positive definite. However, we will show that this dynamical systems is stable in the Lyapunov sense in Section 3 and give corresponding simulation results in Section 5. Example 2. Consider the problem of finding a solution of the following variational inequality: F(x∗ ), x − x∗  ≥ 0,

for all x ∈ K.

The mapping F and the constraint set K are defined by ⎞ ⎛ 3x1 − x11 + 3x2 ⎜ 3x1 + 3x2 ⎟ ⎟ F(x) = ⎜ ⎝ 4x3 + 4x4 ⎠ . 4x3 + 4x4 − 3 and K = {x ∈ R4 |x1 − x2 = 1, x3 + 3x4 ≥ 0,  ≤ x ≤ h¯} where  = (−1, −3, −3, 0)T and h¯ = (2, 5, 8, 10)T . This problem has only one solution x∗ = (0.6319, −0.3681, −3.0000, 3.7500)T . The mapping F is not monotone on K, namely, it does not satisfy the (A2). But we can show the stability of corresponding dynamical system in the sense of Lyapunov in Section 3 and give simulation results in Section 5. The objective of this paper is to improve the existing sufficient conditions for the stability in the sense of Lyapunov and exponentially stability of dynamical systems (3). In addition to removing condition (A2), we can still obtain the stability in the sense of Lyapunov. Our new results include the case that the Jacobian matrix of

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F is continuous and positive definite, namely, F is continuously differentiable, and the Jacobian matrix of F is positive definite, and then exponentially stability of dynamical systems (3) can be obtained. An application to constrained optimization and nonlinear variational inequality is given to show the significance of the results obtained. The organization of this paper is as follows. In the next section, the preliminary information is introduced to facilitate later discussions. In Section 3, we prove the stability in the sense of Lyapunov and exponentially stability of dynamical systems (3). Some extensions are reported in the section 4. Simulation results are presented in Section 5. In the last section, we give the conclusions. And we denote the solution of problem (2) by S∗ . Moreover, we denote the set of equilibrium points of dynamical systems (3) by E ∗ , which is assumed to be nonempty. 2 Preliminaries For later discussion, some definitions and lemmas are introduced. Definition 1. The Jacobian matrix ∇F(u) is said to be positive semi-definite on K if hT ∇F(u)h ≥ 0,

for all u ∈ K, h ∈ Rn .

∇F(u) is positive definite on K if the above inequality holds strictly. Definition 2. The dynamical systems (3) is said to converge globally to the solution set S∗ of (2) if, irrespective of the initial point, the trajectory of the dynamical systemsu(t) satisfies lim dist(u(t), S∗ ) = 0,

(4)

t→∞

where dist(u, S∗ ) = inf∗ u − v . v∈S

It is easy to see that, if the set S∗ has a unique point u∗ , then (4) implies that lim u(t) = u∗ .

t→∞

If the dynamical systems still stable at u∗ in the Lyapunov sense, then the dynamical systems globally asymptotically stable at u∗ . Definition 3. The dynamical system (3) is said to be globally exponentially stable with degree η at u∗ if the trajectory of the system u(t) satisfies u(t) − u∗ ≤ μ exp(−η (t − t0 )),

for all t > t0 ,

where μ is a positive constants dependent on the initial point and η is a positive constants independent of the initial point. It is clear that globally exponential stability is necessarily globally asymptotical stability and that the dynamical systems converges arbitrarily fast. Lemma 1. Assume that K is a closed convex set. Then (v − PK (v))T (PK (v) − x) ≥ 0,

for all x ∈ K, v ∈ Rn ,

and PK (·) is non-expansive, i.e, PK (u) − PK (v) ≤ u − v ,

for all u ∈ Rn , v ∈ Rn .

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Proof. See Ref [12]. Lemma 2. u∗ is the solution of V I(G, K0 ) if and only if u∗ is the equilibrium point of the dynamical systems (3). Proof. See Ref [11].  n The norm is usual 2-norm in R , namely, x = ∑ni=1 x2i . 3 Dynamical systems analysis Theorem 3. Suppose that F is differentiable on an open set including K and ∇F(u) is positive semi-definite on Rn+ . Then for any initial point u(t0 ) ∈ K0 , (1) there exists a unique continuous solution u(t) = (x(t), y(t), z(t)) ∈ K0 for (3) over [t0 , T (u0 )]. (2)In addition, x(t) ≥ 0, y(t) ≥ 0. Proof. See Ref [11]. Now, we give improvements on existing results in the case where F is differentiable, ∇F(u) is positive semi-definite, but F is not necessarily monotone on K. Theorem 4. Suppose that F is continuously differentiable on an open set including K and ∇F(u) is positive semi-definite on Rn+ . If u∗ = (x∗ , y∗ , z∗ ) is an equilibrium point of (3) where x∗ is the solution of (1), then the dynamical systems (3) is stable in the sense of Lyapunov. Proof. First, consider the following function: 1 1 E(u) = −G(u), H(u) − H(u) 2 + u − u∗ 2 . 2 2 by Lemma 1, (v − PK0 (v))T (PK0 (v) − u) ≥ 0, Take v = u − G(u), it follows that

for all u ∈ K0 , v ∈ Rn+m+r .

(u − G(u) − PK0 (u − G(u)))T (PK0 (u − G(u)) − u) ≥ 0, and then

−G(u)T [PK0 (u − G(u)) − u] ≥ PK0 (u − G(u)) − u 2 .

since H(u) = PK0 (u − G(u)) − u, thus −G(u), H(u) ≥ H(u) 2 . By the Theorem 3.2 in [13],

∇E(u) = G(u) − (∇G(u) − I)H(u) + u − u∗,

where ∇G(u) denotes the Jacobian matrix of G, and ⎞ ⎛ ∇F(x) −AT −BT O1 O2 ⎠ , ∇G(u) = ⎝ A B O3 O4 where O1 ∈ Rm×m , O2 ∈ Rm×r , O3 ∈ Rr×m , O4 ∈ Rr×r are zero matrices. Now, du dE = ∇E(u),  dt dt = G(u) − (∇G(u) − I)H(u) + u − u∗, H(u) = G(u) + u − u∗ , H(u) + H(u) 2 − H(u), ∇G(u)H(u),

(5)

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and by the results gave in [14], we obtain that H(u) + u − u∗ , −H(u) − G(u) ≥ 0, which deduces that G(u) + u − u∗ , H(u) ≤ −u − u∗ , G(u) − H(u) 2 . So

dE ≤ −u − u∗ , G(u) − H(u), ∇G(u)H(u). dt

Since ∇F is positive semi-definite, so is ∇G. Thus dE ≤ −u − u∗ , G(u). dt Since by Lemma 2, we obtain that u∗ is the solution of (2), u − u∗ , G(u∗ ) ≥ 0, It follows that

for all u ∈ K0 .

dE ≤ −u − u∗ , G(u) − G(u∗ ). dt

Since ∗

ˆ



u − u , G(u) − G(u ) = ˆ =

1 0 1 0

(u − u∗ )∇G(u + s(u − u∗ ))(u − u∗ )ds (u − u∗ )∇G(u)(u ˆ − u∗ )ds,

where uˆ = u + s(u − u∗ ), we obtain that ∗



ˆ

1



ˆ

1

(x − x )∇F(x)(x ˆ − x )ds + (y − y∗ )T AT (x − x∗ )ds 0 0 ˆ 1 ˆ 1 (z − z∗ )T BT (x − x∗ )ds − (x − x∗ )T AT (y − y∗ )ds + 0 0 ˆ 1 (x − x∗ )T BT (z − z∗ )ds − 0 ˆ 1 (x − x∗ )∇F(x)(x ˆ − x∗ )ds, =

u − u , G(u) − G(u ) =



0

where uˆ = (x, ˆ y, ˆ zˆ). Since ∇F is positive semi-definite, thus dE ≤ −u − u∗ , G(u) − G(u∗ ) dt ˆ 1 (x − x∗ )∇F(x)(x ˆ − x∗ )ds =− 0

≤ 0. So the dynamical systems (3) is stable in the Lyapunov sense.

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Remark 1. Compared with Yashtini and Malek [10], Theorem 4 does not require the additional condition that the mapping F is monotone on K and that the Jacobian matrix of F only positive semi-definiteness instead of positive definiteness. Obviously, our conditions is much weaker than that in [10]. Namely, Theorem 4 removes the additional assumption (A2) and weaken the (A3) into the positive semi-definiteness of the Jacobian matrix of F. In the following, if we strengthen the matrix of the mapping F on K, can we obtain stronger results than that in Yashtini and Malek [10]? The answer is affirmative. Thus, furthermore, we establish the exponentially stability results of dynamical systems (3), where F is continuously differentiable, and the Jacobian matrix of the mapping F is positive definite. Theorem 5. Suppose that F is continuously differentiable on K, ∇F is positive definite on K. Then the solution trajectory of the neural network (3) is exponentially stable at u∗ where u∗ = (x∗ , y∗ , z∗ ) is an equilibrium point of (3). Proof. Consider the Lyapunov function E(u) which is defined in Theorem 4. From the proof of Theorem 4, we know that dE ≤ −u − u∗ , G(u), dt and then ˆ t

E(u(t)) ≤ E(u(t0 )) − By the inequality (5), we obtain that E(u(t)) ≥

t0

1 ∗ 2 2 u − u .

ˆ

∗ 2

u(t) − u ≤ 2E(u(t0 )) − 2 ≤ 2E(u(t0 )) − 2 = 2E(u(t0 )) − 2 By the results of Theorem 4, we obtain that

dE dt

G(u(s)), u − u∗ ds.

t

t

ˆ 0t t ˆ 0t t0

It follows that

G(u(s)), u − u∗ ds G(u(s)) − G(u∗ ), u − u∗ ds (u − u∗ )T ∇G(u)(u ˆ − u∗ )ds.

≤ 0, E(u) is nonincreasing respect to u. It implies that

r u(t) ⊂ Ω = {u ∈ Rn+ × Rm + × R |E(u) ≤ E(u0 )}

and E(uk ) → ∞ whenever uk → ∞ by E(u(t)) ≥ 12 u − u∗ 2 . Then {u(t) = (x(t), y(t), z(t))} and Ω are both bounded. Since ∇F(x) is positive definite on K, and ⎞ ⎛ ∇F(x) −AT −BT O1 O2 ⎠ , ∇G(u) = ⎝ A B O3 O4 where O1 ∈ Rm×m , O2 ∈ Rm×r , O3 ∈ Rr×m , O4 ∈ Rr×r are zero matrices, then ∇G(u) is positive definite on r Rn+ × Rm + × R , and so ∇G(u) is positive definite on Ω. vˆT ∇G(u)vˆ > 0,

for all u ∈ Ω,

v ˆ 2 = 1.

Let g(u) = vˆT ∇G(u)vˆ be a function defined on Ω. Since F is continuously differentiable on K, thus ∇F is continuous on K, it implies that ∇G is continuous on Ω. Then g(u) is continuous on Ω, Thus, there exists σ > 0 such that vT ∇G(u)v ≥ σ v 2 , for all u ∈ Ω, v ∈ Rn+m+r .

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Note that u(t) is bounded, then the following set: Ω1 = {u|u = u(t) + s(u(t) − u∗ ), 0 ≤ s ≤ 1,t ≥ t0 } is bounded and Ω1 ⊂ Ω. Since u(t) + s(u(t) − u∗ ) ∈ Ω1 , then for all t ≥ t0 , (u(t) − u∗ )T ∇G(u(t) + s(u(t) − u∗ ))(u − u∗ ) ≥ σ u(t) − u∗ 2 , It follows that

ˆ

∗ 2

u(t) − u ≤ 2E(u(t0 )) − β

t

t0

u(s) − u∗ 2 ds,

for all t ≥ t0 .

for all t ≥ t0 ,

where β = 2σ . According to the Bellman-Gronwall inequality [15], we obtain that u(t) − u∗ 2 ≤ 2E(u(t0 )) exp

´t t0

−β ds

= 2E(u(t0 )) exp−β (t−t0 ) . It follows that



−β (t − t0 ) , for all t ≥ t0 . 2 Therefore, the proposed neural network is globally and exponentially stable at u∗ . u(t) − u∗ ≤

2E(u(t0 )) exp

Remark 2. Compared with Yashtini and Malek [10], Theorem 5 required the mapping F is continuously differentiable and the Jacobian matrix of F is positive definite which ascertain the exponentially stability of dynamical systems (3). In fact, it is known that if F is continuously differentiable and the Jacobian matrix ∇F(x) is positive definite for all x ∈ K, i.e. d, ∇F(x)d > 0, for all x ∈ K, d ∈ Rn (d = 0), then F is strictly monotone on K, see [16]. Remark 3. According to Theorem 5, we conclude that the output trajectory of the projection neural network can converge to a solution with any given precision ε > 0 within a finite time. In fact, we see that u(t) − u∗ ≤ It follows that



2E(u(t0 )) exp

−β (t − t0 ) , 2

for all t ≥ t0 .

2E(u(t0 )) −β (t − t0 ) ≥ , exp 2 ε 2E(u(t0 )) 2 ). (t − t0 ) ≥ ln( β ε

and then

Thus u(t) − u∗ < ε , provided that 2 t ≥ t0 + ln( β



2E(u(t0 )) ). ε

4 Extensions Consider the following variational inequality problem: ˜ To find x∗ ∈ K, F(x∗ ), y − x∗  ≥ 0,

˜ for all x ∈ K,

where x, A, B, c, d and F are defined in Section 1 and K˜ = {x ∈ Rn |Ax − b ≥ 0, Bx = c, x ∈ X },

for X = {x| ≤ x ≤ h¯}.

(6)

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Corollary 6. x∗ is the solution of (6) if and only if u∗ = (x∗ , y∗ , z∗ )T is the equilibrium point of the neural network (7): ⎛ ⎞ (PX (x − F(x) + AT y + BT z) − x du ⎠, = H(u) = ⎝ (7) (y − Ax + b)+ − y dt −Bx + c where u = (x, y, z)T and PX · : Rn → X is a projection operator which is defined by PX (x) = [PX (x1 ), . . . , PX (xn )]T , ∀i = 1, 2, . . . , n. ⎧ ⎪ xi < i ⎨i , PX (xi ) = xi , i ≤ xi ≤ h¯i . ⎪ ⎩ h¯i , xi > h¯i Inspired by Theorem 4 and Theorem 5, we can get the similar stability results for the neural network (6): Theorem 7. Suppose that F is continuously differentiable on K˜ and ∇F is positive semi-definite for any x ∈ X . Then the neural network (7) is stable in the sense of Lyapunov. Furthermore, if F is continuously differentiable, and ∇F is positive definite on X , then the neural network (7) is globally exponentially stable at u∗ where u∗ is the equilibrium point of the neural network (7). Proof. One can get the results following the similar arguments presented in Theorem 4 and Theorem 5. The proof is completed. Corollary 8. For the set Kˆ = {x ∈ Rn+ |x ∈ X }, (6) becomes the following complementarity problem: x ∈ X , F(x) ≥ 0, x, F (x) = 0,

(8)

and the corresponding neural network for solving (8) can be given by dx = PX (x − F(x)) − x. dt

(9)

Proof. It is trivial. Corollary 9. x∗ is the solution of (8) if and only if x∗ is the equilibrium point of neural network (9). Theorem 10. If F is continuously differentiable and ∇F is positive definite on X . Then the neural network (9) is globally exponentially stable at x∗ where x∗ is the equilibrium point of neural network (9). Proof. See Corollary 1 [17]. 5 Simulation results In order to demonstrate the effectiveness and efficiency of the proposed neural network, we implement it in MATLAB 7.1 to solve Example 1 and Example 2 and a nonlinear variational inequality problem Example 3. We give simulation results in Example 1 with feasible initial point and infeasible initial point. Figures 1 and 2 depict transient behavior based on neural network (3) with the feasible initial point (0.2, 1.0.3, 0.75, −0.3, −0.5)T and the infeasible initial point (0.5, 0.5, −0.1, −0.4, 0.9, −0.5)T , respectively. All simulation results show that the neural network (3) is stable and converges to the optimal solution x∗ = (0.2792, 0.2416, −0.0000, 0)T . And here y∗ = 0, z∗ = 0.9662, thus u∗ = (0.2792, 0.2416, −0.0000, 0, 0, 0.9662)T . Theorem 4 guarantees that the

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Mi Zhou, Xiaolan Liu / Discontinuity, Nonlinearity, and Complexity 5(4) (2016) 341–353 1

z

x(t),y(t),z(t)

0.5 x1 x2

x4 x3 0

−0.5

y

0

5

10

15 Time(sec)

20

25

30

Fig. 1 Transient behavior based on the neural network model (3) with the feasible initial point (0.2, 1.0.3, 0.75, −0.3, −0.5)T in Example 1. 1

z

0.8 y 0.6

x(t),y(t),z(t)

0.4 x4

x1 x2

0.2 x3

0

−0.2

−0.4

−0.6

0

5

10

15 Time(sec)

20

25

30

Fig. 2 Transient behavior based on the neural network model (3) with the infeasible initial point (0.5, 0.5, −0.1, −0.4, 0.9, −0.5)T in Example 1.

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351

stability of the neural network (3), although ∇F(x) is only positive semi-definite on R4+ . The simulation results show that the neural network (3) with any initial point is always convergent to u∗ within a finite time. Now, Fig.3 and Fig.4 give simulation results in Example 2 with the feasible initial point (1, 0, 3, 2, 2, −3)T and the infeasible initial point (0.1, 0.9, 4, 3, −2, −3)T , respectively. Simulation results show that the neural network (3) is stable and converges to the optimal solution x∗ = (0.6319, −0.3681, −3.0000, 3.7500)T . And here y∗ = 0, z∗ = −0.7913, thus u∗ = (0.6319, −0.3681, −3.0000, 3.7500, 0, −0.7913)T . Theorem 4 guarantees that the stability of the neural network, although F(x) is not monotone on K. The simulation results show that the neural network (3) with any initial point is always convergent to u∗ within a finite time.

4 x4 3

(x(t),y(t),z(t))

2

1 x1 y

0 x2 z

−1

−2 x3 −3

0

5

10

15 Time(sec)

20

25

30

Fig. 3 Transient behavior based on the neural network model (3) with the feasible initial point (1, 0, 3, 2, 2, −3)T in Example 2.

Example 3. Consider the following variational inequality problem: F(x∗ ), x − x∗  ≥ 0,

for all x ∈ K.

The mapping F and the constraint set K are defined by ⎞ ⎛ 5x1 + (x1 + 2)2 + x2 + x3 + 10 F(x) = ⎝ 5x1 + 3x2 2 + 10x2 + 3x3 + 10 ⎠ . 10(x1 + 2)2 + 8x2 2 + 4x3 + 3x3 2 and K = {x ∈ R3 |x1 − 2x2 − x3 = 1, x1 + x2 + x3 ≥ 4, x ≥ 0}. Note that ⎞ ⎛ 1 1 9 + 2x1 5 6x2 + 10 3 ⎠ . ∇F(x) = ⎝ 20x1 + 40 16x2 4 + 6x3 Note that F(x) is continuously differentiable and ∇F(x) is positive definite on R3+ . This problem has only one optimal solution x∗ = (2.9998, 0.9998, 0). And here y∗ = 46.6649, z∗ = 4.3332, thus u∗ = (2.9998, 0.9998,

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4

x4

3

(x(t),y(t),z(t))

2

1

x1 y

0

x2 z

−1

−2 x3 −3

0

5

10

15 Time(sec)

20

25

30

Fig. 4 Transient behavior based on the neural network model (3) with the infeasible initial point (0.1, 0.9, 4, 3, −2, −3)T in Example 2.

6

5

4

3

2

1

0

0

20

40

60

80

100

Fig. 5 Convergence behavior of the norm u(t) − u∗ 2 based on the neural network model (3) with 20 random initial points in Example 3.

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0, 46.6649, 4.3332)T . Theorem 5 can be used to ascertain the exponential stability of dynamical systems (3) for solving the problem. Fig.5 shows that this dynamical systems (3) is exponentially stable at u∗ . Fig.5 displays the convergence behavior of the u(t) − u∗ 2 based on neural network (3) with 20 random initial points. 6 Conclusions In this paper, we further study the stability of dynamical systems (3) whose equilibrium points coincide with solutions of variational inequality problems. We obtain results on the stability of such a system in the sense of Lyapunov under the continuously differentiability of the mapping and positive semi-definiteness of the Jacobian matrix of the mapping, and on exponential stability under the condition that the mapping is continuously differentiable and the Jacobian matrix is positive definite. The stability results obtained improve the existing results. In particular, the stability results can apply to the stability results of variational inequalities with linear constraints and bounded constraints and some complementarity inequalities. Simulation results show that our results can be used to solve effectively variational inequality problems and related optimization problems. References [1] Luenberger, D.G. (1973), Introduction to Linear and Nonlinear Programming, Addison-Wesley, Reading, MA. [2] Xia, Y.S. and Wang, J. (2000), On the stability of globally projected dynamical systems, Journal of Optimization Theory and Applications, 106(1), 129–150. [3] Xia, Y.S. (2004), Further results on global convergence and stability of globally projected dynamical systems, Journal of Optimization Theory and Applications, 122(3), 627–649. [4] Kennedy, M.P. and L.O.Chua (1988), Neural networks for nonlinear programming, IEEE Transactions on Circuits and Systems, 35, 554–562. [5] Xia, Y.S. and Wang, J. (1998), A general methodology for designing globally convergent optimization neural networks, IEEE Trans. Neural networks, 9(6), 1331–1343. [6] Malek, A. and Oskoei, H.G. (2005), Primal-dual solution for the linear programming problems using neural networks, Appl.Math.Comput, 169, 451–471. [7] Xia, Y.S. (1996), A new neural network for solving linear and quadratic programming problems, IEEE Trans. Nerual networks , 7(6), 1544–1547. [8] Tao, Q., Cao, J.D., Xue, M.S., and Qiao, H. (2001), A high performance neural network for solving nonlinear programming problems with hybrid constraints, Phys. Lett.A, 288(2), 88–94. [9] Wang, J., Hu, Q., and Jiang, D. (1993), A Lagrangian neural network for kinematics control of redundant robot manipulators, IEEE Trans. Nerual networks, 10(5), 1123–1132. [10] Yashtini, M. and Malek, A. (2007), Solving complementarity and variational inequalities problems using neural network, Appl.Math.Comput, 190, 216–230. [11] Xia, Y.S. (2004), An Extended Projection neural network for constrained optimization, Neural Computation, 16, 863– 883. [12] Kinderlehrer, D. and Stampcchia, G. (1980), An introduction to variational inequalities and their applications, Academic Press, New York. [13] Fukushima, M. (1992), Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems, Mathematical Programming, 53, 99–110. [14] Pang, J.S. (1987), A Posteriori error bounds for the linearly-constrained variational inequality problem, Math. Oper. Res, 12, 474–484. [15] Slotine, J.J. and Li, W. (1991), Applied nonlinear control, Englewood Cliffs, NJ: Prentice Hall. [16] Ortega, J.M. and Rheinboldt, W. C. (1970), Iterative solution of nonlinear in several variables, Academic Press, New York. [17] Xia, Y.S. and Wang, J. (2000), A recurrent neural network for solving linear projection equations, Neural Networks, 13, 337–350.

Discontinuity, Nonlinearity, and Complexity 5(4) (2016) 355–364

Discontinuity, Nonlinearity, and Complexity https://lhscientificpublishing.com/Journals/DNC-Default.aspx

How the Minimal Poincar´e Return Time Depends on the Size of a Return Region in a Linear Circle Map N. Semenova, E. Rybalova, V. Anishchenko† Saratov State University, Saratov, 410012, Russia Submission Info Communicated by Valentin Afraimovich Received 15 December 2015 Accepted 10 April 2016 Available online 1 January 2017

Abstract It is found that the step function of dependence of the minimal Poincar´e return time on the size of a return region τinf (ε ) for the linear circle map with an arbitrary rotation number can be approximated analytically. All analytical results are confirmed by numerical simulation.

Keywords Circle map Poincar´e recurrence Afraimovich-Pesin dimension rotation number Diophantine number Fibonacci stairs ©2016 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Poincar´e recurrence is one of the fundamental features occurring in the time evolution of dynamical systems. Almost every trajectory in the phase space of a system with a fixed measure returns in the vicinity of an initial state. H. Poincar´e called such a trajectory as Poisson-stable [1]. If the system demonstrates chaotic behaviour, then a sequence of Poincar´e recurrences is random and thus can be described by using statistical methods. There are two approaches for analysing of Poincar´e recurrences, namely, local and global ones. Unlike the classical local approach, in which Poincare´e recurrences are calculated in a ε -vicinity of the initial state, the global approach deals with the recurrence characteristics for the whole set. The main characteristic of Poincar´e recurrences in the global approach is the recurrence time dimension which is called the Afraimovich–Pesin dimension (AP dimension) [2, 3]. The return time statistics in the global approach depends on the topological entropy hT . The case of mixing sets (hT > 0) has been studied analytically [2–4] and the results have been confirmed by numerical simulations [5–7]. If hT = 0, then the behaviour is ergodic and without mixing. Such a system can be exemplified by the † Corresponding

author. Email address: [email protected] ISSN 2164 − 6376, eISSN 2164 − 6414/$-see front materials © 2016 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/DNC.2016.12.002

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circle map: Θn+1 = Θn + Δ + K sin Θn mod 2π .

(1)

The variable Θn , which can take a value from zero to 2π , characterizes the rotation angle of the point on a circle with radius 1. The sign mod 2π means that the 2π -fold part of the phase variable is discarded. K is the parameter of nonlinearity, Δ sets the rotation number [8, 9] and is the fixed shift on the circle. The map (1) simulates the dynamics of two-frequency quasiperiodic oscillations in the Poincar´e section of a two-dimensional torus [10–12]. If K = 0, the map is linear: Θn+1 = Θn + Δ mod 2π .

(2)

The analytical results for the linear circle map [3] has been extensively confirmed in numerical experiments in the works [7, 13, 14]. A new dependence of the minimal return time τinf (ε ) on the vicinity size ε has been found. In [13, 14] we call it the “Fibonacci Stairs”. It has been proven that this dependence has a universal geometry for the golden and silver rotation numbers, i.e., the height and the length of each step D are D = ln δ , where δ is the rotation number [13]. For another irrational rotation numbers this universal feature of τinf (ε )dependence does not occur [14]. The aim of the present work is to analyse analytically and numerically the geometrical features of the Fibonacci stairs in a general case of any rotation numbers. 2 System under study In this work we analyse the particular case of the linear shift on the circle (K = 0 in (1)). The rotation number δ is the main characteristic which enables one to diagnose periodic and quasiperiodic regimes. In general, the rotation number is defined as follows: Θn − Θ0 , (3) δ (Δ, K) = lim n→∞ 2π n where Θn is the rotation angle of the circle map (1) or (2). Now we take into account the 2π -fold part. Θ0 is the initial angle, n is the number of iterations. Thus, the rotation number is the mean rotation angle Θ after one iteration of the map. In the linear case, K = 0, the rotation number depends only on the parameter Δ as follows:

δ = Δ/2π .

(4)

1 From a physical standpoint, the rotation number characterizes the ratio of independent frequencies δ = ω ω2 for two-frequency quasiperiodic behavior (see, for instance, [12, 15]). Rational rotation numbers correspond to a periodic sequence Θn = Θn+q , where q is the period of motion. In the case of irrational rotation numbers, this sequence Θn covers the circle uniformly as n → ∞. It corresponds to the two-frequency quasiperiodic regime with an irrational ratio of ω1 and ω2 frequencies.

3 Poincar´e recurrences in the linear circle map In the global approach, the whole set of phase trajectories of a dynamical system is covered with cubes (or balls) of size ε  1. For each covering element ξ j ( j = 1, 2, . . . , m) a minimal return time τinf (ξ j ) of the trajectory to the ξ j neighbourhood is calculated. Then the mean minimal return time τinf (ε ) is found over the whole set of covering elements ξ j . The map (2) produces the set {Θn , mod 2π }. This set is an example of the simplest minimal set with irrational rotation number δ , for which theory of Poincar´e recurrences has been fully developed. In this work we use the following main theoretical results.

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357

It was proved [2, 3] that for the linear circle map (2) d

τinf (ε ) ∼ ε ν (δ ) lnτinf (ε ) ∼

− ν (dδ ) ln ε ,

or (5)

ε  1, d = 1,

where τinf (ε ) is the mean minimal return time which is found over the whole set of covering elements of size ε , ν (δ ) is the maximal rate of Diophantine approximations of an irrational number δ over all possible pairs of p and q, and d is the fractal dimension of the set, which is equal to one. In general for ergodic sets with zero topological entropy it was proved [2, 3] that lnτinf (ε ) ∼ −

d ln ε , ε  1, αc

(6)

where αc is the Afraimovich–Pesin dimension. Comparing (5) and (6) one can obtain that αc = ν (δ ) for the circle shift (2). In such a way the AP dimension is equal to the rate of Diophantine approximations ν (δ ). For Diophantine irrational numbers ν (δ ) = 1 and thus αc = 1. For an irrational rotation number the probability distribution p(Θ) is uniform in the interval [0; 2π ). This implies that in this case the local and global approaches can give equivalent results. Hence we can calculate τinf (ε ) instead of τinf (ε ). The theoretical results corroborated for the circle√map have been confirmed by numerical simulation in our paper [13] for the case of the golden ratio δ = 12 ( 5 − 1). The universal dependence of τinf (ε ) which we referred√to as the “Fibonacci Stairs” has been found. This dependence is shown in Fig. 1 for the golden ratio δ = 12 ( 5 − 1). We have established that the “Fibonacci Stairs” has several features which are as follows.

377

ln((ε))

6

233 144

5

89

D

4

55 34 21

3 2

13

D −5

−4

−3

8 −2

−1

ln(ε) Fig. 1 “Fibonacci Stairs”: Dependence of the minimal return time on the vicinity size for the circle map (2) with √ δ = 12 ( 5 − 1) [13].

1. When ε decreases, the sequence of τinf (ε ) values grows and strictly corresponds to the basic Fibonacci series {Fi } (as indicated in Fig. 1). Each minimal return time which relates to each ith step of the Fibonacci stairs corresponds to the denominator qi of the ith convergent for the fraction pi /qi . For the golden ratio qi = Fi . 2. When ε is varied within any of the stair steps, three return times τ1 < τ2 < τ3 can be distinguished. Additionally, τ1 = τinf . This property follows from Slater’s theorem [15]. 3. The length and height of the steps in Fig.1 depend on the rotation number as D = − ln δ .

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√ The silver ratio corresponds to the rotation number δ = 2 − 1. In this case we obtain the same step function sequence of values τinf (ε ) strictly with all the features of “Fibonacci Stairs”. However, when ε decreases, the √ 3 corresponds to the basic Pell series. In general cases, for example, δ = 2, δ = e or δ = lg 5, the first two universal properties given above are preserved but the third one is violated. We note that in the first property, the sequence of τinf (ε ) for each indicated δ still looks like as a step function but is described by a different law (series) with no special name. Thus, the Afraimovich–Pesin dimension cannot be calculated for the system (2) using (5) because of unpredictability of the next step emergence. In the present work we try to predict analytically the ε values, which correspond to the emergence of new steps, by using the rotation number value and the stepwise configuration of ε ). This enables one to calculate the exact value of AP dimension for any Diophantine rotation the function τinf (√ numbers as δ = 3 2, δ = e or δ = lg 5. 4 Fibonacci Stairs approximation The structure of the “Fibonacci Stairs” is closely related to the theory of convergents and continued fractions. An irrational number is a real number which cannot be written as a fraction p/q, where p and q are natural numbers, 1, 2, . . .. In the general case, an irrational rotation number can be presented in the form of a continued fraction [16]:

δ = a0 +

1 a1 + a

2+

1

(7)

1 1 a3 + ...

This produces a sequence of approximation coefficients {ai }, i ≥ 0. The notation [a0 ; a1 , a2 , a3 , . . .] is an infinite continued fraction representation of the irrational number. The irrational rotation number δ can be approximated by the fraction of two integers pi /qi . This is the method of rational approximations. The ith convergent of the continued fraction δ = [a0 ; a1 , a2 , a3 . . . ] is a finite continued fraction [a0 ; a1 , a2 , . . . , ai ], which value is equal to the rational number pi /qi . The increasing sequences of numerators {pi } and denominators {qi } are called continuants of the ith convergent (7) and can be found using fundamental recurrence formulas: p−1 = 1, p0 = a0 , pi = ai pi−1 + pi−2 ,

(8)

q−1 = 0, q0 = 1, qi = ai qi−1 + qi−2 , where {ai } are natural coefficients of the continued fraction, pi , qi are numerators and denominators of the convergent. It has been found [7,13] that for any rotation number, the dependence τinf (ε ) is a step function and each value τinfi , which corresponds to the ith step, is equal to the denominators of the ith convergent pi /qi of the rotation number δ . Using the equality τinfi (ε ) = qi we obtain the minimal vicinity size which corresponds to this return time τinfi . As noted in the Introduction, after one iteration of the linear circle map (2) the position of the point on the circle changes by 2πδ . The expression τinf (ε ) = qi means that the point returns in the neighbourhood of its initial state after qi iterations, shifting by 2πδ qi . During these iterations the point can make several complete circles and appear to the left or right of the initial state. To take this fact into account we introduce the modulus and subtract the convergent numerator pi which defines the number of complete circles. Thus, the return in the neighbourhood of the initial state x0 takes place at the distance of 2π |δ qi − pi | from the point x0 [17]. Let us consider the case when we start not from the point x0 but from the right boundary of its neighborgood, i.e., from the point x 0 = x0 + εc /2. The return in εc after the minimal number of iterations qi happens near the left boundary of this neighbourhood, i.e., at the point x0 − εc /2. In such a case, as mentioned above, the point shifts by 2π |δ qi − pi | from the initial position x 0 (see Fig. 2). This means that x 0 − 2π |δ qi − pi | = x0 − εc /2

(9)

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359

Fig. 2 Schematic representation of the neighbourhood ε and initial and return points on the circle.

x0 + εc /2 − 2π |δ qi − pi | = x0 − εc /2

(10)

This enables one to derive the expression for calculating the value εi , which corresponds to the left boundary of the stairs step with the minimal return time τinf = qi for any irrational rotation number δ :

εi = εc = 2π |δ qi − pi |,

(11)

where δ is the rotation number, qi is convergent denominator, pi is the convergent numerator. As discussed above, the golden (silver) ratio represents a special case. The universal feature of the staircase dependence is due to the fact that numerators and denominators of convergents have the same definition rules and are elements of the Fibonacci (Pell) series. Thus, for the golden ratio, (11) can be rewritten as follows;

εi = 2π |δ Fi − Fi−1 |, εi ≈

or

2π Li ,

(12)

√ where Fi is the ith Fibonacci number, δ = ( 5 − 1)/2 is the golden ratio, and Li is the ith Lucas number. A more detailed description is given in Appendix 1. Following the same motivation, for the silver ratio we can find

εi = 2π |δ Pi − Pi−1 | or εi ≈

2π Qi ,

(13)

√ where Pi is the ith Pell number, δ = 2 − 1 is the silver ratio, and Qi is the ith Pell-Lucas number. We confirm our analytical results (11)–(13) by√ numerical simulation for the golden and silver ratios (Fig. 3) as well as for more complex Diophantine numbers 3 2, e, and lg(5) which correspond to the absence of universal geometry of the step dependence (Fig. 4). Using (11) we can find the dependence of each step length Di on its number in general (see Appendix 2). For the golden and silver ratios, the length of stairs steps is constant and independent of the step number but is defined by the rotation number (see Appendix 2): Di = const = ln δ .

(14)

Figure 5 illustrates dependences of the step length on the step number for the golden ratio (the universal √ geometry is valid) and for two different values of the rotation number, namely, δ = 3 2 and lg 5 (no universal geometry is observed). Figure 5,a corresponds to the golden ratio and shows that all step lengths are equal.

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Figures 5,b and c indicate that the universal geometry fails for the other rotation numbers. As can be seen from Fig. 5, analytical and numerical results are in full agreement. This means that the universal geometry can be obtained only in cases of the golden and silver ratios for which the step length (Fig. 5,a) does not depend on the step number.

(a) 10

Fibonacci Stairs εi=2π |δFi − Fi−1|

8

εi=2π/Li

6 4 2 0

δ=21/2−1

12

ln (τinf)

12

ln (τinf)

(b)

δ=(51/2−1)/2

10

Stairs εi=2π |δPi − Pi−1|

8

εi=2π/Qi

6 4 2

−8

−6

−4

ln ε; ln εi

−2

0

0

−8

−6

−4

ln ε; ln εi

−2

0

√ √ Fig. 3 Dependences ln τinf (ln ε ) for (a) the golden ratio (δ = ( 5 − 1)/2) and (b) the silver ratio (δ = 2 − 1) are indicated by solid lines, dashed lines with plus points and circle points represent the corresponding approximations using (12) for the golden ratio and (13) for the silver ratio.

(a) 10

ln (τinf)

4 2 −8

−6

−4

ln ε; ln εi

−2

4

0

0

ln (τinf)

4 2

−4

ln ε; ln εi

−2

0

−2

0

Stairs εi=2π |δqi − pi|

8

6

−6

δ=lg5

10

Stairs εi=2π |δqi − pi|

8

−8

(d)

δ=e

10

ln (τinf)

6

2

(c)

0

Stairs εi=2π |δqi − pi|

8

6

0

δ=71/3

10

Stairs εi=2π |δqi − pi|

8

ln (τinf)

(b)

δ=21/3

6 4 2

−8

−6

−4

ln ε; ln εi

−2

0

0

−8

−6

−4

ln ε; ln εi

√ √ Fig. 4 Dependences ln τinf (ln ε ) for four values of the rotation number: (a) δ = 3 2, (b) δ = 3 7, (c) δ = e and (d) δ = lg(5) (solid lines). Dashed lines with circle points show the corresponding approximations using (11)

N. Semenova, E. Rybalova, V. Anishchenko / Discontinuity, Nonlinearity, and Complexity 5(4) (2016) 355–364

(a)

(b)

1/2

δ=(5

0.6

−1)/2

0.55

Analytics

2

Di

0.45

1.5

0.4

1

0.35

0.5

0.3

Numerical simulation

2.5

Analytics

0.5

Di

1/3

δ=2

3

Numerical simulation

361

2

4

6

0

8 10 12 14 16 18 20

1

2

3

i

4

5

6

7

8

9

i

(c)

δ=lg5

3

Numerical simulation Analytics

2.5 2

Di

1.5 1 0.5 0

1

2

3

4

5

6

7

i Fig. 5 Dependences of the step length on the step number for three values of the rotation number: (a) the golden ratio, (b) √ δ = 3 2, and (c) δ = lg 5

5 Conclusion We have shown that the dependence τinf (ε ) has a step structure for any irrational rotation number. The values τinfi , which corresponds to the ith step, are equal to denominators of the ith convergents pi /qi of the rotation number δ . Using τinf (εi ) = qi we find the minimal vicinity size which corresponds to the left boundary of the step (11). Correctness of this approximation is confirmed by numerical simulation not only for the golden and silver ratios, for which the dependence τinf (ε ) is named the “Fibonacci √ Stairs” √ and has several features, but also for more complex values of the rotation number, namely, algebraic ( 3 2 and 3 7) and transcendental (e and lg 5). Using the rotation number value and the step form of the function τinf (ε ) one can predict the critical values εi which correspond to the emergence of new steps. This enables one to calculate the Afraimovich-Pesin dimension √ for Diophantine rotation numbers, for example, for δ = 3 2, δ = e or δ = lg(5). Acknowledgements This work was partly supported by the RFBR (Grant No. 15-02-02288). References [1] Nemytskii, V.V. and Stepanov V.V. (1989), Qualitative Theory of Differential Equations, Dover Publ. [2] Afraimovich, V. (1997), Pesin’s dimension for Poincar´e recurrences, Chaos, 7, 12—20. [3] Afraimovich, V., Ugalde, E., and Urias, J. (2006), Fractal Dimension for Poincar´e Recurrences, Elsevier.

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[4] Afraimovich, V. and Zaslavsky, G. (1997), Fractal and multifractal properties of exit times and Poincar´e recurrences, Phys. Rev. E, 55, 5418–5426. [5] Penn´e, V., Saussol, B., and Vaienti, S. (1998), Fractal and statistical characteristics of recurrence times, J. de Physique (Paris) Proc. of the conference ”Disorders and Chaos”, Rome, 8, 163–171. [6] Anishchenko, V., Astakhov, S., Boev, Y., Biryukova, N., and Strelkova, G. (2013), Statistics of Poincar´e recurrences in local and global approaches, Commun. in Nonlinear Sci. and Numerical Simul., 18, 3423–3435. [7] Anishchenko V., Boev, Y., Semenova, N., and Strelkova, G. (2015), Local and global approaches to the problem of Poincar´e recurrences. Applications in nonlinear dynamics, Phys. Rep., 587, 1–39. [8] Kuznetsov, S. (2001), Dynamical Chaos, Fizmatlit, Moscow (in Russian). [9] Pikovsky A., Rosenblum, M., and Kurths, J. (2002), Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press. [10] Rand, D., Ostlund, S., Sethna, J., and Siggia, E. (1982), Universal transition from quasiperiodicity to chaos in dissipative systems, Phys.Rev.Lett., 49, 132–135. [11] Boyland, P. (1986), Bifurcations of circle maps: Arnold’s tongues, bistability and rotation intervals, Commun. Math. Phys., 106, 353–381. [12] Semenova, N.I. and Anishchenko, V.S. (2015), Fibonacci stairs and the Afraimovich-Pesin dimension for a stroboscopic section of a nonautonomous van der Pol oscillator, Chaos, 25, 073111. [13] Anishchenko V., Semenova, N., and Vadivasova, T. (2015), Poincar´e Recurrences in the Circle Map: Fibonacci stairs., Discontinuity, Nonlinearity and Complexity, 4, 111–119. [14] Semenova, N., Vadivasova, T., Strelkova, G., and Anishchenko, V. (2015), Statistical properties of Poincar´e recurrences and Afraimovich–Pesin dimension for the circle map, Commun. Nonlinear Sci. Numer. Simul., 22, 1050—1061. [15] Slater, N. (1967), Gaps and steps for the sequence nθ mod 1, Proc. Camb. Philos. Soc. 63, 1115–1123. [16] Pettofrezzo A.J., and Byrkit, D.R. (1970), Elements of number theory, Prentice-Hall. [17] Buric, N., Rampioni, A., and Turchetti, G. (2005), Statistics of Poincar´e recurrences for a class of smooth circle maps, Chaos, Solut. & Fractals 23 1829–1840.

APPENDIX

Golden ratio √ In the case of the golden ratio (δ = ( 5 − 1)/2), denominators and numerators of the convergents of δ can be found as qi = Fi and pi = Fi−1 , where {Fi } is the Fibonacci sequence. The golden ratio is a special case when numerators and denominators have the same determination rule and are elements of the same sequence. Thus, we can simplify the expression (11). Each ith Fibonacci number is defined by the following recurrence relation: Fi = Fi−1 + Fi−2 .

(A1)

with the set values F0 = 1, F1 = 1. The ith Fibonacci number can be also found using Binet’s formula: Fi =

ϕ i − (−ϕ )−i , 2ϕ − 1

(A2)

√ where ϕ = ( 5 + 1)/2 is the root of the equation ϕ 2 − ϕ − 1 = 0 and depends on δ as

δ = ϕ −1



ϕ = 1+δ.

(A3)

The value of ε which corresponds to the emergence of a new stairs step and relates to the left boundary of the step with the minimal return time τinf = Fi is

εi = 2π |δ Fi − Fi−1 |

(A4)

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Using (A2) and (A3) this expression can be rewritten as follows:

εi =2π |(ϕ − 1)Fi − Fi−1 | =2π |ϕ Fi − (Fi + Fi−1 )| =2π |ϕ Fi − Fi+1 | ϕ i − (−ϕ )−i ϕ i+1 − (−ϕ )−i−1 − | 2ϕ − 1 2ϕ − 1 ϕ i+1 + (−ϕ )−i+1 − ϕ i+1 + (−ϕ )−i−1 | =2π | 2ϕ − 1 (−1)−i−1 (ϕ −i+1 + ϕ −i−1 ) | =2π | 2ϕ − 1 ϕ 2 + 1 −i−1 ϕ =2π |(−1)−i−1 | × 2ϕ − 1 ϕ 2 + 1 −i−1 =2π ϕ 2ϕ − 1 ϕ2 + 1 =2πϕ −i 2 2ϕ − ϕ ϕ2 − ϕ − 1 + ϕ + 2 =2πϕ −i 2 2ϕ − ϕ − ϕ − 2 + 2 + ϕ ϕ +2 =2πϕ −i ϕ +2 −i =2πϕ =2π |

(A5)

2πϕ −i (ϕ i + (−ϕ )−i) ϕ i + (−ϕ )−i 2π = i (1 + (−1)−i ϕ −2i ). ϕ + (−ϕ )−i =

Since ϕ > 1, the second term between the brackets tends to zero when i → ∞. Thus, lim εi (τinf = Fi ) =

i→∞

2π 2π = , ϕ i + (−ϕ )−i Li

(A6)

where Li is the ith Lucas number. It is defined by the same recurrence relation as the Fibonacci numbers (A1) but with another set values L0 = 2, L1 = 1. The Lucas numbers can be approximately defined by the following formula: (A7) Li = ϕ i + (−ϕ )−i. Calculation of the step length The size of the neighbourhood εi , which corresponds to the left boundary of a step with the minimal return time τinf = qi for any irrational rotation number δ can be found as follows:

εLi = εi = 2π |δ qi − pi | .

(A8)

Similarly we can obtain the value εi−1 . Since the dependence τinf (ε ) is a step-like function, εi−1 is simultaneously the left boundary of the step with the minimal return time τinf = qi−1 and the right boundary of the step with τinf = qi : εRi = εi−1 = 2π |δ qi−1 − pi−1 | (A9)

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Hence, the length of the ith step of the dependence ln τinf (ln ε ) can be calculated as follows: Di = ln εLi − ln εRi = ln

εLi εRi

2π |δ qi − pi | 2π |δ qi−1 − pi−1 | |δ qi − pi | = ln |δ qi−1 − pi−1 | qi |δ − pi /qi | = ln qi−1 |δ − pi−1 /qi−1 | qi . ≈ ln qi−1 = ln

(A10)

Thus, in general the length of stairs steps depends on denominators of convergents of rotation numbers. For the golden ratio, the denominators and numerators of the convergents are related to the Fibonacci series as pi−1 = qi = Fi . It follows that for the golden ratio, Di ≈ ln

qi pi−1 ≈ ln ≈ ln δ . qi−1 qi−1

(A11)

The same motivation can be used for the silver ratio. In this case, the numerators and denominators are connected with the Pell series: pi−1 = qi = Pi . For this reason the step lengths for the golden and silver ratios are constant and independent of the step number. They are defined only by the rotation number ln δ .

Discontinuity, Nonlinearity, and Complexity 5(4) (2016) 365–374

Discontinuity, Nonlinearity, and Complexity https://lhscientificpublishing.com/Journals/DNC-Default.aspx

Reversible Mixed Dynamics: A Concept and Examples S.V. Gonchenko† Nizhny Novgorod State University, Nizhny Novgorod, Russia Submission Info Communicated by A.C.J. Luo Received 27 February 2016 Accepted 24 May 2016 Available online 1 January 2017

Abstract We observe some recent results related to the new type of dynamical chaos, the so-called, “mixed dynamics” which can be considered as an intermediate link between “strange attractor“ and “conservative chaos”. We propose a mathematical concept of mixed dynamics for two-dimensional reversible maps and consider several examples.

Keywords Strange attractor Conservative chaos Elliptic orbit Symmetry-breaking bifurcation

©2016 L&H Scientific Publishing, LLC. All rights reserved.

Introduction In this paper we discuss one very interesting type of chaotic behavior of orbits of dynamical systems, the socalled mixed dynamics, which is connected with the existence of such open regions, in the space of dynamical systems, where systems with the following properties are dense: (i) the system has infinitely many hyperbolic periodic orbits of all possible types (stable, completely unstable, saddle); (ii) the closures of the sets of orbits of different types have a nonempty intersection. In principle, the phenomenon of coexistence and nonseparability of infinitely many periodic orbits of different types is known in chaotic dynamics. In particular, the well-known Newhouse phenomenon [1] relates to the fact that systems having simultaneously infinitely many saddle and asymptotically stable periodic orbits are dense in some open (in C2 -topology) regions of the space of dynamical systems. Recall that Newhouse regions, i.e. those ones where systems with homoclinic tangencies are dense, exist in any neighbourhood of any system with homoclinic tangency [2, 3]. Criteria that systems with infinitely many periodic attractors are dense (and generic) in these regions were given in [4–6]. Moreover, there are Newhouse regions in which such coexisting attractors can be nontrivial, for example, stable invariant tori and even small strange attractors (of Lorenz-like type and other) [7, 8]. With respect to the theory of dynamical chaos, these † Corresponding

author. Email address: [email protected] ISSN 2164 − 6376, eISSN 2164 − 6414/$-see front materials © 2016 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/DNC.2016.12.003

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results constitute now the mathematical basis of the theory of quasiattractors. The later term was introduced by Afraimovich and Shilnikov [9, 10] for a class of numerically observed strange attractors which either contain stable periodic orbits of very large periods or such orbits appear at arbitrarily small perturbations. The class of quasiattractors is very vast, it includes practically all known strange attractors, for example, such as the H´enon attractor, the R¨ossler attractor, attractors in the Lorenz model for values of parameters outside the region of Lorenz attractor existence [11, 12], almost all spiral attractors, attractors in Chua circuits [13] etc.a Note that in the conservative dynamics, the coexistence of infinitely many saddle and elliptic periodic orbits is considered usually as one of fundamental properties of nonintegrable conservative systems. Although conservative Newhouse regions exist [24] (with all the ensuing consequences), many important problems of conservative chaotic dynamics remain open.b The phenomenon of mixed dynamics was discovered by Gonchenko, Shilnikov and Turaev [25–27]. In particular, in the paper [27] for the case of two-dimensional diffeomorphisms, the existence of Newhouse regions was proved in which systems having nontrivial hyperbolic subsets simultaneously with infinitely many periodic attractors and periodic repellers are dense. This result was established in [27] for one parameter families f μ of two-dimensional diffeomorphisms such that (1) the diffeomorphism f0 has a nontransversal heteroclinic cycle containing two fixed (periodic) points O1 and O2 and two heteroclinic orbits Γ12 and Γ21 such that W u (O1 ) and W s (O2 ) intersect transversely at the points of Γ12 and W u (O2 ) and W s (O1 ) touch quadratically at the points of Γ21 , see Fig. 1(b); (2) the family f μ unfolds this tangency in a generic way and (3) |J(O1 )| < 1 < |J(O2 )|, where J(O) is the Jacobian of the map f0 at the point O. Then, the following result was established in [27] (see Theorem 4 there) • In any segment [−μ0 , μ0 ] with μ0 > 0 of values of the parameter μ , there exists a countable sequence of intervals Δ1i accumulating at μ = 0 as i → ∞ such that Δ1i contains dense subsets Bi1 , Bi2 , Bi12 and Bi∗ and the following holds 1) if μ ∈ Bij , j = 1, 2, then the map f μ has a quadratic homoclinic tangency at the point O j ; 2) if μ ∈ Bi12 , then the map f μ has a nontransversal heteroclinic cycle of the initial type, i.e. W u (O1 ) and W s (O2 ) intersect transversely at the points of Γ12 (μ ), where Γ12 (0) = Γ12 , and W u (O2 ) and W s (O1 ) touch quadratically at the points of some heteroclinic orbit Γ˜ 21 (μ ); 3) the set Bi∗ is a residual subset of Δ1i and if μ ∈ Bi∗ , then the map f μ has simultaneously infinitely many periodic attractors and infinitely many periodic repellers which closures contain the points O1 and O2 . Item 3 of this theorem shows that the intervals Δ1i are Newhouse intervals with mixed dynamics. Note that the mixed dynamics can also occur in higher dimensions (≥ 2 for diffeomorphisms and ≥ 3 for flows). The first examples of systems (of any dimension) near which Newhouse regions with mixed dynamics can exist were given in paper [28] by D. Turaev. In the present paper we will discuss mostly the phenomenon of mixed dynamics for two-dimensional reversible maps. Recall that, by definition, a map f is reversible, if f and f −1 are conjugate by means of an involution R, i.e. such a diffeomorphism of the phase space that R2 = Id. Thus, the relation R f = f −1 R and R f n = f −n R hold for all points of the phase space. a Probably, the only exceptions are such strange attractors as hyperbolic and Lorenz ones as well as wild (pseudo)hyperbolic attractors. Recall that the notion of wild hyperbolic attractor was introduced in paper [14] by Turaev and Shilnikov, where an example of wild spiral attractor was constructed. Such an attractor is genuine in that sense that the property “every orbit of attractor has a positive maximal Lyapunov exponent” is fulfilled for all C1 -close systems. Other examples of pseudohyperbolic strange attractors were recently found, e.g. attractors in periodically perturbed systems with the Lorenz attractor [15,16], discrete Lorenz attractors [17–19] and discrete figure-eight attractors [20, 21]. Note that attractors of the two last types can be freely observed in multidimensional diffeomorphisms since they can appear as a result of very simple, natural and universal bifurcation scenarios realizing in framework of one-parameter families [22, 23]. b In particular, the famous Poincar´e conjecture that stable (in fact, elliptic) periodic points are dense in the phase space of nonintegrable Hamiltonian systems is not proven, and there is no any significant progress towards its proof.

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Fig. 1 Examples of two-dimensional diffeomorphisms with nontransversal heteroclinic cycles: (a) a cycle of general type; (b) the simplest cycle with two saddle fixed points, O1 and O2 , and two heteroclinic orbits, Γ12 and Γ21 , such that W u (O1 ) and W s (O2 ) intersect transversely at the points of Γ12 and W u (O2 ) and W s (O1 ) touch quadratically at the points of Γ21 .

Note that nontransversal heteroclinic cycles of such “mixed” type, as in Fig. 1(b), are typical for reversible systems. An example of such a cycle for a two-dimensional reversible map is shown in Fig. 2(a). Therefore, one can conclude that “mixed dynamics” is the universal and fundamental property of reversible chaotic systems. In this case we will call the corresponding Newhouse regions with mixed dynamics absolute Newhouse regions citing the fact that generic systems from these regions have simultaneously infinitely many stable, unstable, saddle and symmetric elliptic periodic orbits [29, 30]. Thus, the reversible mixed dynamics is characterized by inseparability of attractors, repellers and conservative elements in the phase space. As is well-known, reversible systems are often met in applications and they can demonstrate a chaotic orbit behavior. However, the phenomenon of mixed dynamics means that this type of dynamical chaos can not be associated with “strange attractor” or “conservative chaos”. Attractors and repellers have here a nonempty intersection containing symmetric orbits (elliptic and saddle ones) but do not coincide, since periodic sinks (sources) do not belong to the repeller (attractor). Therefore, “mixed dynamics” should be considered as a new form of dynamical chaos, between “strange attractor“ and “conservative chaos”. These and related questions are discussed in the paper. The main attention is paid here to the development of the concept of mixed dynamics for two-dimensional reversible maps. Some elements of this concept are presented in section 1. In section 2 we discuss some examples of applied reversible systems demonstrating mixed dynamics. 1 Towards the concept of mixed dynamics for two-dimensional reversible maps Let f be a two-dimensional reversible map and R be the involution such that R f = f −1 R and R2 = id. We assume that dim Fix(R) = 1. The property of reversibility of f implies a symmetry of the set of orbits. An orbit intersecting the set Fix(R) (or the set Fix(R f n ) for any n) is called symmetric. Any symmetric periodic orbit possesses the following property: if it has a multiplier λ , then λ −1 is also its multiplier. Thus, in the case of two-dimensional reversible maps, a symmetric periodic orbit has multipliers λ and λ −1 . Moreover, such an orbit with multipliers e±iϕ ,

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where ϕ = 0, π , is, essentially, elliptic one, since the principal hypotheses of the KAM-theory hold here [31]. These properties make reversible and conservative systems to be related. Concerning non-symmetric orbits, whose points, by definition, do not intersect the set Fix(R) (as well as the sets Fix(R f n ) for any n), they can be, in principle, of arbitrary types. This property of reversible systems makes them related to systems of general type. However, for any non-symmetric orbit, there exists a symmetric to it orbit with “opposite” dynamical properties. It means that if a periodic orbit has multipliers λi , then the symmetric to it orbit will have multipliers λi−1 . We say that symmetric to each other orbits compose a symmetric couple of orbits. The same as for the dissipative case, in the space of reversible systems, Newhouse regions, i.e. such open regions in which reversible systems with both symmetric and non-symmetric homoclinic tangencies are dense, exist near any system with a symmetric homoclinic tangency. The proof of this fact is quite standard, see e.g. [32, 33]. However, there is one nontrivial moment related to the proof that these regions are absolute Newhouse regions (recall that in these regions there are dense (and generic) systems having infinitely many coexisting periodic attractors, repellers, saddles and elliptic orbits and the closure of the sets of the orbits of different types has a nonempty intersection). The existence of such absolute Newhouse regions was proved in [29, 30, 33] for some cases of one-parameter families unfolding generally symmetric couples of heteroclinic and homoclinic tangencies. In [32] this result was proved for Cr -perturbations with r ≤ ∞ conserving the reversibility. However, this problem, called in [30] the Reversible Mixed Dynamics conjecture (RMD-conjecture), remains widely open for multidimensional case and even for one parameter families of reversible two-dimensional maps. In the latter case, the main problem consists in the study of global symmetry breaking bifurcations, i.e. such global bifurcations that lead to the birth of a symmetric couple of nonconservative periodic orbits (i.e., of type “attractor-repeller” or “saddle(J > 1)-saddle(J < 1)”).

Fig. 2 Examples of two-dimensional reversible maps with symmetric homoclinic and heteroclinic tangencies. Maps with symmetric nontransversal heteroclinic cycles are shown in figs (a) and (b): here (a) O1 = R(O1 ) and J(O1 ) = J(O2 )−1 < 1, (b) J(O1 ) = J(O2 ) = 1. Maps with symmetric homoclinic tangencies are shown in figs (c)–(e): here the point O is symmetric in all cases; the homoclinic orbit is symmetric in the cases (c) and (e) where examples with a quadratic and a cubic homoclinic tangencies are sown, resp.; (d) an example of reversible map with a symmetric couple of quadratic homoclinic tangencies to O.

We note that the main local symmetry breaking bifurcations are well-known, see e.g. [34], these are, first of all, pitch-fork bifurcations of various types. Concerning global symmetry breaking bifurcations, they have been studied only for some partial cases of two-dimensional reversible maps. In particular, such bifurcations were investigated for the cases (a), (b) and (d) of Fig. 2 in [29], [30] and [33], respectively. Note that global symmetry breaking bifurcations for the homoclinic cases (c) and (e) of Fig. 2 are still not studied (in the framework of general one parameter unfoldings). We see that the main peculiarity of mixed dynamics, in distinct of dissipative and conservative chaos, consists

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in the fact that “attractor” and “repeller” intersect here but do not coincide. Indeed, by any definition, “attractor” is a closed invariant set that has to contain all stable periodic points, analogously, “repeller” should contain all completely unstable periodic orbits. Then, the mixed dynamics implies automatically the required intersection. Evidently, we need to give in this situation more or less adequate definition for “attractor” and “repeller”. For the case of two-dimensional reversible maps with dim Fix(R) = 1 we can define these invariant sets using the notion of ε -trajectories. Recall the corresponding definitions. Definition 1. Let f : M → M be a diffeomorphism defined on some manifold M and let ρ (x, y) be the distance between the points x, y ∈ M. A sequence of points xn ∈ M such that

ρ (xn+1 , f (xn )) < ε , n ∈ Z is called an ε -orbit of the diffeomorphism f . If n ∈ {0, 1, 2, ...} we speak on an ε + -orbit and if n ∈ {0, −1, −2, ...} on an ε − -orbit. Definition 2. We will call a point y achievable from a point x via ε -orbits (ε -achievable) if for any ε > 0 there exist an ε -orbit of the point x passing through the point y. A closed invariant set is called chain-transitive if any two its points are mutually achievable. Now we can define attractor following [14, 35]. Definition 3. An attractor of a point x is a closed invariant set belonging to the set of its achievable via ε + -orbits and which is chain-transitive and stable. The definition for repeller is the same for f −1 . So that attractor AS (resp., repeller RS ) of some set S is a union of the corresponding attractors (resp., repellers) of all its points. Concerning the type of stability for attractor we will use the so-called total stability or stability under permanent perturbations. This type of stability is posed between the Lyapunov stability and the asymptotic stability and, besides, it is well adapted to the notion of ε -orbit. Definition 4. A closed invariant set A is called total stable, if given δ0 > δ1 > 0, there exists ε > 0 such that no ε -orbit starting in the δ1 -neighbourhood of the set A which leaves its δ0 -neighbourhood. Note that, in the reversible case, if a point x belongs to the domain of attraction of some periodic sink ps , then Ax = ps and Rx = R(ps ). The situation can be more complicated when x is saddle or elliptic periodic point, or homoclinic/heteroclinic point, then an attractor (repeller) of such a point can be not trivial. Definition 5. Let f be a R-reversible two-dimensional diffeomorphism and dim Fix(R) = 1. The sets A = AFixR and R = RFixR are called an R-attractor and an R-repeller of f . Evidently, R(A ) = R. The cases when A ∩ R = 0/ are well-known, in these cases the attractor and repeller lie in different parts of the phase space, see e.g. Fig. 3a). If f is an area-preserving and reversible map, then A = R = M. The most interesting cases are those where A ∩ R = 0/ and A = R. Probably the first such case was observed in [36] which was labeled as “the conservative chaos coexists with the dissipative behavior”. Now we can say that a kind of mixed dynamics was observed in [36] when the sets A and R are essentially different, like as in Fig. 3b). Recently a new type of mixed dynamics was discovered when reversible attractor and repeller almost coincide, see e.g. [37, 38]. Schematically this situation can be represented as in Fig. 3c), when the chaotic set becomes bigger comparing with the case of Fig 3b), due to appearance of symmetric homoclinic and heteroclinic orbits of all possible types.

2 Examples In this section we illustrate our theoretic consideration by means of two examples of reversible systems from applications: the Pikovsky-Topaj model of coupling rotators and a nonholonomic model of Celtic stone.

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Fig. 3 Schematic pictures of various types of chaotic behavior in two-dimensional reversible maps: (a) attractor and repeller are separated, A ∩ R = 0; / (b) “the conservative chaos coexists with the dissipative behavior”, A ∩ R = 0/ and there exist an adsorbing domain containing a strange attractor (a symmetric to it strange repeller exists also); (c) reversible mixed dynamics, A and R almost coincide.

2.1

Pikovsky-Topaj model of coupling rotators.

In paper [37] the following model was proposed of a system of symmetrically coupled 4 rotators whose frequencies differ on 1 ψ˙ 1 = 1 − 2ε sin ψ1 + ε sin ψ2 (1) ψ˙ 2 = 1 − 2ε sin ψ2 + ε sin ψ1 + ε sin ψ3 ψ˙ 3 = 1 − 2ε sin ψ3 + ε sin ψ2 , where ψi ∈ [0, 2π ), i = 1, 2, 3, are cyclic variables (in fact, ψi = Ψi+1 − Ψi , where Ψ j is the phase of j-th rotator). Thus, the phase space of system (1) is the three-dimensional torus T3 . Note that system (1) is reversible with respect to the involution R:

ψ1 → π − ψ3 , ψ2 → π − ψ2 , ψ3 → π − ψ1 ,

(2)

i.e. the system (1) is invariant under the coordinate change (2) and time reversal t → −t. It was shown in [37] that, at sufficiently small ε , the dynamics of system (1) looks very close to the conservative one, i.e., for the corresponding Poincar´e map Tε of the section ψ2 = π /2 by orbits of system (1), elliptic islands are clearly observed and the average divergence equals to zero up to the numeric accuracy. However, with ε increasing, this “conservativity” is destroyed definitely which shows itself in the fact that, for example, the average divergence can slightly differ from zero (even on values of order 10−3 ). One more interesting (nonconservative) effect was observed in [37] when the authors tried to construct numerically the invariant measure for the map Tε . Of course, iterations of the initial measure (uniformly distributed on the line Fix(R) : ψ1 + ψ3 = π , ψ2 = π /2) are converged to some limit. However, the limits t → +∞ and t → −∞ for the same initial measure are different (that is visually observed beginning from the values of ε ≈ 0.45. This situation is impossible when the invariant measure exists and is absolutely continuous. However, it can be easily explained if one assumes that the reversible mixed dynamics presents here. In this case, iterations of the initial measure are concentrated on the reversible attractor as the number iterations k tends to +∞ or on the reversible repeller as k → −∞, see Definitions 3 and 5 in Sec.2. Such defined attractor and repeller have a nonempty intersection (they coincide in the case when the invariant measure exists, e.g. in the area-preserving case), since both contain the set of self-symmetric nonwandering orbits. However, they do

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Fig. 4 Phase portraits of the Poincar´e map of system (1) with ε = 0.49 for iterations of points of Fix(R) = {η = π }. (a) 4000 forward iterations and 2000 last iterations are shown (reversible attractor, the average divergency div = −0.00122) and (b) 4000 backward iterations and 2000 iterations are shown (reversible repeller, div = 0.00122 ). We see that the reversible attractor and repeller intersect but do not coincide.

not coincide as a whole, since the attractor contains periodic sinks and does not contain periodic sources, while repeller contains sources and does not contain sinks. 2.2

A nonholonomic model of Celtic stone

Recall that, in the rigid body dynamics, the Celtic stone is a top for which one of the principal inertial axes is vertical and the other two axes are horizontal and they are rotated by some angle with respect to the geometrical axes. A nonholonomic model of Celtic stone is a mathematical model which assumes that both the stone and the plane are absolutely rigid and rough, i.e. the stone moves along the plane without slipping and, moreover, the friction force has zero momentum. This means that the full energy is conserved which is a certain disadvantage of the model. However, it is well known that the nonholonomic model allows one to explain the main phenomenon of the Celtic stone dynamics – the nature of reverse, i.e., rotational asymmetry, which results in the fact that the stone can rotate freely in one direction (e.g. clockwise) but “does not want” to rotate in the opposite direction (counterclockwise). In the latter case it performs several rotations due to inertia, then stops rotating and starts oscillating, after that it changes the direction of rotation and finally continues rotating freely (clockwise). A mathematical explanation of this phenomenon seems now simple enough. The fact is that, like most of the well-known nonholonomic mechanical models, the Celtic stone model is described by a reversible system, i.e., a system that is invariant with respect to the coordinate and time change of the form X → R X , t → −t, where R is an involution, i.e. a specific diffeomorphism of the phase space such that R 2 = Id. However, in the case of Celtic stone, this system is, in general, neither conservative nor integrable, although it possesses two independent integrals, see more details in [39]. Because of this, the system can possess, on the common level set of the integrals, asymptotic stable and completely unstable solutions, stationary (equilibria), periodic (limit cycles) solutions etc., R-symmetric with respect to each other. Then, for example, a stable equilibrium corresponds to a stable vertical rotation of the stone, and an unstable equilibrium symmetric with respect to it corresponds to an unstable rotation in the opposite direction. Nevertheless, the motion of the Celtic stone is still regarded in mechanics as one of the most complicated and poorly studied types of rigid body motion. Moreover, this is one of the few types of motion in which chaotic

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Fig. 5 Some types of chaos in a nonholonomic model of Celtic stone.

dynamics was observed [38, 40, 41]. Some examples of such chaotic dynamics are presented in Fig. 5 taken from [38]. The first two examples correspond to strange attractors observed in the three-dimensional Poincar´e map. In Fig.(a) an example is shown of “not structured” attractor (in black) and the symmetric to it repeller (in grey) which are certainly separated (the situation is the same as in Fig. 3(a)). In Fig.(b) there is shown an example of a spiral attractor, since it contains a saddle-focus equilibrium. In Figs. (c) and (d) some examples of reversible mixed dynamics are shown. In Fig.(c) the attractor (in red) and repeller (in grey) are shown together so that the common figure appears pink. In Fig.(d) another type of mixed dynamics is represented when elements of conservative dynamics, like chaotic tori, are shown itself very clearly. Acknowledgements The author thanks D. Turaev for very useful remarks. This work is particularly supported by RSciF-grant 1441-00044 and RFBR-grants 16-01-00364 and 14-01-00344. Section “Examples” is carried out by RSciF-grant 14-12-00811. References [1] Newhouse, S. (1974,) Diffeomorphisms with infinitely many sinks, Topology, 13, 9–18. [2] Newhouse, S.E. (1979), The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms, Publ. Math. IHES, 50, 101–151. [3] Gonchenko, S.V., Shilnikov, L.P., and Turaev, D.V. (1993), On the existence of Newhouse regions near systems with non-rough Poincare homoclinic curve (multidimensional case), Rus. Acad. Sci.Dokl.Math., 47, No.2, 268-283. [4] Gonchenko, S.V., Shilnikov, L.P., and Turaev, D.V. (1993), Dynamical phenomena in systems with structurally unstable

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Poincare homoclinic orbits, Russian Acad. Sci. Dokl. Math., 47, No.3, 410–415. [5] Gonchenko, S.V., Shilnikov, L.P., and Turaev, D.V. (1996), Dynamical phenomena in systems with structurally unstable Poincar´e homoclinic orbits, Chaos, 6, 15–31. [6] Gonchenko, S.V., Shilnikov, L.P., and Turaev, D. (2008), On dynamical properties of multidimensional diffeomorphisms from Newhouse regions, Nonlinearity, 21(5), 923–972. [7] Gonchenko, S.V., Shilnikov, L.P., and Turaev, D. (2009), On global bifurcations in three-dimensional diffeomorphisms leading to wild Lorenz-like attractors, Regular and Chaotic Dynamics, 14(1), 137–147. [8] Gonchenko, S.V. and Ovsyannikov, I.I. (2013), On Global Bifurcations of Three-dimensional Diffeomorphisms Leading to Lorenz-like Attractors, Math. Model. Nat. Phenom., 8(5), 80–92. [9] Afraimovich, V.S. and Shilnikov, L.P. (1983), in Nonlinear Dynamics and Turbulence (eds G.I.Barenblatt, G.Iooss, D.D.Joseph (Boston,Pitmen). [10] Afraimovich, V.S. (1984), Strange attractors and quaiattractors, Nonlinear and Turbulent Processes in Physics, ed. by R.Z.Sagdeev, Gordon and Breach, Harwood Academic Publishers, 3, 1133–1138. [11] Afraimovich, V.S., Bykov, V.V., and Shilnikov, L.P. (1980), On the existence of stable periodic motions in the Lorenz model, Sov. Math. Survey, 35, No. 4(214), 164–165. [12] Bykov, V.V. and Shilnikov, A.L. (1989), On boundaries of existence of the Lorenz attractor, Methods of the Qualitative Theory anf Bifurcaion Theory: L.P. Shilnikov Ed., Gorky State Univ., 151–159. [13] Shilnikov, L.P. (1994), Chua’s Circuit: Rigorous result and future problems, Int.J. Bifurcation and Chaos, 4(3), 489– 519. [14] Turaev, D.V. and Shilnikov, L.P. (1998), An example of a wild strange attractor, Sb. Math., 189(2), 137–160. [15] Sataev, E.A. (2005), Non-existence of stable trajectories in non-autonomous perturbations of systems of Lorenz type, Sb. Math., 196, 561–594. [16] Turaev, D.V. and Shilnikov, L.P. (2008), Pseudo-hyperbolisity and the problem on periodic perturbations of Lorenz-like attractors, Rus. Dokl. Math., 467, 23–27. [17] Gonchenko, S., Ovsyannikov I., Simo, C., and Turaev, D. (2005), Three-dimensional Henon-like maps and wild Lorenz-like attractors, Int. J. of Bifurcation and chaos, 15(11), 3493–3508. [18] Gonchenko, S.V., Gonchenko, A.S., Ovsyannikov, I.I., and Turaev, D. (2013), Examples of Lorenz-like Attractors in Henon-like Maps, Math. Model. Nat. Phenom., 8(5), 32–54. [19] Gonchenko, A.S. and Gonchenko, S.V. (2015), Lorenz-like attractors in a nonholonomic model of a rattleback, Nonlinearity, 28, 3403–3417. [20] Borisov, A.V., Kazakov, A.O., and Sataev, I.R. (2014), The reversal and chaotic attractor in the nonholonomic model of Chaplygin’s top, Regular and Chaotic Dynamics, 19(6), 718–733. [21] Gonchenko, A. and Gonchenko, S. (2016), Variety of strange pseudohyperbolic attractors in three-dimensional generalized H’enon maps, Physica D, 337, 43–57. [22] Gonchenko, A.S., Gonchenko, S.V., and Shilnikov, L. P. (2012), Towards scenarios of chaos appearance in threedimensional maps, Rus. Nonlin. Dyn. 8(1), 3–28. [23] Gonchenko, A.S., Gonchenko, S.V., Kazakov, A.O., and Turaev, D. (2014), Simple scenarios of oncet of chaos in three-dimensional maps, Int. J. Bif. And Chaos, 24(8), 25 pages. [24] Duarte, P. (2000), Persistent homoclinic tangencies for conservative maps near the identity, Ergod. Th. Dyn. Sys., 20, 393–438. [25] Gonchenko, S.V. (1995), Bifurcations of two-dimensional diffeomorphisms with a nonrough homoclinic cotour, Int. Conf. on Nonlinear Dynamics, Chaotic and Complex Systems, Zakopane, Poland, 7-12 Nov., 1995. [26] Gonchenko, S.V., Shilnikov, L.P., and Turaev, D.V. (1996), Bifurcations of two-dimensional diffeomorphisms with non-rough homoclinic contours, J.Techn.Phys., 37(3-4), 349–352. [27] Gonchenko, S.V., Shilnikov, L.P., and Turaev, D.V. (1997), On Newhouse regions of two-dimensional diffeomorphisms close to a diffeomorphism with a nontransversal heteroclinic cycle. Proc. Steklov Inst. Math., 216, 70–118. [28] Turaev, D.V. (1996), On dimension of nonlocal bifurcational problems, Int.J. of Bifurcation and Chaos, 6(5), 919–948. [29] Lamb, J.S.W. and Stenkin, O.V. (2004), Newhouse regions for reversible systems with infinitely many stable, unstable and elliptic periodic orbits, Nonlinearity, 17(4), 1217–1244. [30] Delshams, A., Gonchenko, S.V., Gonchenko V.S., Lazaro, J.T., and Sten’kin, O.V. (2013), Abundance of attracting, repelling and elliptic orbits in two-dimensional reversible maps, Nonlinearity, 26(1), 1–35. [31] Sevryuk, M. (1986), Reversible Systems, Lecture Notes in Mathematics, 1211, (Berlin, Heidelberg, New York: Springer-Verlag). [32] Gonchenko, S.V., Lamb, J.S.W., Rios, I., and Turaev, D. (2014), Attractors and repellers near generic elliptic points of reversible maps, Doklady Mathematics, 89(1), 65–67. [33] Delshams A., Gonchenko S.V., Gonchenko M.S. and L´azaro, J.T. (2014), Mixed dynamics of two-dimensional reversible maps with a symmetric couple of quadratic homoclinic tangencies, arXiv:1412.1128v1, 18 pages (to appear in Nonlinearity). [34] Lerman, L.M. and Turaev, D.V. (2012), Breakdown of Symmetry in Reversible Systems, Regul. Chaotic Dyn., 17(3–4),

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318–336. [35] Ruelle, D. (1981), Small random perturbations of dynamical systems and the definition of attractors, Comm. Math. Phys., 82, 137–151. [36] Politi, A., Oppo, G.L., and Badii, R. (1986), Coexistence of conservative and dissipative behaviour in reversible dynamicla systems, Phys. Rev. A, 33, 4055–4060. [37] Pikovsky, A. and Topaj, D. (2002), Reversibility vs. synchronization in oscillator latties, Physica D, 170, 118–130. [38] Gonchenko, A.S., Gonchenko, S.V., and Kazakov, A.O. (2013), Richness of chaotic dynamics in nonholonomic models of Celtic stone, Regular and Chaotic Dynamics, 15(5), 521–538. [39] Borisov, A.V. and Mamaev, I.S. (2001), Dynamics of rigid body, RCD-press: Moscow-Izhevsk, 384pp. (in Russian) [40] Borisov, A.V. and Mamaev, I.S. (2002), Strange attractors in celtic stone dynamics, in book “Nonholonomic Dynamical Systems”, Moscow-Izhevsk, RChD-press, 293–316 (in Russian). [41] Borisov, A.V. and Mamaev, I.S. (2003), Strange Attractors in Rattleback Dynamics, Physics-Uspekhi, 46(4), 393–403.

Discontinuity, Nonlinearity, and Complexity 5(4) (2016) 375–395

Discontinuity, Nonlinearity, and Complexity https://lhscientificpublishing.com/Journals/DNC-Default.aspx

We Speak Up the Time, and Time Bespeaks Us Dimitri Volchenkov†, Anna Cabigiosu, Massimo Warglien Center of Exellence – Communication Technology, Bielefeld University, Universitaetsstr. 25, 33615 Bielefeld, Germany Dept.of Management, Ca’ Foscari University, Venice, Italy Submission Info Communicated by Valentin Afraimovich Received 20 January 2016 Accepted 14 March 2016 Available online 1 January 2017 Keywords Temporal patterns of human-communication Communication preferences Structure of communication in-networks of agents

Abstract We have presented the first study integrating the analysis of temporal patterns of interaction, interaction preferences and the local vs. global structure of communication in networks of agents. We analyzed face-to-face interactions in two organizations over a period of three weeks. Data on interactions among ca 140 individuals have been collected through a wearable sensors study carried on two start-up organizations in the North-East of Italy. Our results suggest that simple principles reflecting interaction propensities, time budget and institutional constraints underlie the distribution of interaction events. Both data on interaction duration and those on intervals between interactions respond to a common logic, based on the propensities of individuals to interact with each other, the cost of interrupting other activities to interact, and the institutional constraints over behavior. These factors affect the decision to interact with someone else. Our data suggest that there are three regimes of interaction arising from the organizational context of our observations: casual, spontaneous (or deliberate) and institutional interaction. Such regimes can be naturally expressed by different parameterizations of our models. ©2016 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Time is a fundamental dimension of social interaction. All interactions unfold in time, but most of our representations of interactions lift such temporal aspects to capture only a compressed snapshot of them. While the question of who interacts with whom has received broad attention, the time properties of interaction remain to a large extent underexplored. In a static perspective, the key question is who interacts with whom. In a temporal perspective, the fundamental questions are: when someone interacts with someone else, how frequently, and for how long. Communication is essentially a social process, any change of which immediately alters the nature of groups and, perhaps, the form of government [1]. The regularly renewable process of communication between the group members plays the essential role in continuous functioning of social institutions, serving as a mechanism for strengthening social integrity and group functional stability, as well as a moral sign of group solidarity. † Corresponding

author. Email address: [email protected]

ISSN 2164 − 6376, eISSN 2164 − 6414/$-see front materials © 2016 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/DNC.2016.12.004

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Organizations implement the codes and conventions maintaining the group’s institutions by coordinating communications between the group members and shaping the durations of intervals between all three types of communication in them. The design of complex organizations implies the division of labor, grouping tasks based on similarity in function, and their subsequent coordination and integration. The division of labor allows creating specialized units that are relatively autonomous and allows for creating economies of scale, for localized adaptation within problematic parts of the organization, while simultaneously buffering the unaffected parts. Therefore, it is critically important to study the principles of communication in organizations, in order to understand the nature of institutional longevity. The possibility to observe the unfolding of human behavior in time has dramatically expanded due to the diffusion of digitally networked activities and the availability of wearable sensors. This has opened new opportunities to trace with good accuracy how humans interact in time. However, most often the time dimension of interactions has been flattened: the cumulative duration of interactions has been frequently used to measure the strength of social ties in a static portrait of social networks. Time of interaction has been used as a proxy of the strength of relations. Yet, how the temporal distribution of interactions relates to the network structure of interacting agents is a basic question that has not yet been addressed. We analyze face-to-face interactions in two organizations over a period of three weeks. Data on interactions among ca 140 individuals have been collected through a wearable sensors study carried on two start-up organizations in the North-East of Italy. We develop a simple explanation of such results and models that fit our observations. We claim that both data on interaction duration and those on intervals between interaction respond to a common logic, based on the propensities of individuals to interact with each other, the cost of interrupting other activities to interact, and the institutional constraints over behavior. These factors affect the decision to interact with someone else. Our data suggest that there are three regimes of interaction arising from the organizational context of our observations: casual, spontaneous (or deliberate) and institutional interaction. Such regimes can be naturally expressed by different parameterizations of our models. Furthermore, we analyze how temporal patterns of contact in relation with the structure of the social network of communicating agents and in relation with the properties of information transmission in such networks, two new phenomena emerge. First, the duration of interactions between pairs of agents displays a non monotonic relation with the interaction preferences of each of them, as measured by mutual information. Second, by comparing the statistics of recurrence times and first-passage times of random walks over the biggest connected components of the graphs for different communication durations, we can assess the quality of global connectedness of the working team for the interactions of functionally important durations. Both phenomena can be explained by the existence of the same different regimes of interaction that explain interaction duration and intervals. 2 Employees’ interaction process within organizations Researchers have so far analyzed communications within organizations employing a static approach and network analysis techniques that describe who speaks with whom, the communication content and length. Communication characteristics are often related with performance variables [5, 18] and allow a comparison between the formal and informal organization [17]. The coexistence of formal organizational structures and informal networks in relationships between top managers involved in strategic decision processes was analyzed within the network approach [15]. The results reveal the relevance of informal cooperation ties and that formal horizontal ties are much more likely to be disregarded than formal vertical ties and that a significantly greater number of informal ties are built and maintained in a vertical rather than a horizontal direction. Kratzer et al [12], analyzing R&D projects, also find a relevant misalignment between the formal and informal organizational structure which has an inverted U-shaped relationship with the firm’s creativity and a negative effect on firm’s performance. Other

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authors have analyzed the genesis of networks [11, 19]. As Ahuja et al [2] emphasized, understanding network dynamics is important for several reasons but the most critical is that the understanding of network outcomes is only partial without an appreciation of the genesis of the network structures that resulted in such outcomes. In their review, [3] found that interpersonal relationships are influenced by actor similarity (i.e. age, sex, education, prestige, social class, tenure, and occupant), personality, proximity, organizational structure, and environmental factors (e.g. mergers and acquisitions, downsizing). Inter–unit relationships are influenced by interpersonal ties (often created by units’ leaders), functional ties (explained by complementary resources, network centrality, etc.), organizational processes and control mechanisms. The modularity literature suggests that inter–unit relationships are influenced by the product/task architecture. This literature has developed the within– firm mirroring hypothesis, which focuses on the relationship that exists between modularity in design and NPD activities within the firm, i.e. between modularity in design and interactions between development teams that design specific product modules [4, 13]. Following the within–firm mirroring hypothesis, modular products are developed by loosely coupled development teams focused on the design of one or only a few modules. Interactions among teams follow the boundaries of modules and the technology shapes the organization of NPD activities [16]. NPD team members interact closely within each team and the higher the modularity level of the component they develop, the lower the information and knowledge sharing with other teams. The organizational and geographic boundaries of teams should overlap with modules boundaries and communication patterns between teams are driven by interdependencies between modules [6, 7, 9, 10]. All in all, scholars are aware that the formal organizational structure, the tasks assigned to employees and informal relationships affect employees’ interactions. Nevertheless, we still know little about interaction processes in terms of frequency and length and about how these dynamics are affected by the formal organization. 3 Data collection for the communication study Data collection was carried out in June and July 2010 in two firms: H-farm and H-art. While legally distinguished, the two organizations have a same origin and are located in distinct buildings in the same area in the country outside Treviso in Italy. H-farm is a venture incubator founded in 2005. H-farm’s mission is to encourage the creation of projects aimed at simplifying the use of digital tools and services by people and companies, helping them transform their processes into digital workflows. In 2010 H-farm had 75 employees and hosted 9 start-ups that included 54 team members. H-farm’s staff, which supports start-ups’ development, had 21 employees. H-farm and the start-ups all have a functional structure and all start-ups have dedicated space. Since 2009, H-art works in the media industry and provides to multiple brands creative and innovative marketing plans. In 2010 H-art had 71 employees. H- art has a modified functional structure in which employees belong to functions and are assigned, at the same, to multiple projects. H-art and H-farm employees and start-ups’ members were asked to wear the radio-frequency identification sensors reported on occasions of physical proximity. Twelve sensor readers were placed all over the workplace, allowing its full coverage. We monitored face to face interactions for 24 hours per day and 7 days per week over the 4- weeks observation period. Our analysis is based on 18 working days. We did not consider the first and the last days of observation because during these days we distributed and collected, respectively, sensor badges to employees. We also did not consider weekends since there is no interaction over those days. We decided to consider a 12-hour time window interval (from 8.30 a.m. to 8.29 p.m.) due to the fact that these times are the average of, respectively, the first and last communication of each day. RFIDs report an interaction or a tick if they are at least one meter close to each other. The distance was chosen after a field observation of work activities and layout in the two firms. This distance excludes the recording of an interaction when employees are facing each other sitting at their desks during individual work. Thus it allows to selectively record only face to face interactions which happen when at least one of the two interlocutors have solved from his individual work station.

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We ended-up with two adjacency matrixes, one for H-art and one for H-farm, for each minute of the analyzed days. The first column and row of these matrixes list, for the H-art’s matrix the sensors of H-art’s employees and for the H-farm’s matrix the sensors of H-farm’s employees and of H-farm’s incubators members. The entries are 0;1 and represent the lack (”0”) of interaction between two sensors or the existence of an interaction or a tick (”1”). Interactions are bidirectional and matrixes are symmetric. We further recoded data to measure the duration of each interaction. Field observation suggested that when in a single interaction there was a one minute interruption between two ticks, this interruption represented noise in the data recording rather than a separation between different interaction episodes. The H–farm is a typical venture incubator founded with a mission to encourage the creation of projects aimed at simplifying the use of digital tools and services by people and companies, and helping them to transform their processes into digital workflows. In the time of study, the start–ups hosted by the H–farm have a functional structure and dedicated desks. In contrast to it, the H– art works in the media industry and provides creative and innovative marketing plans to multiple brands. It had a modified functional structure in which employees belonged to functions and were assigned, at the same, to multiple projects managed by a staff, the Alpha team. While H–farm’s organizational structure design is compatible with a representation of its architecture a la Simon [23] where the start-ups and their functions are loosely-coupled nested sub-systems, H–art’s organization increases cooperation among functions thus questioning the possibility to observe loosely-coupled nested sub-systems. 4 Statistics on interactions in organizations 4.1

Statistics on size of communicating groups

The communication processes in the both working teams were remarkable for the absence of a characteristic size of communicating groups (see Fig. 1). The distributions of the number of joint communication events are strongly skewed, with the long right tails decreasing with the size of communication groups approximately following the power laws (see the trend lines show in Fig. 1). The impression of a power law in the distributions of communicating group sizes can result from the superposition of different behaviors. In particular, approximate power laws can be generated by the combination of many different exponentials [24–26]. We suggest this might be indeed the case of our observations. In general, smaller groups of team members communicated more frequently than larger groups. Meetings involving a considerable part of the entire working team were the rare events (especially in the H–farm). It is remarkable that there is an evident difference between the power exponents characterizing the steepness of slope in the distribution tails. Perhaps, this difference arises due to the variance at the organization structure and pursued goals between H-farm and H-art. 4.2

Intervals between interactions

The durations of intervals between sequent communication act is an important characteristic of organized interaction, allowing us to judge on the degree of personal commitment to take part in business and social interactions, as well as on the faculty of team members to dynamically schedule the emerging communication into the current working timetable. Some striking regularities appear that are reflected in our observations. A typical pattern of communication activity demonstrated by a team member contains the bursts of communication activity separated by the relatively long breaks sometimes lasting longer than two hours, as shown in Fig. 2. The distribution of intervals between the sequent communication events for all members of both organizations is summarized in the following chart given in the log-log scale (Fig. 3). The distribution is remarkably skewed, indicating a significant proportion of the abnormally long periods of inactivity. The data for the shortest and most probable intervals between sequent communications is well fitted by the very common normal probability distribution, characterized by the mean interval between sequent communications 2.0 min with the

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Fig. 1 Statistics of communication group size. Error bars (along the horizontal axis) with the 5% value are given for the selected chart series to show the statistically significant difference between the both empirical distributions. The solid trend line (fitting the data for H-art) is N = 571032 · s−4.793, with the goodness-of-fit linear regression R2 = 0.98. The dotted trend line (fitting the data for H-farm) is N = 681423 · s−5.717, with the goodness-of-fit linear regression R2 = 0.98.

Fig. 2 A typical pattern of communication activity of a group member. The horizontal axis denotes time in minutes and each vertical line corresponds to a communication event. The upper diagram shows an individual pattern of communications acts during the entire period of observation. The lower diagram represent the enlargement of a short period of the recorded communication activity. The interval between two consecutive lines is the inter-event time.

standard deviation 1.78 (see Fig. 3, the solid trend line). The normal distribution of interval durations lasting not longer than 4 min can be interpreted as an average outcome of many statistically independent processes that determine the majority of short interruptions in communication. While the value of the normal distribution should be practically zero when the duration of interval lies more than a few standard deviations away from the mean, the distribution of intervals lasting longer than 4 min exhibits the long right tail, indicating the effect of management strategies for interaction resumption after the longer interaction breaks. The right tail of the distribution displays a crossover between the algebraic decay (fitting the data well for the intervals between 4 to 20 min) and the Zipf asymptote (that fits the data best for the longer intervals of 25 - 120 min long) observed in many types of data studied in the physical and social sciences [27]. Finally, an exponential cut-off of the distribution for the intervals longer than two hours is obviously due to the ”finite size effects” – the large fluctuations that occur in the tail representing large but rare events. Below, we propose a simple model of the decision to interact after a break that is analogous to the

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Fig. 3 The distribution of intervals between the sequent communication events for all members of the both working teams (in √ the log-log scale). The first fitting curve (black solid line) stays for the normal distribution, exp(−(t − ν )2 /2σ 2)/ 2πσ 2 /2, characterized by the mean interval between sequent communications ν = 2.0 min with the standard deviation σ = 1.78. The crossover in the distribution right tail occurs between the algebraic decay shown by the (red) dashed trend line ∼ 1/(t + 1)(t + 2) and the apparently Zipf’s asymptote shown by the (blue) solid line ∼ t −1−ε , with ε = 10−4 .

probability model of subsistence under uncertainty which we discussed in the first chapter. The model starts from the obvious remark that it takes at least two to speak. In the model, interaction is the result of two parameters, one regulating the willingness to propose an interaction, the other one the willingness to accept it. Let us assume that the propensity of an individual to be engaged into an interaction act can be characterized by a certain threshold xc ∈ [0, 1]. If the potential partner is able to motivate her at time t to interact by providing a strong enough reason, xt ≥ xc , she accepts the invitation to interact, but evades it otherwise. We assume that at each moment of time the motivation degree varies, and if considered over the working team is a random variable distributed in the interval [0,1], with respect to some given probability distribution function Pr{x < u} = F(u). We also think of the threshold xc as being chosen once, randomly from the interval [0,1], with respect to some given probability distribution function Pr{x < u} = G(u). The proposed decision making model is based on a number of essential simplifications. First, it is difficult if ever possible to reliably estimate the instantaneous motivation degree xt and the way how such a motivation degree can be expressed, as it might depend upon the permanently variable interaction context and can involve many personal factors, being beyond the scope of any reasonable modeling. Second, in contrast to the instantaneously varying motivation degree xt , the threshold value xc filtering out the unimportant interactions is likely determined by the pressure exerted by competing activities and the opportunity cost they generate. We can think that the value xc might be quite high if we are pressed by a heavy schedule, having a lot to do that day, or, on the contrary, it might be relatively low during leisure time. No matter, whether the threshold value xc is high or low, we assume that it is virtually invariable (at least during day time of observation) in comparison to the highly variable motivation degree xt . From the analysis of the probability model of subsistence under uncertainty, we know that the probability of observing an interval of length t between sequent interactions is ˆ 1 dG(u)F t (u) (1 − F(u)) . (1) Π(t) = 0

In particular, if both probability distributions F and G are taken to be the invariant measures of a map of the

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interval [0, 1], dF(u) = (1 + α )uα , dG(u) = (1 + β )(1 − u)β , for α > −1 and β > −1, the probability (1) reads as follows Γ(2 + β )Γ(1 + (1 + α )t) Γ(2 + β )Γ(1 + (1 + α )(t + 1)) − (2) Π(t) = Γ(2 + β + (1 + α )t) Γ(2 + β + (1 + α )(t + 1)) where Γ(x) is the Gamma function. For instance when both probability distributions are taken to be uniform, dF(x) = dG(x) = dx, the probability Π(t) exhibits an algebraic decay, Π(t) =

1 1 ≈ 2, (t + 1)(t + 2) t

t  1.

(3)

The algebraically decaying function (3) describes the statistics of intervals between sequent communications quite well for the intermediate values of the intervals between 4 to 20 min (see Fig. 3, the dashed trend line) but fails to explain the data for the longer intervals lasting from 30 to 130 min. It is remarkable that the slowly decaying far-right tail of the distribution for longer intervals well fitted by Zipf’s asymptote can be explained as a limiting case of the same simple model for communication decision making – for institutional (mandatory) communications. Namely, in the limiting case of ultimately high threshold filtering out the unimportant interactions xc → 1, only mandatory (institutional) communications are attended. Mandatory communications may include urgent, exigent contacts made in emergency, as well as some common rites and rituals that serve important functions for all team members. In most of organizations, attending a mandatory meeting is a must that is difficult to evade even on the days off. Following the discussion related to the probability model of subsistence under uncertainty, we can choose the distribution of xc → 1 to be the spike–like probability distribution focused at 1, Gε (u) = 1 − (1 − u)ε ,

ε > 0,

(4)

so that the corresponding probability density over the interval ]0, 1] is dGε (u) = ε (1 − u)1−ε du. Then, for any choice of the probability F, the probability of the interval between the sequent mandatory communication acts is dominated by the Zipf asymptote as t  1, Π2 (t) ≈

t −1−ε , ζ (1 + ε )

ε > 0,

(5)

where ζ (x) is the Riemann zeta function. It is clear that for long enough time intervals t  1 the slowly decaying Zipf asymptote Π2 (t) ∝ t −1−0.01 effectuates the crossover between the trends Π1 (t) and Π2 (t) visible in Fig. 3 (the solid trend line). It is also worth mentioning that the normal distribution of interval durations lasting not longer than 4 min can be naturally interpreted in the framework of proposed model as unmanaged short intervals characterized by the very low threshold xc → 0, so that any interaction can be resumed after a short occasional break not exceeding 4 min. Given the probability of communication resumption the same for all participants in all trials, a frequency distribution of the possible number of successful communication acts in a given number of trials is the binomial distribution, being best approximated by the normal distribution if the chance to be engaged into a brief interruption is close to a fair coin tossing. 4.3

Interaction durations

The distribution of communication durations is strongly skewed either. In general, brief communications are much more common than longer ones - and the shortest communication events (of 1 min) are the most frequent among all interactions (see Fig. 4). The statistics of communication durations indicates a significant proportion of long interactions. Communications of the shortest and most probable durations (1-2 min) may also be fitted by the normal probability distribution with the standard deviation (Fig. 4, the dashed trend line) though the two points are not enough for a reliable fitting. It is however obvious that the distribution of interactions

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Fig. 4 The distribution of communication durations for all members of both organizations is given in the log-log scale (by dots) along with three trend lines.

with durations exceeding 2 min has the right tail indicating the effect of interaction management strategies. Similarly to the statistics of intervals, the right tail of the distribution (Fig. 4) displays a crossover between the asymptotically algebraic decay (fitting the data well for the intermediate durations of 2 to 20 min) and the Zipf asymptote shown by the solid trend line (that fits the particular data points for the communication durations of 13 - 120 min). Finally, exceptional (unique) occasions of very long interactions constitute outliers of the duration statistics. The distribution of interaction durations can be interpreted with the help of a threshold model for decision making that is similar to one we used for the distribution of intervals between the sequent communication events. Assume that different pairs of individuals have different ”propensities” for interacting with each other. Such propensities should be taken as broader than simple ”liking”: they may be due to homophily, task complementarities, organizational roles, spatial proximity or many other factors. These propensities affect the relative likelihood that A interacts with B rather than A with C. At the same time, the duration of interactions with others can be limited by considerations of cost, by interrupting events, or other cause of ”hazard” of the interaction. The huge potential variety of factors limiting the continuation of interactions suggests to treat those causes statistically in terms of an ensemble of random variables. Let us assume that the propensity of an individual to keep the current interaction going can be characterized by a certain threshold yc ∈ [0, 1]. If the communication partner challenges the already heavy schedule of the individual at time t by yt ≥ yc , the current interaction stops but keeps going otherwise. The proposed threshold model mimics the continuous decision making process on unceasing interaction. We further assume that at each moment of time the parameter yt varies, being a random variable distributed over the interval [0, 1] with respect to some probability distribution Pr{x < u} = F(u), and the critical threshold value yc is chosen randomly once from the interval [0, 1] with respect to another probability distribution Pr{x < u} = G(u) and kept unchanged during interaction. The statistic of communication events for the intermediate interaction durations between 2 to 20 min (Fig. 4) is best fitted by the probability function (2) with α = β = 1.0 in the probability densities dF and dG. The corresponding trend is shown in (Fig. 4) by the dash-dotted line and can be approximated asymptotically for t  1 by the cubic hyperbola, Π(t) ∝ t −3 . The choice α = β = 1.0 in the model indicates that the high values of tolerance threshold yc are increasingly more probable than lower values, but the high values of the motivation parameter yt are decreasingly less probable than lower values: the communication process is statistically ”sticky”

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in intermediate times. Eventually, in the case of ultimately high tolerance threshold yc → 1 that can be modeled by the spike–like probability distribution focused at 1, the probability of interaction duration t follows the Zipf Law asymptote ∝ t −1−0.09 dominating the statistics for longer interaction durations as shown in (Fig. 4) by the solid trend line. The Zipf asymptote may correspond to the protracted institutional interactions, for which no characteristic time limits are imposed. 4.4

On three statistically different types of interaction

We conclude the analysis of temporal patterns of interaction with the remark on three statistically different types of interaction reflecting the different valuation and management strategies applied to time intervals of different duration. These three interaction regimes can be parsimoniously represented by different distributions of communication durations and intervals between sequent communication events. Short time intervals (interaction durations and intervals between sequent communications alike) largely remain unmanaged and unregulated. Short occasional breaks in communication are tolerated. On the one hand, short interactions are unavoidable as soon as a person may randomly bump on someone, on the other hand they might be so undemanding that one can hardly reject them – everybody can be engaged into a brief communication at every moment of time, so that we call them casual interactions. Time intervals of intermediate durations (lasting up to 20-25 minutes) are thoroughly managed by individuals demonstrating the high propensity to keep the current interaction going while filtering out the potentially unimportant forthcoming communications. We call such interactions spontaneous, as they are motivated by the propensity to interact with others. Finally, where the Zipf’s Law manifests itself, we suggest that a logic of institutional interaction prevails, where top-down, almost mandatory interaction occurs. The simple threshold models for the decision to interact and to keep the current interaction going support the proposed taxonomy. 5 Time and social structure of interactions Until now we have analyzed time patterns of interactions, abstracting from the concrete relational structure within which they were taking place. Still, interpreting the distribution of interaction durations and intervals has required to introduce heterogeneous propensities of individuals to interact with each other. In this section, we take a closer look to the finer texture of the relational network of agents, and how it interlaces with the temporal unfolding of interactions. The non-monotonic relationships between interaction time and different network metrics suggest that the three regimes of interaction found in our former analysis may contribute to explain how relational structures and temporal patterns of interaction affect each other. We have already introduced a notion of interaction propensity. Not all agents are equally likely to interact with each other in structured contexts such as organizations, for a host of reasons including personal preferences, task requirements, organizational roles. It is a reasonable conjecture that such propensities may affect the duration of interactions. Each individual should be expected to spend longer time when interacting with other individuals with which she has a higher propensity to interact. At the same time, there are competing demands over each individual time budget that may limit this effect - one cannot spend infinite time with other persons she likes. Beyond a reasonable time limit, one may expect that other institutional factors may become the dominant driver of time allocation. For example, in the organizations we studied periodic collective meetings can force face-to-face interactions for protracted time. In order to analyze how interaction propensities and the duration of interaction affect each other, we use mutual information [28] as a statistical measure of pairwise interaction propensities. Given a random event XA that a subject A is presently communicating (with anybody) during time t described by the probability function Pt (XA ) and a random event XB that another subject B is communicating during

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same time characterized by the probability function Pt (XB ), it is possible to analyze the pairwise interaction preferences of A and B, as well as of the entire working team, with the help of the mutual information as introduced in [28], Pt (XA , XB ) , (6) I(t) = ∑ IAB (t) = ∑ Pt (XA , XB ) log2 Pt (XA )Pt (XB ) {A,B} {A,B} where the summation is performed over all possible pairs of individuals {A, B}. If during the observation period A and B participated in meetings independently, Pt (XA , XB ) = Pt (XA )Pt (XB ), then the amount of mutual information IAB (t) associated to such a pair is zero. As the amount of mutual information in a communicating pair obviously reaches the maximum when while XA takes part in a communication event, XB always does either (as, perhaps, they speak to each other), this value allows for assessing the degree of communication preferences in each pair and, if being summed over all communicating pairs, the degree of communication preferences within the entire working team. The mutual information can be analyzed for every communication duration serving a measure of how much knowing the fact of that A is communicating during time t would reduce uncertainty about that B is communicating, provided the joint probability Pt (XA , XB ) for A and B is known. We have used the mutual information in order to analyze interaction preferences in communications of every duration. In Fig. 5, we have shown the dependence of mutual information upon the durations of communication acts observed High values of mutual information show that team members demonstrate a high degree of selec-

Fig. 5 Mutual information vs. communication duration. The trend lines (given by the cubic splines optimally fitting the collected data points) are shown to facilitate understanding of the dependencies.

tivity while choosing an interaction partner, and vice versa, the interaction partners would be selected at random if the level of mutual information is minimal. The performed analysis of mutual information shows that the degree of selectivity in both companies monotonously increases with the interaction duration, until their maximum values are attained, for durations ranging between 10 and 25 min, and then falls down rapidly to the minimal values. For particularly long interactions, the values of mutual information is particularly small, as the statistical contribution from uniquely rare long conversations occurred between pairs of individuals was insignificant. Thus, the structure of interactions in the two organizations reveals an essentially high degree of selectivity

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Fig. 6 The communication preference pyramid for H-farm and H-art.

for the interactions whose duration is concentrated in the interval from a few minutes to several tens of minutes. Reciprocally, the duration of interactions reveals the weight of relational ties within the organizations (Fig. 6). In the communication graphs where each vertex represents an individual, and edges connecting the vertices are characterized by weights assigned accordingly to the intensity of communication between the two (for instance, the probability of communication). Instead of a single, static communication graph, in our dynamic approach we have analyzed an ensemble of graphs, in which the probabilities of communications in all pairs of interlocutors are described by an individual graph, for each communication duration. The collected empirical data shows convincingly that the shortest communication events lasting one minute are ubiquitous, as encompassing all employees and perhaps serving the basic communication needs within a working team (Fig. 7.a). The communication graphs that describe the probabilities of pairwise communications of the longer durations are more sparse but also more rich in structure, as accounting for a good deal of the personal and working communication preferences. For example, they can include micro-communities, consisting of just a few permanently communicating partners loosely connected (in the sense of communication probability) with other members of the working teams. In particular, the communication graphs for the longer durations can contain a number of connected components of different sizes, sometimes including either a single pair of interlocutors, or just an unconnected vertex (if the corresponding subject did never take part in a communication event of that duration during the entire observation period) (see Fig. 7.b). Interactions of different durations may possess the very different structural properties and generate different graphs. Again, the data suggests the existence of three regimes. Lowest duration interactions correspond to low mutual information, the shortest interaction events lasting few minutes are ubiquitous, and correspond to random, occasional or aborted encounters - it takes a short interaction also to say that you currently have no time available. Intermediate duration is where the graph of interactions strongly reflects pairwise interaction propensities. It is more sparse but has more structure. This corresponds to the regime that we have labeled as spontaneous interaction. Above approximately 20 mins, there is a decline in mutual information that we interpret as the result of a substitution of motivations to interact - from spontaneous to institutional. The graph of interactions is structured mostly by persistent chains of interactions reflecting collective work meetings where agents are sitting close to each other in a meeting room (as clearly suggested by the longest chain in Fig. 7.c).

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a.)

b.)

c.)

Fig. 7 Interaction networks between the HA–firm members corresponding to 3 different sample durations: a.) 1 min; b.) 10 min; c.) 20 min.

6 Time, interaction synchronization, and information transmission We turn our analysis in terms of the information transmission properties of face to face interactions. We look at interactions as communication episodes. The main objective of analysis is to understand how the ”local”, individual interaction propensities described by the connectivity of subjects as nodes of a communication graph determine the ”global”, connectedness property of the entire communication process described by the ensemble of communication graphs for all communication durations. In order to address this problem in relation to all communication graphs, let us consider a model of simple random walks, a statistical metaphor of message transmission in a working team. We suppose that a message (requiring t time units to be transmitted) is passed on by each subject X to another one – Y , selected at random among all available companions accordingly to the connection probability (t) TXY determined by the communication graph of communication duration t. We can characterize a degree of variability in individual (local) communication preferences by the minimal amount of information required to record the choice of a partner Y for communication made by X in order to pass a message, (t)

(t) (t)

(t)

hX = − ∑ πX TXY log2 TXY ,

(7)

{Y }

(t)

(t)

where πX is a stationary distribution of the random walk, the left eigenvector of the matrix TXY belonging to the maximal eigenvalue 1, and, as usual, we suppose that 0 · log 0 = 0. Then the minimal amount of information required to record a single random transition of a message in the entire communication graph correspondent to the duration t is defined by the entropy rate of random walks [29], H (t) = −



{X,Y }

(t) (t)

(t)

πX TXY log2 TXY ,

(8)

summed over all pairs of interlocutors. The entropy rate reaches the maximal value if subjects have no communication preferences, transmitting the message equiprobably to any other member of the working team, but it takes the minimal values when a connected component of the communication graph constitutes a chain, in which the only forthcoming communication partner is available. In homogeneous graphs where all vertices and all transitions between them are supposed to be equiprobable, the transmission of a message can be viewed as a sequence of statistically independent transmission events, so that its entropy rate is the same as entropy of any individual member in the communicating team. The serial quantities,   (t)  (t)  2 2 (t) → Y log2 Pr X − →Y , (9) H2 = − ∑ Pr X − {X,Y }

(t)

H3 = −



{X,Y }

  (t)  (t)  3 3 Pr X → − Y log2 Pr X → − Y ,...

(10)

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  n where Pr(t) X − → Y denotes the probability to observe a path of length n connecting X and Y (which can be calculated readily, as the transition probabilities in the random walks is independent of n) sequentially define the Shannon entropy over the n- blocks [29], quantifying the amounts of information required to record a block of 2, 3, . . . random transmissions of the message in the communication graph. The complimentary information on a global connectedness of communication graphs can be obtained by analyzing the level of correlations between infinitely long paths (along which a message would be transmitted) with the use of the excess entropy [30], (t)

E (t) = lim (HN − N · H (t) ), N→∞

(11)

expressing the amount of information required to describe the additional structural irregularities of message transmission that cannot be explained statistically by a simple superposition of individual communication propensities while considering increasingly longer paths of message transmission. If the excess entropy is zero, the interaction process is perfectly synchronized within a single stream of sequential communication events. However, the large values of excess entropy indicate that the process of message transmission cannot be synchronized within a single communication stream in the same time slot. For the group members have rather different individual interaction propensities during the different intervals of time, the several independent interaction streams are required in order to synchronize them simultaneously. Therefore, by juxtaposing the entropy rates expressing the connectivity property with respect to the random walks on the entire communication graph and the excess entropy of random walks describing correlations of the very long message transmission paths in that, we can get an insight into complexity of interaction schedules and understand how the communication process works structuring temporal interactions within the working teams and improving its communication integrity (see Fig. 8). We used the diagram showing the entropy rates vs. the excess entropy earlier, in purpose of studying the graphs and their subgraphs at different scales [31]. The data on the values of excess entropy and of entropy rates

Fig. 8 The excess entropy of random walks vs. the entropy rate of random walks for the ensembles of communication graphs corresponding to the different communication durations. The trend lines are shown to facilitate understanding of the dependencies.

for the messages transmitted by random walkers on the communication graphs (Fig. 8) show that the difference of institutional structures is crucially important for complexity and heterogeneity of interactions.

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Interactions in the H–art, having a modified functional structure, look more natural, as the low values of the excess entropy are associated with the low values of the entropy rate, and interaction complexity assessed by the excess entropy grows steadily with the improvement of the local connectivity in interaction graphs. However, the level of complexity of interactions in the H–farm, having a traditional functional structure, is relatively high already for low entropy rates indicating the importance of institutional ties for structuring interactions. Nevertheless, independently of the difference in functional structure, the level of interaction complexity remains bounded by approximately 3.2 bits for the entropy rate of 10 bits, uniformly for both firms. In a completely open communication environment where everyone of 73 employees in the working team can talk to each other, the entropy rate of 10 bits means that the probability of interaction between any pair of members would amount to 0.78. Therefore, by promoting subjects to become more open and flexible while choosing a partner for transmitting messages, we can promote complexity of interactions in the entire group. However if the entropy rate exceeds 10 bits (i.e., subjects communicate all together virtually at every occasion), the level of interaction complexity within the teams decays rapidly, reducing the communication process to attending at general meetings. We conclude the discussion on structural properties of communication graphs by

Fig. 9 The box plots represent the distributions of the ratio the first passage time to a node and the recurrence time to it in the biggest connected components of the communication graphs for the different communication durations.

comparing the statistics of recurrence times and first-passage times of random walks over the biggest connected components of these graphs observed for the different durations of interactions. We can say that a group exists as a single communication entity if it constitutes a good enough transmission media for messages addressed to every group member. The ability of the group to transmit messages directly, in short enough time, reveals the level of its global connectedness with respect to interactions of a given duration, as the message can follow all paths available in the interaction network at once although some paths are more probable than others. On the contrary, a message can spread over the graph, literary speaking at random, as a rumor, due to the individual interaction propensities of group members. We expect that for some window of interaction durations the quality of global connectedness should be superior to the net effect of local connectivity of individual interaction propensities. Any vertex X in a finite weighted undirected connected graph can be characterized in relation to the nearest neighbor random walks defined on that by the recurrence time to it (how long one must wait to revisit the vertex), RX = πX−1 , where πX is the stationary distribution of random walks on X , [32, 33]. For the finite connected undirected weighted graphs, the stationary distribution πX is nothing else but the connectivity of the vertex X

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normalized to the total weight of all edges in the graph and therefore characterizes the local property of vertex, independently of the connectivity of other vertices [32]. In the model of message transmission by random walks, the recurrence time can be thought of as a characteristic time of feedback for a message or rumor circulation, accounting for the expected number of transmission acts required for a message to return to its source - given that the random walks carrying the rumor can revisit any other vertex many times. However, another characteristic time might be of greater importance in the context of communication process: that is the first- passage time to a group member – the expected number of transmissions required for a message issued by any other group member (chosen at random among all collaborators with respect to the stationary probability π ) in order to reach the addressee for the first time, following a self-avoiding random walk, in which revisiting of vertices is not allowed. In the first-passage time to X , all possible transmission paths of any length concluding at are taken into account (excluding the paths comprising loops), yet some of them are considered to be more probable then others, as being weighted by the probability of being traversed by a random walker. The applications of the first-passage times for the analysis of structure of graphs and databases and the calculation methods of the first-passage times are discussed in details in [32]. a finite connected undirected graph, the Given the a transition matrix TXY describing a random walk on  −1  first-passage times can be calculated as the diagonal elements ΦX = L XX of the multiplication group inverse (so called Drazin’s generalized inverse) of the Laplace operator LXY = δXY − TXY describing the correspondent diffusion process on the graph [32]. In contrast to the recurrence time to X , the first-passage time to it characterizes the role of X with respect to entire graph structure, as all infinity of candid paths of all lengths concluding at X is taken into account. For homogeneous graphs of regular structure, the first -passage times are approximately equal to the corresponding recurrence times [32], however for heterogeneous graphs of complex structure ample with cycles the values of first passage times can depart from recurrence times substantially, spotting structurally integrated and structurally isolated vertices [33]. In particular, a random walker would be trapped in the sites X , for which RX < ΦX , and would virtually fly by the sites where RX > ΦX . In the context of the random walk model of communication process, the ratio of both characteristic times ΦX /RX calculated over a communication graph can spot the key team members playing the important roles in communications of the given duration. In order to characterize the structural properties of the biggest connected components of communication graphs of every duration, we have summarized the data on the distributions of ratios ΦX /RX for all X in the form of box plots quite useful to compare similar data sets. Each box plot shown in Fig. 9 comprises a central line showing the median of the data, a lower line showing the first quartile, and an upper line showing the third quartile. Two lines extending from the central box of maximal length 3/2 the interquartile range (if it does not extend past the range of the data). Finally, outliers indicate the data values that lie outside the extent of the previous elements. It is remarkable that in the communication graphs for the durations not exceeding 15 min the recurrence times to the most of the team members are typically longer than the correspondent first passage times, ΦX /RX < 1, indicating that these graphs are very well integrated. The level of connectedness in graphs for short communication durations systematically surpasses the level of local connectivity based on the individual communication preferences of the group members. The deficiency of first passage times compared to the values of recurrence times is minimal in the fully connected weighted graph observed for the shortest communication durations of 1 min, and gradually increases in the biggest connected components of communication graphs for longer durations. Nevertheless, in all these graphs, there is always a few structurally well integrated interlocutors (indicated by the upper outliers in the box plots shown on Fig. 9) for which the first passage times are approximately equal of even exceed the recurrence times.

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Multiple cycles and structural heterogeneity are gradually effaced from the biggest connected components of communication graphs observed for the durations longer than 12 min – they are structurally dominated by the chain segments due to the high level of selection in personal and working communication, promoting the directed propagation of messages over the circulation of rumors. For communication durations longer than 15 min when the level of communication preferences estimated by the value of mutual information (Fig. 5) reaches its maximum, the first passage times dominated the recurrence times for most of communicating subjects. We find here again (inverted) the non monotonic shape characterizing the relations of mutual information to interaction duration, suggesting that a same underlying principle unifies pur different observation. 7 Communication pyramid and longevity of institutions At the origin of human race, in the first family or a group, a single person (a parent) spoke, and others imitated him – and started to talk to each other after numerous imitations [1]. A word that began as the word of a leader ordering, warning, threatening, or condemning belonged to a realm of mandatory communications that was not reciprocal, as assuming neither objection, nor reply. Then, after being copied and repeated, it became the word of a deliberate communication between equal peers, each having an equal chance to influence the other. They were approving the initial word until, finally, it turned into the word of a spoken language pertinent to casual communications. To some extent, the words of institutional mandatory interaction prevails over those of both deliberate and casual communications by stimulating, enslaving, and providing a contextual frame for them that is crucially important for effectuating the change of opinions and behavior of people, as well as for enshrining the required public opinion. The main feature of deliberate communication is that it occurs by mutual agreement between partners that is the central point of the model for communication decision making discussed by us in the previous section. Intentionally interacting peers have equal rights either to accept communication or reject it. On the contrary, by requiring obedience to a group discipline, mandatory communications maintain the superiority of authority and once established rules over the individual rights of group members, thus transforming a group of people into an organization that can be integrated by chains of commands. Our results reported in the previous section show that while equality of deliberate communications requires active maintenance from every member of a group (by filtering out the unimportant interaction motives), mandatory communications would reward renunciation of dominance with a sense of full social acceptance, engendering the strong pull of social solidarity in group members that can be expressed in special rituals, rites, and other common social events. It is therefore virtually important for an organization to find an optimal balance between these two types of communications. In view of that we can summarize graphically our observations on intervals between interactions with as an ”onion dome” reflecting different types of interaction – in the form of an sequent interval population pyramid, in which every axially centered horizontal bar indicates a fraction of sequent interactions, sorted accordingly intervals between them (see Fig. 10). The pyramid comes in at the form of a ”onion dome”, with a very wide bulge at the base, corresponding to the casual interactions dominating others in number, and rapidly contracting upwards, from deliberate to mandatory communications. The contraction rates in the deliberate and mandatory levels of the pyramid are different. In deliberate communications maintained by every member of the working teams, an inter–event interval that lasts twice as long, occurs on average quarter as often. And in mandatory communications contemplating the social solidarity in group members an interval that lasts twice as long, occurs twice as rare. The slow decay rates of the distributions of intervals between the sequent intentional communications provide a statistical ground for extraordinary longevity of organizations and institutions.

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Fig. 10 The interval dome: the majority of experimentally observed interactions were casual, being characterized with the average time break about 2 min long between the sequent communication acts. Spontaneous interactions enjoy a rule of thumb: an interval that lasts twice as long, occurs quarter as often. Finally, institutional interactions are subjected to another rule of thumb: an interval between them that lasts twice as long, occurs twice as rarely

For neither deliberate, nor institutional mandatory communications possess any characteristic time scale,

∑ tP2(t)  ∑ tP1(t) → ∞,

t≥0

t → ∞.

(12)

t≥0

Both deliberate and institutional mandatory communications in organizations once interrupted, can be resumed at anytime. 8 Discussion and conclusion Our results show that in both H-art and H-farm the relationship between the frequency and size of teams during interaction processes follow a power law. Interestingly enough the α exponents of the two power laws are higher than that usually found in the literature that documents two universality classes characterized by exponents α = 1 [8, 20, 21] and α = 3/2 [14, 21]. The exponents regard models focused on single individual dynamics while organizations connect people in social networks and several of their activities are not performed independently. These works concerned not the personal communications teteˆ a-t ` ete, ˆ but rather impersonal exchange of messages, e-mails, and letters: people were not speaking to each other, but writing each other instead. Consequently, their models were the versions of the so-called preferential attachment approach of Barabasi, in which an individual can contact (send a message to) everybody else with some probability. However, in case of live conversations if one employee is speaking to someone else, she could hardly talk to anybody else simultaneously - and these ”difficulty” of conducting multiple conversations is expressed in the extraordinary high exponent observed. Hence, our results seem suggesting that in organizations such as firms, interactions are clustered in smaller units than those observed when individual interactions were considered. Within firms, individuals can manage less interactions contemporarily involving multiple individuals (team-based interactions) than the number of one-to-one interactions they can usually effort. Individuals prefer shorter meetings (spontaneous interactions). These results may be driven both by the formal organization and by a sort of span of manageable complexity. First, we observe often smaller and frequent meetings because big meetings are only those formally planned and scheduled by the organization. Formal organizations plan large meetings that are

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scheduled by top managers to share data, results and set goals. Large meetings are scheduled periodically but they are not selected to manage operative daily activities by employees. Daily activities are managed relying on more lean and small meetings. Second, during meetings employees exchange data, information and knowledge and the need to absorb and understand such communications may generate an upper limit to effective interactions events. Thus, individuals prefer interactions in smaller teams and of shorter lengths. Finally, individuals have constraints about available time to attend meetings and this may explain the relevance of shorter and smaller events. The exponents of the H-farm and H-art’s power laws are different and the H-farm exponent is higher suggesting that H-farm is more fragmented than H-art. H-farm is an incubator of stand-alone start-ups that have their own staffs and physical locations. Coherently in H-farm we observe more frequently smaller teams. H-art has a more complex organization. H-art has a modified functional structure in which employees belong to functions and are assigned, at the same, to multiple projects. Furthermore a dedicated team supervises the H-art’s on-going projects. Hence in H-art larger groups are more frequent (the exponent (α ) of H-art is smaller): H-art displays a more complex and less bounded organizational structure in which multiple geometries of larger teams do emerge. Finally H-art’s employees work in an open space that facilitates interactions. Overall, in the paper we identify three typologies of interactions in both H-art and H-farm: a.) casual interactions that are very short, often one minute, involve small teams and are ubiquitous. Very short interactions may represent casual events, denied meetings or quick question and answer interactions; b.) spontaneous or deliberate interactions, for which the pressure to meet is higher, that involve medium size teams and medium length conversations. These meetings are more likely to reflect individual preferences about how to manage tasks. Spontaneous events happen daily and concern how employees manage their work and/or prefer to interact among them. Time is a scarce resource and individuals prefer medium size and length meetings in which they carefully select the identity of their colleagues. These events have a frequency that follows this rule: conversation that last twice as long occurs quarter as often. These events are those with the highest selectivity (or mutual information) of collocutors, which is associated to events of about 10-20 minutes. Casual and institutional events have the lowest mutual information. Also these are the most efficient and structured events for which the first passage time is longer than the recurrence time; c.) institutional interactions include large teams, long and rare events (an event that last twice as long occurs twice as rarely). While the above results concern both H-art and H-farm, some differences between the two exist. For both Hfarm and H-art mutual information grows till a pick, of about 10 minutes, and then goes down but in H-farm mutual information is higher also for shorter interactions and goes down rapidly after 10 minutes; in H-art mutual information is lower for shorter events and after the pick, at about 10 minutes, it remains higher than in H-farm before falling down. Again, while a common trend exists, two different organizational structures seem explaining non-identical interaction patterns. In H-art shorter communications may be less informative because generated by the open space and by the complexity of the organizational structure in which individuals contemporarily belong to functions and projects and the combinatorial possibilities of their meetings (at least the shortest ones) is higher. On the contrary, H-farm is more clustered and modular thus also shorter events help predicting collocutors. In both cases individuals are more willing to engage in shorter interaction processes while longer events are few. Furthermore in H-art mutual information persists for longer meetings than in Hfarm suggesting that H-art needs more time to manage its daily operative activities: the higher complexity of H-art’s organizational structure may be reflected in higher interaction process efforts as measured by the time spent interacting with selected colleagues. Overall different organizational structures generated different interaction processes. The most modular organization requires shorter and smaller meetings while the most complex organization needs additional communication efforts captured by, other things being equal, longer and bigger meetings.

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393

We also find a positive correlation between individuals’ interaction openness and the length of transmission paths featuring the communication structure in the entire working team. H-art’s transmission paths benefit the most from interaction openness: in organizations that are less structured and modular openness of interactions may have stronger effects on the length of communication processes. Finally we find that individuals are highly interconnected among them for short-medium communications, not exceeding 15 minutes, while communications become more structured and efficient after this threshold. In H-farm interconnection is higher for shorter events while at about ten minutes H-art appears as being more interconnected. The efficiency of communication patterns increase after a certain length, i.e. longer communications become more structured. This result may be related to the firms’ tendency to structure longer and bigger meetings. Coherently H-art that has the most complex organization displays the higher levels of interconnection at ten minutes. Overall, our results suggest that individuals’ interaction process within firm is more complex than interaction processes previously studied. Power laws still explain such processes but in a new way: higher exponents are required and individuals can handle a lower level of complexity than in previously observed one-to-one interaction processes. Our results suggest that simple principles reflecting interaction propensities, time budget and institutional constraints underlie the distribution of interaction events. As a result, the duration of interactions (as well as interval between interactions) reveal deep aspects of social systems. Not only interaction duration reveals a multiplicity of regimes affecting interaction parameters, but it also offers differentiated windows over different social network structures corresponding to such regimes. Interestingly, the level of complexity of different organizational structures affects interaction processes. In more complex and less bounded organizational structures multiple geometries of larger teams requiring longer meetings can emerge. Also, individuals may be involved in more complex problems of collocutors’ selection because more complex and open organizations increase the combinatorial possibilities of individuals. While longer events are imposed, shorter communication processes are voluntary. Indeed, individuals in complex organization spend more time in larger meetings and face more complex issues about how to allocate their time to collocutors. On the contrary, modular organizations require shorter and smaller meetings and available collocutors are ex-ante defined by the organizational structure [16]. Hence, the higher the organizational complexity the higher the communication effort of individuals and the resources to be allocated to interaction processes. Certainly, important aspects of the interaction process may have been affected by the setting of our observations. Our focus on two business organizations may have led to a stronger emphasis on institutional factors of communication than one might find in less structured contexts. However, as no human interaction is devoid of any institutional constraints, we expect our result set hold in a variety of interaction contexts. Our work also extends the range of tools available for analyzing the dynamic properties of interaction. In particular, we have demonstrated that mutual information can be useful for assessing pairwise interaction propensities, the entropy rate is helpful for rating interaction flexibility of group members while choosing a partner for communication, and the excess entropy can be viewed as a measure of complexity and heterogeneity of interactions. Finally, we have shown that by comparing the statistics of recurrence times and first-passage times of random walks over the biggest connected components of the graphs for different communication durations, one can appraise the quality of global connectedness of the working team for the interactions of functionally important durations. Future studies may deepen the role of the formal and informal organization, as separated variables, in shaping communication processes. Particularly, building on the extant social network literature, future studies may try to understand if those variables, such as hierarchical relationships and friendship [3, 17], that affect cumulative interactions also determine interaction processes.

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Acknowledgements The research leading to these results has received funding from the European Union’s Seventh Framework Programme (FP7/2007-2013) under grant agreement no. 318723: Mathematics of Multi-Level Anticipatory Complex Systems (MatheMACS). D.V. acknowledges the support from the Cluster of Excellence Cognitive Interaction Technology ’CITEC’ (EXC 277), Bielefeld University (Germany). A.C. acknowledges the support from the Ca’Foscari University of Venice. References [1] Moscovici, S. (1985), L’Age des foules: un trait´e historique de psychologie des masses, Fayard, 1981 / The age of the crowd: a historical treatise on mass psychology. Cambridge, Cambridge University Press. [2] Ahuja, G., Soda, G., and Zaheer, A. (2012), The Genesis and Dynamics of Organizational Networks, Organization Science, 23(2), 434–448. [3] Brass, D.J., Galaskiewicz, J., Greve, H.R., and Tsai, W. (2004), Taking stock of networks and organizations: a multilevel perspective, Academy of Management Journal, 47(6), 795–817. [4] Colfer, L. and Baldwin, C. (2010), The Mirroring Hypothesis: theory, evidence and exceptions, Harvard Business School Working Papers, No. 10-058. [5] Allen, J., James, A. D., and Gamlen, P., (2007), Formal versus informal knowledge networks in R&D: a case study using social network analysis, R&D Management, 37(3), 179–196. [6] Amrit, C.and Van Hillersberg, J., (2008), Detecting coordination problems in collaborative software development environments, Information Systems Management, 25,57–70. [7] Baldwin, C.Y. and Clark, K.B. (2000), Design rules - Vol. 1: ”The power of modularity”, Cambridge, Massachusetts, MIT Press. [8] Barbasi, A.-L. and Bonabea, E. (2003), Scale-Free Networks, Scientific American, May, 50–59. [9] Brusoni, S. and Prencipe, A. (2001), Unpacking the black box of modularity: technologies, products and organizations, Industrial and Corporate Change, 10(1),179–205. [10] Cabigiosu, A., Zirpoli, F., and Camuffo, A.(2013), Modularity, interfaces definition and the integration of external sources of innovation in the automotive industry, Research Policy, 42, 662–675. [11] Kossinets, G. and Watts, D.J. (2006), Empirical analysis of an evolving social network, Science, 311(5757), 88–90. [12] Kratzer, J., Gem¨unden, H.G., and Lettl, C. (2008), Balancing creativity and time efficiency in multi-team R&D projects: the alignment of formal and informal networks, R&D Management, 38(5), 538–549 . [13] MacCormack, A.D, Rusnak, J., and Baldwin, C.Y, (2012), Exploring the duality between product and organizational architectures: a test of the ”mirroring” hypothesis, Research Policy, 41, 1309–1324. [14] Oliveira, J.G. and Barabasi, A.-L. (2005), Nature 437, 1251. [15] Rank, O.N. (2008), Formal structures and informal networks: Structural analysis in organizations, Scandinavian Journal of Management, 24(2), 145–161 . [16] Sanchez, R. and Mahoney, J.T. (1996), Modularity, flexibility, and knowledge management in product and organization design, Strategic Management Journal,17, 63–76. [17] Soda, G., Zaheer A., ”A network perspective on organizational architecture: performance effects of the interplay of formal and informal organization.” Strategic Management Journal 33(6), 751-771 (2012). [18] Sparrowe, R.T., Liden, R.C., Wayne, S.J., and Kraimer, M.L. (2001), Social Networks and the Performance of Individuals and Groups, Academy of Management Journal, 44(2), 316–325. [19] Tsai, W. (2000), The formation of intraorganizational linkages, Strategic Management Journal, 21(9), 925–939. [20] Vazquez, A. (2005), Exact Results for the Barabasi Model of Human Dynamics, Phys. Rev. Lett., 95, 248701. [21] Vazquez, A., Oliveira, J.G., Dezsæ, Z., Goh, K.-I., Kondor, I., and Barabasi, A.-L. (2006), Modeling bursts and heavy tails in human dynamics, Phys. Rev. E, 73, 036127. [22] Vazquez, A. and J.G. Oliveira (2009), Impact of interactions on human dynamics, Physica A 388, 187–192. [23] Simon, H.A. (1962), The Architecture of Complexity, Proceedings of the American Philosophical Society 106 (6), 467–482. [24] Beylkin, G. and Monzon, L., (2005), On approximation of functions by exponential sums, Appl. Comput. Harmon. Anal. 19, 17–48. [25] Matth¨aus, F., Mommer, M.S., Curk, T., and Dobnikar, J. (2011), On the Origin and Characteristics of Noise-Induced L´evy Walks of E. Coli, PLoS ONE 6(4): e18623. doi:10.1371/journal.pone.0018623. [26] Petrovskii, S., Mashanova, A., and Jansen, V.A.A. (2011), Variation in individual walking behavior creates the impression of a L´evy flight”, Proceedings of the National Academy of Sciences 108 (21), 8704–8707. [27] Powers, D.M.W., (1998), Applications and explanations of Zipf’s law, Association for Computational Linguistics,

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Stroudsburg, PA, USA, pp. 151–160 . [28] Arndt, C. (2004), Information Measures, Information and its Description in Science and Engineering (Springer Series: Signals and Communication Technology), ISBN 978-3-540-40855-0. [29] Cover, T. and Thomas, J., (1991), Elements of Information Theory, John Wiley and Sons, Inc., ISBN 0-471-06259-6. [30] Crutchfield, J. P. and Young K. (1989), Inferring statistical complexity, Phys. Rev. Lett. 63, 105–108 . [31] Volchenkov, D. (2014), Path integral distance for the automated data interpretation, Discontinuity, Nonlinearity, and Complexity 3(3), 255–279 . [32] Volchenkov, D.and Ph. Blanchard, (2011), Introduction to Random Walks and Diffusions on Graphs and Databases, in Springer Series in Synergetic 10, Berlin, Heidelberg , ISBN 978-3-642-19591-4 . [33] Volchenkov, D. and Ph. Blanchard,(2008), ”Intelligibility and first passage times in complex urban networks”, Proc. R. Soc. A 464 2153-2167; doi:10.1098/rspa.2007.0329 . [34] Volchenkov, D., (2013), Markov Chain Scaffolding of Real World Data, Discontinuity, Nonlinearity, and Complexity 2(3) 289-299 — DOI: 10.5890/DNC.2013.08.005.

Discontinuity, Nonlinearity, and Complexity 5(4) (2016) 397–406

Discontinuity, Nonlinearity, and Complexity https://lhscientificpublishing.com/Journals/DNC-Default.aspx

On Quasi-periodic Perturbations of Duffing Equation A.D. Morozov†, T.N. Dragunov Institute of IT, Mathematics and Mechanics, Lobachevsky University of Nizhny Novgorod, 23 Gagarin Ave, Nizhny Novgorod, 603950, Russia Submission Info Communicated by Valentin Afraimovich Received 13 January 2016 Accepted 14 March 2016 Available online 1 January 2017 Keywords Resonance Quasi-periodic Motion Invariant Tori Bifurcation

Abstract Quasi-periodic two-frequency perturbations are studied in a system which is close to a nonlinear two-dimensional Hamiltonian one. The example of Duffing equation with a saddle and two separatix loops is considered. Several problems are studied: dynamical behavior in a neighborhood of a resonance level of the unperturbed system, conditions for the existence of resonance quasi-periodic solutions (two-dimensional resonance tori), global behavior of solutions inside domains separated from the unperturbed separatrix. In a neighborhood of the unperturbed separatrix the problem of relative position of stable an unstable separatrix manifolds is studied, conditions for the existence of doubly asymptotic solutions are found. ©2016 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Consider the system

∂H + ε g(x, y,t) ∂y ∂H + ε f (x, y,t), y˙ = − ∂x x˙ =

(1)

where ε is a small positive parameter. Suppose that inside the domain G ⊂ R2 its Hamiltonian H(x, y) and functions g and f are continuous and uniformly bounded in G with their partial derivatives upto order two and quasi-periodic in t. Suppose that the unperturbed system has a cell D ⊂ G which is partitioned by closed phase curves. Suppose also that the boundary of the cell D contains a separatrix loop for a saddle equilibrium. The first problem is to study behavior of solutions inside the cell D. The case of periodical perturbation was considered in various publications, e.g. [1], [2]– [5] and references therein. The second problem is to analyze the distance between separatrix manifolds Wεs and Wεu which coincide in the unperturbed system. This problem is solved for the periodical perturbation in [6]. † Corresponding

author. Email address: [email protected] ISSN 2164 − 6376, eISSN 2164 − 6414/$-see front materials © 2016 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/DNC.2016.12.005

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Below we consider these problems for the following example x˙ = y, y˙ = x − x3 + ε (p1 y + F(t)),

(2)

in two cases: Case 1: F(t) = p2 sin ω1t + p3 sin ω2 t; Case 2: F(t) = p2 sin ω1t sin ω2 t. Here pk , k = 1, 2, 3 are parameters. The system is equivalent to the equation x¨ − x + x3 = ε (p1 x˙ + F(t)).

(3)

Suppose that frequencies ω1 , ω2 are incommensurable. This implies that the perturbation is a quasi-periodic function in t.

Fig. 1 Phase portrait of system (2) at ε = 0.

Problems of the existence of quasi-periodic and almost periodic solutions of Duffing equation with one saddle equilibrium subjected to a quasi-periodic and almost periodic forcing were considered in [7] (see also [8]). The existence of complex dynamics in Duffing-like equations was discussed in various publications, e.g. [2] [5], [9]- [12]. The case of periodic in t perturbations for Duffing – Van der Pol equation was studied in [13]. System (2) with ε = 0 has in its phase space three cells partitioned by closed phase curves: two cells D˜ ± corresponding to values of energy integral h ∈ (−0.25, 0) and D˜ 0 corresponding to h > 0. Energy integral has 2 2 4 the form y2 − x2 + x4 = h. Denote by D∗ the domain which is the result of subtracting a neighborhood of the unperturbed separatrix (x, y) : −δ < h < δ from R2 where δ is sufficiently small positive constant. Denote by D± the domain corresponding to h ∈ (−0.25, −δ ) and denote by D0 the domain corresponding to h > δ . Figure 1 illustrates the phase portrait of the unperturbed system. The original system (2) should be considered in extended phase space R3 . Closed phase curves of the unperturbed system correspond there to cylinders which are invariant integral surfaces.

2 On behavior of solutions in the domain D∗ Changing in (2) variables x, y to action and angle variables I, θ we obtain the following system I˙ = ε F1 (I, θ , θ1 , θ2 ) θ˙ = ω (I) + ε F2(I, θ , θ1 , θ2 )

θ˙1 = ω1 θ˙2 = ω2 ,

(4)

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where

F1 = [p1 y(I, θ ) + F(t)]xθ (I, θ ),

F2 = −[p1 y(I, θ ) + F(t)]xI (I, θ ).

399

(5)

Phase space of this system is R1+ × T 3 . When ε = 0 the four-dimensional phase space exfoliates into three-dimensional tori T3 with conditionally periodic motion with frequencies ω , ω1 , ω2 . When ε = 0 invariant tori break down because of • the presence of resonances nω (I) = mω1 + kω2 ,

(6)

where n, m, k are relatively prime integer numbersa ; • the presence of dissipation. Consider the case of resonance. From (6) we can calculate resonance values of action I = Inmk . Making the change in (4) √ θ = v + (mθ1 + kθ2 )/n, I = Inmk + μ u, μ = ε . (7) we obtain that in a neighborhood Uμ × T 2 , where Uμ = {(I, θ ) : (Inmk −C μ < Inmk < Inmk +C μ , 0 ≤ θ < 2π ,C = const > 0} system (4) transforms into u˙ =μ F1 (Inmk , (v + (mθ1 + kθ2 )/n, θ1 , θ2 )+ ∂ F1 (Inmk , (v + (mθ1 + kθ2 )/n, θ1 , θ2 ) u + O(μ 3 ), + μ2 ∂I v˙ =μ b1 u + μ 2 (b2 u2 + F2 (Inmk , (v + (mθ1 + kθ2 )/n, θ1 , θ2 )) + O(μ 3 ), θ˙1 =ω1 ,

(8)

θ˙2 =ω2 . Note. We should visualize the behavior of solutions of the original system (2) in the extended phase space Uμ × R1 , i.e. in a neighborhood of the resonance cylinder of the unperturbed system. Since θ1 = ω1t, θ2 = ω2 t first two equations of system (8) have a standard form and it is applicable for Krylov and Bogolyubov averaging method. Suppose that relation ω1 /ω2 is in a sense hardly approximated by rational numbers. Due to absence of relation between ω1 and ω2 functions F1 and F2 have the following property: the average in time is equal to average in angle variables θ1 , θ2 . The least common period of functions F1 and F2 in θ1 and θ2 is equal to 2π n. Using results of [5] and neglecting terms O(μ3 ) the averaged system may be reduced to the form u˙ = μ A(v, Inmk ) + μ 2 p1 u, v˙ = μ b1 u + μ 2 b2 u2 ,

(9)

where b1 = d ω (Inmk )/dI, b2 = d 2 ω (In11 )/2dI 2 , 1 A= 2 2 4π n

ˆ

2π n ˆ 2π n 0

F1 (Inmk , (v + (mθ1 + kθ2 )/n, θ1 , θ2 )d θ1 d θ2 .

(10)

0

Using the ”slow time” τ = μ t we can rewrite it as u˙ = A(v, Inmk ) + μ p1 u, v˙ = b1 u + μ b2 u2 . a Generally,

there are two integer vectors satisfying (6)

(11)

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The least period of function A(v, Inmk ) is equal to 2π /n [5]. Therefore system (9) is defined on a cylinder them, A (v0 ) = 0. Roots (u, v mod 2π /n). It may have two simple equilibrium points. Let (v0 , 0) be one of

of the characteristic equation for (11) have the form λ1,2 = α ± β , α = μ p1 /2, β = μ 2 p21 + 4b1 A (v0 )/2. If b1 A (v0 ) > 0 then the equilibrium is of saddle type. If b1 A (v0 ) < 0 then the equilibrium is focus and it is stable for p1 < 0. So when p1 = 0 then simple equilibria of the averaged system (9) correspond to two-dimensional invariant tori with quasi-periodic winding motion in the original four-dimensional system, i.e. quasi-periodic solutions with frequencies ω1 , ω2 . This is the result of the second theorem of Bogolyubov [14]). Such tori we will call resonance tori. System (9) has different form depending on the domain where it is calculated. Due to the symmetry of the unperturbed system with respect to change x → −x it is sufficient to study only domains D+ and D0 . 2.1

Resonances in the domain D+

2.1.1

Case F(t) = p2 sin ω1 t + p3 sin ω2 t

Let us find A(v, Inmk ) in D+ . To do this substitute (5) into (10) then obtain 1 A= 2 2 4π n

ˆ

2π n ˆ 2π n 0

0

(p1 y(θ , Inmk ) + p2 sin θ1 + p3 sin θ2 )xθ (θ , Inmk )d θ1 d θ2 ,

(12)

where

θ = v + (mθ1 + kθ2 )/n, y(θ , Inmk ) = ω (Inmk )xθ , x(θ ) = x1 dn(Kθ /π ),  √ √ √ √ ω = π x1 / 2K, x1 = 1 + 1 + 4h, k2 = (2 1 + 4h)/(1 + 1 + 4h).

(13)

Here K(k) is the complete elliptic integral of the first kind, k is its module, k = k(hnmk ), hnmk is the resonance value of energy integral which is determined by expression nω (hnmk ) = mω1 + kω2 . Rewrite A(v) in the form A(v) = p1 S0 + p2 S1 + p2 S3

(14)

where S0 = (2/(3(2 − k2 )3/2 ))[2(k2 − 1)K + (2 − k2)E] ≥ 0, E is the complete elliptic integral of the second kind, 1 S1 = 2 2 4π n 1 S2 = 2 2 4π n

ˆ

2π n ˆ 2π n 0

ˆ

0 2π n ˆ 2π n

0

0

xθ (θ , Inmk ) sin θ1 d θ1 d θ2 ,

xθ (θ , Inmk ) sin θ2 d θ1 d θ2 .

Using the expansion 2π π + dn(ϕ ) = 2K K



jπϕ aj ∑ 1 + a2 j cos K , j=1

√ π K( 1 − k2 ) a = exp(− ), K(k)

(15)

we find that S1 = S2 = 0. Consequently the averaged system (11) with p1 = 0 does not have equilibrium points and resonance levels I = Inmk are passable. Then we conclude that there are no two-dimensional invariant tori in domains D± .

A.D. Morozov, T.N. Dragunov / Discontinuity, Nonlinearity, and Complexity 5(4) (2016) 397–406

2.1.2

401

Case F(t) = p2 sin ω1 t sin ω2t

In this case A(v) = p1 S0 + p2 S1 , where S1 =

1 4π 2 n2

ˆ

2π n ˆ 2π n

0

0

(16)

xθ (θ , Inmk ) sin θ1 sin θ2 .d θ1 d θ2

Finally √ A(v; Inmk ) = p1 S0 + p2 2

an (ω1 + ω2 ) sin nv 1 + a2n

(17)

when k = m = 1 and A(v; Inmk ) = p1 S0 when m, k are different from 1.

U

2.4

2.4

1.6

1.6

0.8

0.8

0.0

U

0.0

-0.8

-0.8

-1.6

-1.6

-2.4

-2.4

-2.4

-1.6

-0.8

0.0

0.8

1.6

2.4

-2.4

-1.6

-0.8

V

0.0

0.8

1.6

2.4

V

(a)

(b)

Fig. 2 Phase portraits: (a) partly passable resonance, (b) passable resonance.

Phase space of this system in the stripe (−π /n, π /n] corresponding to the period of A(v; Inmk ) has two equilibrium points: a saddle and a focus when √ |p1 S0 | < |p2 2

an (ω1 + ω2 )|. 1 + a2n

(18)

The focus is stable when p1 < 0. These two equilibria correspond to unstable and stable two-dimensional tori of the original system. Figure 2.1.2 shows phase portraits of system (9). Note that the coefficient before sin nv in (17) is fast decreasing with n while S0 has a fixed value. This leads to existence of only finite number of partly passable resonances, i.e. resonances that have in their neighborhood simple equilibrium points of the averaged system (9). So the following theorem is true. Theorem 1. When ε > 0 is sufficiently small, p1 = 0 and F(t) = p2 sin ω1 t + p3 sin ω2 t then there are no twodimensional resonance invariant tori in a neighborhood of Uμ × T 2 in the system (2). When F(t) = p2 sin ω1 t sin ω2 t and conditions (18) are satisfied then there are two two-dimensional resonance tori: a torus which is stable when p1 < 0 and an unstable saddle torus T2 i.e. there exists quasi-periodic in t solution with periods 2π n/ω1 , 2π n/ω2 .

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2.2

Resonances in the domain D0

Case F(t) = p2 sin ω1 t + p3 sin ω2 t The unperturbed system in the cell D˜ 0 , (h > 0) has periodic solution x(θ ) = x1 cn(2Kθ /π ) (see e.g. [5]), where 2.2.1

ω =π





x21 − x22 /2

 2K,

x1,2 =

√ 1 ± 1 + 4h,

√ √ k2 = (1 + 1 + 4h)/2 1 + 4h.

(19)

The function A(v) we can represent as before in the form (14). Then we obtain S0 =

4 3(2k2 − 1)3/2

[(1 − k2 )K + (2k2 − 1)E] > 0.

Using the expansion cn(ϕ ) =

2π kK



1

a j− 2 πϕ ∑ 1 + a2 j−1 cos(2 j − 1) 2K , j=1

(20)

we get S1 = 0, S2 = 0. Then A(v, Imnk ) = p1 S0 does not depend on v and the averaged system (9) does not have equilibria. All resonance levels I = Imnk are passable. 2.2.2

Case F(t) = p2 sin ω1 t sin ω2t

In this case we find

√ an/2 (ω1 + ω2 ) sin nv, A(v; Inmk ) = p1 S0 + p2 2 1 + an

(21)

when n is odd and k = m = 1. When n is even or k and m are different from 1 then A(v; Inmk ) = p1 S0 . So the condition of the existence of equilibrium in (9) is √ an/2 (ω1 + ω2 )|. |p1 S0 | < |p2 2 1 + an

(22)

The second approximation of the averaging method gives a system in the form (9). Consequently its analysis is similar to the study of the domain D+ . So for resonance levels in the domain D0 theorem 1 is valid. Note that in the domain D0 resonance tori may exist only for odd values of n while in domains D± resonance tori may exist for all positive integer values of n. 2.3

Non-resonance case

Let us show that in D∗ × T 2 there are no three-dimensional tori T3 . Assume that for I = I∗ the undamped frequency ω (I∗ ) is incommensurable with frequencies ω1 , ω2 and the following condition is true for any m, n, k ∈ Q, m > 0 and certain positive C, τ : |ω∗ − (nω1 + kω2 )/m| > C/mτ . Making in (4) the change I = I∗ + μ r, we result in r˙ = μ F1 (I∗ , θ , θ1 , θ2 ) + O(μ 2 ), θ˙ = ω (I∗ ) + μ br + O(μ 2 ),

θ˙1 = ω1 , θ˙2 = ω2 . If we fix an integer N and use Fourier expansion of the function F1 F1 (I∗ , θ , θ1 , θ2 ) =

N



m,n,k=−N

Fmnk (I∗ )expi(mθ +nθ1 +kθ2 ) + RN (I∗ , θ , θ1 , θ2 )

(23)

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403

and make the following change in (23) Fmnk (I∗ ) expi(mθ −nθ1 −kθ2 ) + RN (I∗ , θ , θ1 , θ2 ), mω∗ − nω1 − kω2

N



r = u − iμ

m, n, k = −N

(24)

m2 + n2 + k2 = 0 then we get the following system u˙ = μ (p1 S0 (I∗ ) + RN (I∗ θ , θ1 , θ2 )) + O(μ 2 ), θ˙ = ω∗ + μ bu + O(μ 2 ),

θ˙1 = ω1 , θ˙2 = ω2 .

(25)

In general, the series in the change (24) with N = ∞ is diverging. For considered case |p1 S0 | > maxθ ,θ1 ,θ2 |RN | if N is sufficiently large. Since S0 = 0, non-resonance levels I = I∗ are passable when p1 = 0 and so the tori T 3 do not exist in a neighborhood of such levels I = I∗ . Thus, the behavior of solutions in domains D± , D0 in the case 1 does not differ significantly from the autonomous case when p2 = p3 = 0, p1 = 0. But for the case 2, there may exist partly passable resonances and resonance two-dimensional tori corresponding to two-frequency modes.

3 Splitting of separatrix Consider system (1). Without loss of generality assume that the saddle of unperturbed system lies at the origin. Denote by W0s ,W0u the stable and unstable integral manifolds of saddle (0, 0) of the system (1) when ε = 0. In (1) closed phase curves are projections of cylinders in R2 × R and the separatrix loop is the projection of the cylinder separatrix manifold. Suppose that (26) g(0, 0,t) = 0, f (0, 0,t) = 0 ∀t ∈ R1 . According to [15, 16] for sufficiently small ε there exist manifolds Wεs ,Wεu such that lim Wεs = lim Wεu = 0,

t→∞

sup τ ∈[t,∞)

t→−∞

W0s (τ ) −Wεs (τ ) =

O(ε ),

sup W0u (τ ) −Wεu (τ ) = O(ε ).

τ ∈(−∞,t]

Let us find the distance Δ between manifolds Wεs and Wεu of system (1). In the case of periodical perturbation this distance is determined by Melnikov formula [6]. Its derivation was performed by using Poicar`e method of small parameter requiring analyticity of the right-hand side of system (26). Sanders [15] extended applicability of Melnikov formula to sufficiently smooth systems. The existence of quasi-periodical solutions and integral manifolds was considered by Hale [16]. Following [6, 15] we find Δε (t0 ) = ε Δ1 (t0 ) + O(ε 2 ), ˆ

where Δ1 (t0 ) =



−∞

( f (xs , ys ,t)xθ − g(xs , ys ,t)yθ )dt,

(27)

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xs = x(t − t0 ), ys = x(t − t0 ) is the solution of unperturbed system on the separatrix. Come back to the equation (3). Similar equation in the case 2 was considered in [9] where with commensurable ω1 and ω2 Melnikov formula was used without fixing saddle periodic motion and perturbation with quasi-periodic function F(t) was studied only numerically. To fix the saddle periodic motion perform a change x = ξ + ε x1 (t) + O(ε 2 ). Then neglecting terms O(ε 2 )) we obtain the following equation: ξ¨ − ξ + ξ 3 = ε (p1 ξ˙ − 3ξ 2 x1 (t)), (28) where x1 (t) = −

p2 p3 sin (ω1 t) − sin (ω2 t), 1 + ω12 1 + ω22

for the case 1, and x1 (t) =

p2 cos (ω1 − ω2 )t cos (ω1 + ω2 )t [ + ], 2 2 1 + (ω1 − ω2 ) 1 + (ω1 + ω2 )2

for the case 2. Equation (28) satisfies condition (26). Then ˆ Δ1 (t0 ) = where





([p1 ξ˙s (t − t0 ) − 3ξ 2 (t − t0 )x1 (t)]ξ˙s (t − t0 )dt,

√ ξs (t) = ± 2/ cosh t,

√ ξ˙s (t) = ∓ 2 sinh(t)/ch2 (t)

(29)

(30)

is the solution of the unperturbed equation on the separatrix. Here plus sign corresponds to the right separatrix loop and minus sign corresponds to the left separatrix loop. In the case 1 calculation of the integral results in Δ1 (t0 ) = 4p1 /3 + p2 B1 cos (ω1 t0 ) + p3 B2 cos (ω2 t0 ), where

(31)

√ √ 3 2 3 2 , B2 = . B1 = (1 + ω12 ) cosh(πω1 /2) (1 + ω22 ) cosh(πω2 /2)

In the case 2 the result is √ 4 π 2 [B1 sin (ω1 + ω2 )t0 + B2 sin (ω1 − ω2 )t0 ], Δ1 (t0 ) = p1 + p2 3 2 where B1,2 =

(32)

(ω1 ± ω2 ) . cosh ((ω1 ± ω2 )π /2)

Here plus sign corresponds to B1 and minus sign is for B2 . So if the perturbation is quasi-periodic in t then Melnikov function is quasi-periodic too.  √ Formula (31) implies that if |p1 | < (3/4) p22 B21 + p23 B22 for the case 1 and |p1 | < (3/8) 2π |p2 | B21 + B22 for the case 2 then function Δ1 (t0 ) is sign-alternating and consequently Wεs ∩ Wεu = 0. / Then there exist homoclinic points which have doubly-asymptotic solutions in (x, y,t) space passing through them. The structure of a neighborhood of such solutions was studied in [17].

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405

Fig. 3 Dependency of undamped frequency ω on energy integral value h and resonance values 3-1-1, 2-1-1 for ω 1 = 1, √ ω2 = 3.

t

400 y 1.6

300 1.2

0.8

200 0.4

-1.6

-1.2

-0.8

-0.4

0.4

0.8

1.2

1.6

x

100

-0.4

y -0.8

0.8

-1.2

-1.6

-1.2

-0.8

-0.4

0 -0.8

-1.6

0.4

0.8

1.2

1.6

x

-1.6

t

400 y 0.8

300 0.6

0.4

200

0.2

-1.2

-0.8

-0.4

0.4

0.8

1.2

x

100

-0.2

y -0.4

0.4

-0.6

-1.2

-0.8

-0.4

0 -0.4

0.4

0.8

1.2

x

-0.8

-0.8

Fig. 4 Images of a partly passable resonance level (p 1 = −0.01, p2 = 0.6, ω1 = 1, ω2 =

√ 3, ε = 0.005).

4 On global behavior of solutions Due to the absence of resonance tori in the case 1 quality behavior of solutions does not differ significantly from the autonomouos case. In the case 2 the number of resonance levels I = Inmk with resonance tori is bounded so

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we can also establish global behavior of solutions in D∗ . Conditions (18), (22) for the existence of equilibria in averaged systems imply the existence of resonance tori only for several first values of n. Using formulas for the undamped frequency (13), (19) we can plot its dependency on h (Fig. 3). By fixing ω = (ω1 + ω2 )/3) we get two values of h311 . We can also select suitable values for the parameters p1 , p2 so that quasi-periodical solutions (two-dimensional resonance tori) appear in a neighborhood of level h = h311 . While visualizing solutions it is naturally to consider extended three-dimensional phase space of the original system (2), i.e. the behavior of solutions in Uμ × R1 . So if there exists a resonance torus then projections of 2 2 4 solutions onto phase plane should densely paint a ring containing the resonance curve y2 − x2 + x4 = hn11 . Figure 4 presents projection of solutions onto plane (x, y) and corresponding three-dimensional picture. Acknowledgments Our work was partially supported by the RFFR grant No 14-01-00344, RSCF, grant No 14-41-00044 and the Ministry of Education and Science of Russian Federation, Project 1410.

References [1] Afraimovich, V.S. and Shil’nikov, L.P. (1974), On small periodic perturbations of autonomous systems, Dokl. Akad. Nauk SSSR (Russia), 214(4), 739–742. [2] Morozov, A.D. (1976), On total qualitative investigation of the Duffing equation, J. Differentsialnye uravnenia (Russian), 12(2), 241–255. [3] Morozov, A.D. and Shil’nikov, L.P. (1983), On nonconservative periodic systems similar to two-dimensional Hamiltonian ones, Prikl. Mat. i Mekh. (Russian), 47(3), 385–394. [4] Wiggins, S. (1990), Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer, Berlin. [5] Morozov, A.D. (1998), Quasi-conservative systems: cycles, resonances and chaos, World Sci.: Singopure, in ser. Nonlinear Science, ser. A, V. 30, 325 p. [6] Mel’nikov, V.K. (1963), On stability of a center under periodic in time perturbations, Works of Moscow Math. Soc., 12, 3–52. [7] Berger, M.S. and Chen, Y.Y. (1992), Forced Quasiperiodic and Almost Periodic Oscillations of Nonlinear Duffing Equations, Nonlinear Analysis, Theory, Methods and Applications, 19(3), 249–257. [8] Liu, B. and You, J. (1998), Quasiperiodic solutions of Duffing’s Equations, Nonlinear Analysis, 33, 645–655. [9] Ravichandran, V., Chinnathambi, V. and Rajasekar, S. (2007), Homoclinic bifurcation and chaos in Duffing oscillator driven by an amplitude-modulated force, Physica A, 376, 223–236. [10] Grischenko, A.D. and Vavriv, D.M. (1997), Dynamics of Pendulum with Quasi-periodic excitation, J. Theor. Phys. (in Russian), 67(10), [11] Jing, Z.J., Huang, J.C., and Deng J. (2007), Complex dynamics in three-well duffing system with two external forcings, Chaos, Solitons and Fractals, 33, 795–812. [12] Spears, B.K., Hutchings, M., and Szeri, A.J. (2005), Topological Bifurcations of Attracting 2-Tori of Quasiperiodically Driven Oscillators. J. Nonlinear Sci, 15, 423–452. [13] Morozov, A.D. and Kostromina, O.S. (2014), On Periodic Perturbations of Asymmetric Duffing-Van-der-Pol Equation, International Journal of Bifurcation and Chaos, 24(5). [14] Bogolyubov, N.N. and Mitropolsky, Yu.A. (1958), Asymptotical methods it the theory of nonlinear oscillations (in Russian), Fizmatgiz, Moscow. [15] Sanders, J.M. (1980), Melnikov’s method and averaging, SIAM J. Math. Anal. 11, 750–770. [16] Hale, J.K. (1963), Oscillations in nonlinear systems, McGRAW-Hill Book company Inc., New York, Toronto, London. [17] Shilnikov, L.P. (1967), On a Poincar`e-Birkhoff problem, Math. USSR Sb., 74(3), 378–397.

Discontinuity, Nonlinearity, and Complexity 5(4) (2016) 407–414

Discontinuity, Nonlinearity, and Complexity https://lhscientificpublishing.com/Journals/DNC-Default.aspx

μ A Study of the Dynamics of the Family fλ ,μ = λ sin z + z−k π where λ , μ ∈ R \ {0} and k ∈ Z \ {0}

Patricia Dom´ınguez†, Josu´e V´azquez, Marco A. Montes de Oca Facultad de Ciencias F´ısico Matem´aticas, Benem´erita Universidad Aut´onoma de Puebla, Puebla, Pue. CP. 72595, Mexico Submission Info Communicated by Valentin Afraimovichs Received 18 January 2016 Accepted 15 March 2016 Available online 1 January 2017 Keywords

Abstract In this article we investigate the dynamics of the meromorphic family μ fλ ,μ (z) = λ sin z + z−k π , λ , μ ∈ R \ {0} and k ∈ Z \ {0}. We show that for some parameters λ , μ the Stable set contains an attracting component which is multiply connected and completely invariant. We give a definition of a cut of the space of parameters, with μ and kπ fixed, and show examples of a cut and the Stable and Chaotic sets related to the cut, for some λ given.

Iteration Meromorphic function Stable set Chaotic set ©2016 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Iterations of transcendental entire functions were initially studied by Fatou [1] and Baker [2] [3], some families of this class of functions are λ ez and λ sin z. The dynamics of such families are completely different and have been investigated by several mathematicians such as Devaney [4], Bathachayya [5], Dom´ınguez and Sienra [6]. If we add a (non omitted) pole in the families given above the dynamics will change dramatically since we will have a set of preimeges of ∞. In this article we will be interested in the dynamics of the following class of functions.  | f is transcendental meromorphic and has at least one pole which is not omitted}. M = {f : C → C If f is a function in class M the sequence formed by its iterates is defined and denoted by f n := f ◦ f n−1 , n ∈ N, and f 0 := Id where ◦ denotes composition. We say that z0 is a periodic point of f if f p (z0 ) = z0 for some p ∈ N, when p = 1 the point z0 is called fixed point. If f ∈ M the classification of a fixed point z0 of period p is given by: (a) super-attracting if |( f p ) (z0 )| = 0; (b) attracting if 0 < |( f p ) (z0 )| < 1; (c) repelling † Corresponding

author. Email address: [email protected] ISSN 2164 − 6376, eISSN 2164 − 6414/$-see front materials © 2016 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/DNC.2016.12.006

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if |( f p ) (z0 )| > 1; (d) rationally indifferent if |( f p ) (z0 )| = 1 and ( f p ) (z0 ) is a root of unit; and (e) irrationally indifferent if |( f p ) (z0 )| = 1, but ( f p ) (z0 ) is not a root of unit. Dynamics of meromorphic functions in class M initially were studied by Baker, Kotus and Yinian in [7], [8], [9] and [10]. The Stable set F( f ) of a function in class M is defined by the set of points z ∈ C such that the sequence { f n }n∈N is well defined and normal in some neighbourhood of z. The Chaotic set is the complement of the Stable set and we shall denote it by J( f ). Some properties of J( f ) and F( f ) for functions in class M are: (a) F( f ) is open and J( f ) is closed. (b) J( f ) is perfect and non empty. (c) F( f ) and J( f ) are completely invariant under f . (d) F( f ) = F( f n ) and J( f ) = J( f n ) for all n ∈ N. (e) J( f ) is the closure of the set of all repelling periodic points of f . The classification of a component U in the Stable set can be periodic, pre-periodic or wandering. • If f n (U ) ⊂ U for some integer n ≥ 1, then U is called a periodic component of F( f ). The minimum n is the period of the component. In particular, if n = 1, then such a component U is said to be an invariant component or a fixed component. • If f m (U ) is periodic for some integer m ≥ 0, then U is called a pre-periodic component of F( f ). • If U is neither periodic nor pre-periodic, then U is a wandering component. If U is periodic component of F( f ) of period p the classification of the periodic component is given as follows for functions in class M . 1. If U contains an attracting periodic point z0 of period p and f np (z) → z0 for z ∈ U as n → ∞, then U is called the attracting component. 

2. If ∂ U contains a periodic point z0 of period p and f np (z) → z0 for z ∈ U as n → ∞. Then ( f p ) (z0 ) = 1 if  z0 ∈ C. For z0 = ∞ we have (g p ) (0) = 1 where g(z) = f (11 ) . In this case, U is called either a Leau domain or parabolic component.

z

3. U is called a Siegel disc if there exists an analytic homeomorphism ϕ : U → D, where D is the unit disc such thatϕ ( f p (ϕ −1 (z))) = e2π iα z for some α ∈ R \ Q. 4. U is called a Herman ring if there exists an analytic homeomorphism ϕ : U → A, where A is an annulus A = {z : 1 < |z| < r}, r > 1, such that ϕ ( f p (ϕ −1 (z))) = e2π iα z for some α ∈ R \ Q. 5. U is called a Baker domain if there exists z0 ∈ ∂ U such that f np (z) → z0 , for z ∈ U as n → ∞, but f p (z0 ) is not defined. The set of singular values of a function f in class M are the critical and asymptotic values of f . We recall  is an asymptotic value of f if there is a that a critical value is the image of a critical point, and a point a ∈ C path γ (t) → ∞ as t → ∞, such that f (γ (t)) → a as t → ∞. When ∞ is an asymptotic value for functions in class M it belongs to the Chaotic set. Definition 1. The class B is the set of functions f ∈ M of bounded type, this is the set B consists of functions  f ∈ M for which all singular values are contained in a bounded set in C.

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In Section 2 we prove Theorem 1 which is a generalization of the following result given by Dom´ınguez in [11]. Result. Let λ , μ ∈ R such that 0 < λ < 1 and μ > 0 sufficiently small. The family fλ ,μ (z) = λ sin z + z−μπ has an attracting completely invariant component in the Stable set which is multiply connected. Theorem 1. If λ , μ are real parameters such that 0 < |λ | < 1 and μ > 0 sufficiently small, then the family μ fλ ,μ (z) = λ sin z + z−k π , k ∈ Z \ {0}, has an attracting completely invariant component U in the Stable set, such component is multiply connected. μ Corollary 2. For λ = 1 and 0 < μ ∈ R sufficiently small the family f μ (z) = sin z + z−k π , k ∈ Z \ {0}, has an attracting completely invariant component U in the Stable set which is multiply connected. μ Corollary 3. The family fλ ,μ (z) = λ sin z + z−k π , k ∈ Z \ {0}, belongs to the class B.

2 Proofs of Theorem 1 and Corollaries 2 and 3 μ If we plot the graph of the real function fλ ,μ (x) = λ sin x + x−k π for some real parameters 0 < λ < 1 and 0 < μ sufficiently small, taking any k ∈ Z \ {0}, it is not difficult to see that there are two fixed points one is attracting and the other is repelling which belong to the Stable set and the Chaotic set respectively. In what follows we will prove Theorem 1 by using the proof in [11] with some changes since now the pole for the family fλ ,μ (z) is kπ , for any k ∈ Z \ {0} .

Proof of Theorem 1. We shall take the case when 0 < λ < 1 since for the case −1 < λ < 0 the proof is analogue. 

Let λ , μ ∈ R such that 0 < λ < 1 and μ > 0 sufficiently small. Following the idea in [11] take λ and α so   that λ < λ < 1, 0 < α < 1 and λ | cos z| < λ for z ∈ H = {z : |Im z| < α }.  Let r > 0 such that r < α and assume that μ is so small that λ α + μr < α . Now let T be as follows, see Figure 1. T = H ∩ {z : |z − kπ | > r, for some k ∈ Z \ {0} fixed}.

iα T r

λ

λ

0





Fig. 1 The set T , where kπ = 2π for k = 2.

Claim. There is an attracting invariant component U of the Stable set which contains T .

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First we shall show that for any z = x + iy ∈ T the difference | fλ ,μ (z) − λ sin x| is bounded by one. We split up the proof into two cases. μ (1) When y = 0 we have | fλ ,μ (z) − λ sin x| ≤ λ | sin x − sin x| + |z−k π| =

μ r

< 1.

(2) When y = 0 we have the following: | fλ ,μ (z) − λ sin x| ≤ λ | sin z − sin x| +

μ μ  μ = λ α | cos z| + < λ α + < α < 1. |z − kπ | r r

In both cases we observe that λ sin x ∈ [−λ , λ ] and since we are taking 0 < λ < 1, so it follows that fλ ,μ (z) is bounded in a ball with center λ and radius 1, this is B(λ , 1) ⊂ T , even more the family fλ ,μ (z) is uniformly bounded. Then by Montel’s Theorem fλ ,μ (z) is normal in T , thus T belongs to an invariant component U of the Stable set. It follows that the family fλ ,μ (z) belongs to a compact subset S ⊂ T in which fλn,μ (z) → p, where p is finite and belongs to the closure of S, so p must be an attracting fixed point of the family. Thus there exists an attracting invariant component U , which contains T , of the Stable set. μ Claim. All the finite critical values of fλ ,μ (z) = λ sin z + z−k π , where k ∈ Z \ {0}, are contained in U .

Observe that on any path γ which tends to ∞ we have μ (z − kπ )−1 → 0 and fλ ,μ (z) has a limit, say L, if and only if λ sin z → L. This is possible only for L = ∞. Thus apart from ∞ all singular values of fλ ,μ (z) are finite critical values of fλ ,μ (z) which come from the calculation of the solutions of the equation fλ ,μ (z) = 0. In what follows we shall take two cases: (i) If fλ ,μ (z) = 0 and |z − kπ | > t = π4 , then |λ cos z| − |μ (z − kπ )−2 | ≤ |λ cos z − μ (z − kπ )−2 | = 0. Thus we have | cos z| < μλ −1 t −2 < 16μλ −1 π −2 < 2μλ −1 , and sin z = ±(1 + η ), where |η | < 2μ 2 λ −2 (if μ was originally chosen small enough). For any such z we have | fλ ,μ (z) − λ sin z| < 4μ /π , and making some calculations we obtain: | fλ ,μ (z) ± λ | < 2μ 2 λ −1 + 4μ /π . Thus we conclude that fλ ,μ (z) ∈ T ⊂ U , if μ was chosen small enough. (ii) If fλ ,μ (z) = 0 and |z − kπ | ≤ t = π4 , then |λ cos z| = |μ (z − kπ )−2 |. Thus

√ μ μ ≤ |(z − kπ )2 | = | | ≤ 2μ /λ , π /4 (λ cos z) (λ e )

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since √ | cos z| ≥ | cos x cos hy| ≥ 1/ 2 Thus

and

| cos z| < e|y| .

1√  |z − kπ | < 2 4 μ 1/λ , √  | sin z| < 2 μ 1/λ , √ √ μ |z − kπ |−1 < μ λ e.

Therefore √ √ √ √ μ | < 2 μ λ + μ λ e. z − kπ Thus fλ ,μ (z) ∈ T ⊂ U , provided μ was chosen small enough. From (i) and (ii) we have that all the finite critical values of the family fλ ,μ (z) are in U . The claim is proved. | fλ ,μ (z)| = |λ sin z| + |

Claim. There are not Siegel discs or Herman rings. We shall denoted by E1 the set of the singular values of fλ ,μ (z) which consists of a countable subset of T whose closure is compact in T , together with ∞. The same is true for the following sets:  (a) E = ∞j=0 fλj ,μ (E1 \ B j ), where B j = {z : fλj ,μ is not meromorphic at z}. (b) E  = { points which are either accumulation points of E or singularities of some branch of fλ−n ,μ for infinitely many values of n} and  (c) E ∪ E  = ∞j=0 fλj ,μ (E1 \ {∞}) ∪ {p, ∞}. From (c) it follows that there are no Siegel discs or Herman rings since the boundary of a Siegel disk or a Herman ring should be contained in E ∪ E  by Theorem 8.2 in [11] or Theorem 7.1.4 in [12]. Thus the only cyclic component of the Stable set is U . Claim. U is completely invariant. We know that all the finite singular values of fλ ,μ (z) are critical values which are contained in U . Take a −1 point z0 in U and a branch g of fλ−1 ,μ such that g(z0 ) ∈ U . For any z1 in U and any branch h of fλ ,μ at z1 we can reach h(z1 ) by analytic continuation of g along a path γ from z0 to z1 . Now γ is homotopic to a path γ1 in C \ E1 ( f ) from z0 to z1 , and the continuation of g along γ1 is h at z1 . But g(γ1 ) belongs to the Stable set and hence g(γ1 ) ⊂ U . Thus U is completely invariant. Claim. There are not wandering components. The possible constant limits of sequences fλn,μ in the components of the Stable set are p and ∞. Thus the only possible components other than U should be wandering components in which f n → ∞ as n → ∞ by definition of such components. We shall show that no such components exist. Without lost of generality we shall assume that the pole of the family is π , taking k = 1. Suppose that there is a wandering component, G say, such that fλn,μ (G) does not meet D(π , r), 0 < r < α < 1 for any n ∈ N, since fλn,μ (G) → ∞. We must have Im fλn,μ → ∞ in G and hence also fλ ,μ ( fλn,μ ) → ∞ and ( fλn,μ ) → ∞ in G. It follows from Bloch’s theorem that fλn+1 ,μ (G) contains some disc D(a, 4π ), where |Im a|

can be taken arbitrarily large, but then on the horizontal diameter of D(a, π ) it is not difficult to see that fλn+2 ,μ (G) contains some real points, which must be in U . This is impossible, so there is no such domain G.

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Claim. The component U is multiply connected. The Stable set consists of one completely invariant component U which is attracting and multiply-connected since kπ , for k ∈ Z \ {0}, is not in the Stable set. Thus the proof of Theorem 1 is completed. Proof of Corollary 2. Take λ = 1, 0 < μ sufficiently small, α < 1, H = {z : |Imz| < α } and T as in the proof of Theorem 1. It is not difficult to prove that | f μ (z) − sin x| < 1 for any z = x + iy ∈ T . Thus f μ (z) is uniformly bounded in a ball with center 1 and radius 1, say B(1, 1) , such that B(1, 1) ⊂ T . Then it follows as in the proof of Theorem 1 that f μ (z) belongs to a compact subset of T . Thus T belongs to an invariant component U of the Stable set in which f μn (z) → p, where p is an attracting fixed point of f μ (z). We also can prove by following (i) and (ii) in the proof of Theorem 1 that | f μ (z) ± 1| < 2μ 2 + 4μ /π and

√ √ | f μ (z)| < 2 μ + μ .

Thus f μ (z) ∈ T ⊂ U , provided μ was chosen small enough, which means that all the finite critical values of f μ (z) are U . To prove that U is not wandering and the only cyclic component of the Stable set which is completely invariant and multiply connected follows straight forward from the proof of Theorem 1. Proof of Corollary 3. By (i) and (ii) in the proof of Theorem 1 we conclude that all the finite singular values are very close to ±λ and 0, therefore we can take a bounded set C which contains ±λ , 0 and ∞ in the sphere. μ Thus the family fλ ,μ (z) = λ sin z + z−k π , k ∈ Z \ {0}, belongs to the set B given in Definition 1. 3 Cuts of the space of parameters If we study a family with one parameter, λ say, and one critical point of the function, then we can follow the orbit of such point under iteration and get the parameter plane, for instance this is the case of the family gλ (z) = z2 + λ , λ ∈ C. For the family gλ (z) the only critical point is zero, so there is not problem to plot the parameter plane which is the well known Mandelbrot set. μ The family fλ ,μ (z) = λ sin z + z−k π , k ∈ Z \ {0}, has three parameters and the family has more than one critical point, see proof of Theorem 1, thus we cannot plot a plane of parameters but we can plot a cut of the space of parameters. In what follows we shall fix the two parameters μ and kπ in the family fλ ,μ and follow the orbit of a critical point. Fixing the parameters μ = μ0 and kπ = z0 in the family fλ ,μ (z) we have the following expression:

fλ ,μ0 (z) = λ sin z +

μ0 . z − z0

We define a cut of the space of parameters of the family in (1) as follows: M = {λ ∈ R : | fλn,μ0 (critical point)| is bounded}.

(1)

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413

For instance if we take μ0 = 0.5 and z0 = 2π Figure 2 shows the set M for the family fλ ,0.5 (z) = λ sin z +

0.5 . z − 2π

(2)

Fig. 2 A cut of the space of the parameters with μ = 0.5 and the pole in 2π

The color on black in Figure 2 contains the parameters λ ∈ (−1, 0) ∪ (0, 1] of Theorem 1 and Corollary 2. Given a value to the parameter λ in M we can obtain the Stable and Chaotic sets. For example if we take λ = 1 in (2) the Stable set, attracting completely invariant and multiply connected, is on black in Figure 3. The Chaotic set is the boundary of the Stable set and we can see that it is not connected in C.

Fig. 3 The Stable set on black and a hole in 2π

Observation: If we give different values to μ and kπ of the ones given in (2), in the original family fλ ,μ (z) = μ λ sin z + z−k π , k ∈ Z \ {0}, we will have different cuts of the space of parameters and therefore different Stable and Chaotic sets for λ given.

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4 Conclusions The study of the family fλ ,μ (z) given in the abstract of this document depends of three parameters λ , μ and kπ , k ∈ Z \ {0}, so we cannot give general results related to the Stable and Chaotic sets but we can give partial results as Theorem 1 and Corollary 2 related to the dynamics for some cuts of the space of parameters. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

Fatou, P. (1926), Sur L’ it´eration des Fonctions Transcendentes Entier`es, Acta Math., 47, 337–370. Baker, I.N. (1959), Fix Points and Iterates of Entire Functions, Math. Z., 71, 146–153. Baker, I. N. (1975), The Domains of Normality of an Entire Funtion, Ann. Acad. Sci. Fenn. Ser. A. I. Math., 1, 277–283. Devaney, B. (1984), Julia Sets and Bifurcation Diagrams for Exponential Maps, Bulletin of the American Mathematical Society, 11, 167–172. Bhattacharyya, P. (1969), Iteration of Analytic Functions. PhD Thesis, University of London. Dom´ınguez, P. and Sienra, G. (2002), A Study of the Dynamics of the Family λ senz, International Journal of Bifurcation and Chaos, 12, 2869–2883. Baker, I. N., Kotus J. and Yinian L¨u. (1991), Iterates of Meromorphic Functions I, Ergodic Theory Dynamical Systems, 11, 241–248. Baker, I.N., Kotus J. and Yinian L¨u. (1990), Iterates of Meromorphic Functions II: Examples of Wandering Domains, J. London. Math. Soc., 42, 267–278. Baker, I.N., Kotus J. and Yinian L¨u. (1991), Iterates of Meromorphic Functions III: Preperiodic Domains, Ergodic Theory Dynamical Systems, 11 (2), 603–618. Baker, I.N., Kotus J. and Yinian L¨u. (1992), Iterates of Meromorphic Functions IV: Critically Finite Functions, Results in Mathematics, 22, 651–656. Dom´ınguez, P. (1998), Dynamics of Transcendental Meromorphic Functions, Ann. Acad. Sci. Fenn. Ser. A. I. Math., 23, 225–250. Herring, M.E. (1994), An Extension of the Julia-Fatou Theory of Iteration, PhD Thesis, University of London.

Discontinuity, Nonlinearity, and Complexity 5(4) (2016) 415–425

Discontinuity, Nonlinearity, and Complexity https://lhscientificpublishing.com/Journals/DNC-Default.aspx

New Results on Exponential Stability of Fractional Order Nonlinear Dynamic Systems Tianzeng Li1,2†, Yu Wang1,3 , Yong Yang 1 School

of Science, Sichuan University of Science and Engineering, Zigong 643000, China Detecting and Engineering Computing, Zigong 643000, China 3 Artificial Intelligence Key Laboratory of Sichuan Province, Zigong, 643000, China 2 Sichuan Province University Key Laboratory of Bridge Non-destruction

Submission Info Communicated by A.C.J. Luo Received 14 December 2015 Accepted 18 April 2016 Available online 1 January 2017 Keywords Mittag-Leffler function Nonlinear dynamic system Fractional order Fractional comparison principle

Abstract In this letter stability analysis of fractional order nonlinear systems is studied. An extension of Lyapunov direct method for fractional order systems is proposed by using the properties of Mittag-Leffler function and Laplace transform. Some new sufficient conditions which ensure local exponential stability of fractional order nonlinear systems are proposed firstly. And we apply these conditions to the Riemann-Liouville fractional order systems by using fractional comparison principle. Finally, three examples are provided to illustrate the validity of the proposed approach.

©2016 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction In recent years, studies of fractional order nonlinear systems have attracted increasing interests from scientists and engineers. There are two essential differences between fractional order derivation and integer order derivation. Firstly, the fractional order derivative is concerned with the whole time domain for a mechanical or physical process, while the integer order derivative indicates a variation or certain attribute at particular time. Secondly, the fractional order derivative is related to the whole space for a physical process, while the integer order derivative describes the local properties of a certain position. It is the reason that many real world physical systems are well characterized by the fractional order state equations [1-4], such as fractional order Lotka-Volterra equation [1] in biological systems, fractional order Sch¨odinger equation [2] in quantum mechanics, fractional order Langevin equation [3] in anomalous diffusion, fractional order oscillator equation [4] in damping vibration and so on. In particular, stability is one of the most fundamental and important issues for fractional order systems. There are some works about stability of fractional order systems in recent years. The necessary and sufficient † Corresponding

author. Email address:[email protected] ISSN 2164 − 6376, eISSN 2164 − 6414/$-see front materials © 2016 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/DNC.2016.12.007

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stability conditions for linear fractional order differential equations and linear time-delayed fractional differential equations have already been obtained in [5-7]. In [8] the authors investigated the stability of n-dimensional linear fractional order differential systems with order 1 < α < 2. However, only under some special circumstances or in certain cases, the practical problems may be regarded as linear systems. Therefore, stability of nonlinear dynamics not only is of great significance, but also has important values in application. In [9], the stability of fractional nonlinear time-delay systems for Caputo’s derivative is investigated, and two theorems for Mittag-Leffler stability of the fractional order nonlinear time-delay systems are proved. The finite-time stabilization of a class of multi-state time delay of fractional nonlinear systems was proposed in [10]. In [11,12], the authors studied the stability of fractional nonlinear dynamic systems using Lyapunov direct method with the introductions of Mittag-Leffler stability and generalized Mittag-Leffler stability notions. In [13], some new sufficient conditions ensuring asymptotical stability of fractional-order nonlinear system with delay are proposed firstly. In this paper, by the properties of Mittag-Leffler function, Laplace transform, and some fractional order inequalities, some new sufficient conditions for the local (global) exponential stability of fractional nonlinear systems with order 0 < α < 1 are proposed firstly. For extending the application of fractional calculus in nonlinear system, we introduce the fractional comparison principle and some properties of Mittag-Leffler function. Then the application of Riemann-Liouville fractional order systems is extended by using fractional comparison principle and Caputo fractional order systems. Finally, three examples are provided to illustrate the proposed approach. This paper is organized as follows: In Sect. 2 the preliminaries are presented. Main results are discussed in Sect. 3. In the Sect. 4, three examples are used to illustrate the validity and feasibility of the proposed method. Finally, conclusions are in Sect. 5. 2 Fractional order derivatives and Mittag-Leffler functions 2.1

Definition of fractional derivatives and Mittag-Leffler functions

Fractional calculus plays an important role in modern science [14-18]. There are some definitions for fractional derivatives. In this paper, we give three commonly used definitions [16]: Gr¨unwald-Letnikov(GL), RiemannLiouville(RL), and Caputo definition. Definition 1. ([14,16]) The fractional integral a Dt−α of function f (t) is defined as follow: ˆ t 1 −α D f (t) = (t − τ )α −1 f (τ )d τ , a t Γ(α ) a ´∞ where fractional order α > 0 and Γ(z) = 0 t z−1 e−t dt is the gamma function. Definition 2. ([14,16]) The Riemann-Liouville derivative with order α of function f (t) is defined as ˆ dn t d n −(n−α ) 1 RL α D f (t) = D f (t) = (t − τ )n−α −1 f (τ )d τ , a t a dt n t Γ(n − α ) dt n a

(1)

(2)

where n − 1 < α < n, n ∈ Z + . Also, there are other definitions of fractional derivative introduced by Caputo and Gr¨unwald-Letnikov. Definition 3. ([14,16]) The Caputo derivative with order α of function f (t) is given as ˆ t n 1 −(n−α ) d C α f (t) = (t − τ )n−α −1 f (n) (τ )d τ , a Dt f (t) = a Dt dt n Γ(n − α ) a where n − 1 < α < n, n ∈ Z + .

(3)

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417

α The formulas for Laplace transform of Riemann-Liouville derivative RL a Dt f (t) and Caputo fractional derivaC α tive a Dt f (t) have the following forms [16]: n−1

α α k RL α −k−1 f (t)]t=0 , L{RL a Dt f (t); s} = s F(s) − ∑ s [a Dt

(4)

k=0

and n−1

L{Ca Dtα f (t); s} = sα F(s) − ∑ sα −k−1 f (k) (0),

(5)

k=0

where n − 1 ≤ α < n and F(s) = L{ f (t); s} =

´∞ 0

e−st f (t)dt.

Definition 4. ([14,16]) The Gr¨unwald-Letnikov derivative with order α of function f (t) is defined as GL α a Dt

−α

f (t) = lim h =

  α ∑ (−1) r f (t − rh) r=0 n

r

h→0 nh=t−a m f (k) (a)(t − a)−α +k



k=0

+

Γ(−α + k + 1)

1 · Γ(m − α + 1)

ˆ a

t

(t − τ )n−α f (m+1) (τ )d τ ,

(6)

where m < α < m + 1. Remark 1. Throughout studying some papers, we obtain the following conclusions. Gr¨unwald-Letnikov definition is suitable for numerical calculations, Riemann-Liouville definition plays an in important in theory analysis, and Caputo definition is well used since its Laplace transform allows for initial conditions taking the same forms as those for integer order derivatives, which have clear physical interpretations and have a wide range of application in the process of factual modeling. More differences of the three definitions in theory and application will be found in [16,17,18]. As a generalization of the exponential function which is frequently used in the solutions of integer-order systems, the Mittag-Leffler function is frequently used in the solutions of fractional systems. The definitions and properties are given in the following. Definition 5. ([16]) The Mittag-Leffler function is given as Eα (z) =



zk

∑ Γ(kα + 1) ,

(7)

k=0

where α > 0 and z ∈ C. The generalization of Mittag-Leffler function with two parameters is wildly used and defined as follows[16]: Eα ,β (z) =



zk

∑ Γ(kα + β ) ,

k=0

where α > 0, β > 0 and z ∈ C. Remark 2. If β = 1, we have Eα ,1 (z) = Eα (z), especially, E1,1 (z) = E1 (z) = ez .

(8)

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2.2

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Properties of fractional derivatives and Mittag-Leffler functions

In this section, we give some important properties of the fractional derivatives and the Mittag-Leffler functions which will be used in the following. Lemma 1. Let α ∈ (0, 1) and f (0) ≥ 0, then C α 0 Dt

α f (t) ≤ RL 0 Dt f (t),

(9)

α where C0 Dtα and RL 0 Dt are, respectively, the Caputo and Riemann-Liouville fractional derivatives. −α

f (0)t α C α Proof. By using the definitions of fractional derivatives, we have RL 0 Dt f (t) = 0 Dt f (t)+ Γ(1−α ) . Since α ∈ (0, 1) α f (0) ≥ 0, we have the conclusion C0 Dtα f (t) ≤ RL 0 Dt f (t).

Lemma 2. If C0 Dtα x(t) ≥ 0 and x(0) ≥ 0, 0 < α < 1, then x(t) ≥ 0. Proof. We suppose that C0 Dtα x(t) = f (t, x) ≥ 0. Using the equivalent Volterra integral equation [16], we have ˆ t 1 (t − τ )α −1 f (τ , x(τ ))d τ . (10) x(t) = x(0) + Γ(α ) 0 Since t − τ ≥ 0, Γ(α ) > 0 and f (t, x) > 0, we can get x(t) ≥ x(0) ≥ 0, i.e. x(t) ≥ 0. Theorem 3. (Comparison Theorem) Let 0 < α < 1 and x(0) = y(0), then we have x(t) ≥ y(t), if C0 Dtα x(t) ≥ C Dα y(t). 0 t Proof. The fractional differentiation and fractional integration are linear operations, then C0 Dtα (x(t) − y(t)) ≥ 0. By the Lemma 2 we can easily get x(t) − y(t) ≥ 0, i.e. x(t) ≥ y(t). Lemma 4. ([14,19]) Considering the Laplace transform of Mittag-Leffler function with two parameters, we have L{t β −1 Eα ,β (−λ t α )} =

1 sα −β , (R(s) > |λ | α ), α s +λ

(11)

where t and s are, respectively, the variables in the time domain and Laplace domain, R(s) stands for the real part of s, λ ∈ R and L{·} denotes the Laplace transform. Proof. By the definitions of Laplace transform and Mittag-Leffler function, we obtain ˆ ∞ ˆ ∞ ∞ ∞ (−1)k λ k t α k (−1)k λ k β −1 α −st β −1 Eα ,β (−λ t )} = e t L{t ∑ Γ(kα + β ) dt = ∑ Γ(kα + β ) 0 e−st t α k+β −1dt 0 k=0 k=0 =



∞ (−1)k λ k Γ(α k + β ) λ = ∑ Γ(kα + β ) sα k+β ∑ (−1)k s−β ( sα )k k=0 k=0

=

sα −β , sα + λ

(12)

1

where R(s) > |λ | α . Lemma 5. ([20,21]) For the Mittage-Leffler function Eα (At α ), there exists finite real constant KEα ≥ 1 such that for any 0 < α < 1, Eα (At α ) ≤ KEα eAt , where A ∈ R.

(13)

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419

Proof. The proof of this Lemma can be found in [16]. Lemma 6. ([21,22]) Let α > 0, u(t) is a nonnegative function locally integrable on [0, T ) and a(t) is a nonnegative, nondecreasing continuous function defined on [0, T ), a(t) < M (constant), and suppose z(t) is nonnegative and locally integrable on [0, T ) with ˆ z(t) ≤ u(t) + a(t)

t 0

(t − τ )α −1z(τ )d τ ,

(14)

on this interval. Then ˆ z(t) ≤ u(t) +

t 0



(Γ(α )a(t))k (t − τ )kα −1 u(τ )]d τ . Γ(k α ) k=1

[∑

(15)

Moreover, if u(t) is a nondecreasing function on [0, T ), we have z(t) ≤ u(t)Eα (Γ(α )a(t)t α ).

(16)

3 Fractional order extension of Lyapunov direct method 3.1

Fractional order nonlinear systems

Firstly, we consider the Caputo fractional nonlinear systems [16,22] C α a Dt x(t)

= f (t, x(t))

(17)

with the initial condition x0 = x(a), where α ∈ (0, 1), f : [a, ∞) × Ω → Rn is piecewise continuous in t and satisfies the local Lipschitz condition with respect to x, and Ω ∈ Rn is a domain which contains the origin x = 0. If f (t, x∗ ) = 0, the constant x∗ is called the equilibrium point of Caputo fractional nonlinear system (17). Without loss generality, we suppose that the equilibrium point be x = 0. In fact the real-valued function f (t, x) in system (17) is locally bounded and satisfies the local Lipschitz condition with respect to x, which implies the uniqueness and existence of the solution to the fractional order system (17)[16]. In the following, we give the relationship between the Lipschitz condition and fractional nonlinear system. Lemma 7. Considering the real-valued continuous f (t, x) in system (17), we obtain ||a Dt−α f (t, x(t))|| ≤ a Dt−α || f (t, x(t))||

(18)

where α > 0 and || · || denotes an arbitrary norm. Proof. It follows the definition of fractional integral (1) that ˆ t 1 f (τ , x(τ )) d τ || ||a Dt−α f (t, x(t))|| = || Γ(α ) a (t − τ )1−α ˆ t f (τ , x(τ )) 1 || ||d τ ≤ Γ(α ) a (t − τ )1−α ˆ t 1 || f (τ , x(τ ))|| = dτ Γ(α ) a (t − τ )1−α = a Dt−α || f (t, x(t))||.

(19)

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Theorem 8. For the fractional nonlinear system (17) with a = 0, let x = 0 be the equilibrium point. And f (t, x) is piecewise continuous with respect to t and satisfies the Lipschitz condition on x with Lipschitz constant L, then the solution of system satisfies ||x(t)|| ≤ ||x(0)||KEα eLt , where KEα ≥ 1 is a real constant which is obtained by Lemma 5. Proof. Applying the fractional integral operator 0 Dt−α to both sides of (17), one can have x(t) − x(0) = 0 Dt−α f (t, x(t)).

(20)

It follows the Lipschitz condition and Lemma 7 that ||x(t)|| − ||x(0)|| ≤ ||x(t) − x(0)|| = ||0 Dt−α f (t, x(t))|| ≤ 0 Dt−α || f (t, x(t))|| ≤ L0 Dt−α ||x(t)||.

(21)

From (21), we easily get ||x(t)||

≤ ||x(0)|| + L0 Dt−α ||x(t)||

L = ||x(0)|| + Γ(α )

ˆ

t 0

(t − τ )α −1||x(τ )||d τ .

(22)

By the Lemma 5 and Lemma 6, there exists a constant KEα ≥ 1 such that ||x(t)|| ≤ ||x(0)||Eα (Lt α ) ≤ ||x(0)||KEα eLt .

(23)

Finally, the solution of system satisfies ||x(t)|| ≤ ||x(0)||KEα eLt . 2 3.2

Fractional order extension of Lyapunov direct method

It is well known that Lyapunov stability provides an important tool for stability analysis in nonlinear systems. We primarily study the Lyapunov direct method which involves finding a Lyapunov function for a given nonlinear system. If there exists such a function, the system is stable. Applying Lyapunov direct method is to search for an appropriate function. However, Lyapunov direct method is a sufficient condition. It means that if one cannot find a Lyapunov function, the system may still be stable and one cannot claim that the system is not stable. In the following, we extend the Lyapunov direct method to the fractional order nonlinear system and give some sufficient conditions of stability for fractional order systems. Theorem 9. Let x = 0 be an equilibrium point for fractional nonlinear system (17), and Ω ∈ Rn is a domain containing the origin. And V (t, x(t)) : [0, ∞) × Ω → R is a continuously differentiable function and satisfies locally Lipschitz with respect to x such that

α1 ||x(t)||a ≤ V (t, x(t)) ≤ α2 ||x(t)||ab ,

(24)

C β 0 Dt V (t, x(t))

(25)

≤ −α3 ||x(t)||ab ,

where x ∈ Ω,t ≥ 0, 0 < β < 1, αi (i = 1, 2, 3), a and b are arbitrary positive constants. Then x = 0 is locally exponentially stable. If the assumptions hold globally on Rn , then x = 0 is globally exponentially stable. β

Proof. By the equations (24) and (25), one can easily get C0 Dt V (t, x(t)) ≤ −α3 α2−1V (t, x(t)). So there is a nonnegative function W (t) satisfying C β 0 Dt V (t, x(t)) +W (t)

= −α3 α2−1V (t, x(t)).

(26)

sβ V (s) −V (0)sβ −1 +W (s) = −α3 α2−1V (s),

(27)

Taking the Laplace transform of equation (26), we have

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421

where nonnegative constant V (0) = V (0, x(0)) and V (s) = L{V (t, x(t)); s}. Then V (s) =

V (0)sβ −1 −W (s) . sβ + αα32

(28)

If initial condition has x(0) = 0, namely, V (0) = 0, the solution of (17) is x = 0. If initial condition has x(0) = 0, namely, V (0) > 0. Since V (t, x(t)) > 0 is locally Lipschitz with respect to x, it follows from the fractional uniqueness and existence theorem [16] and the inverse Laplace transform that the unique solution of (26) is V (t, x) = V (0)Eβ (−

α3 β α3 t ) −W (t) ∗ [t β −1 Eβ ,β (− t β )]. α2 α2

(29)

Because both W (t) and t β −1 Eβ ,β (− αα32 t β ) are nonnegative functions, we have V (t, x) ≤ V (0)Eβ (− αα32 t β ). And it follows Lemma 5 that there exist a constant KEβ ≥ 1 such that α

− α3 t

V (t, x) ≤ V (0)KEβ e

2

.

(30)

Then we substitute equation (30) into (24), ||x(t)|| ≤ ( Let M = (

V (0)KEβ

α1

V (0)KEβ

α1

1

α

− aα3 t

)a e

2

.

(31)

1

) a ≥ 0, then α

− aα3 t

||x(t)|| ≤ Me

2

,

(32)

where M = 0 holds if and only if x(0) = 0. Hence we can obtain that x = 0 is locally exponentially stable. If the assumptions hold globally on Rn , then x = 0 is globally exponentially stable. 2 Theorem 10. Let f (t, x) satisfy the Lischitz condition with Lipschitz constant L for fractional nonlinear system (17). And V (t, x(t)) : [0, ∞) × Ω → R is a continuously differentiable function and satisfies the local Lipschitz condition with respect to x such that

α1 ||x(t)||a ≤ V (t, x(t)) ≤ α2 ||x(t)||, dV (t, x(t)) ≤ −α3 ||x(t)||, dt where x ∈ Ω,t ≥ 0, αi (i = 1, 2, 3) and a are arbitrary positive constants. Then ||x|| ≤ ( x = 0 is locally exponentially stable. Proof. It follows the properties of Caputo derivative and Lemma 6 that C 1−α V (t, x(t)) 0 Dt

dV (t, x(t)) dt ≤ −α30 Dt−α ||x(t)|| α3 ≤ − 0 Dt−α || f (t, x(t))|| L α3 ≤ − ||0 Dt−α f (t, x(t))|| L α3 ≤ − ||x(t)||, L = 0 Dt−α

(33) (34) V (0)KE1−α α1

1

α

− aLα3 t

)a e

2

, i.e.

422

Tianzeng Li, Yu Wang, Yong Yang / Discontinuity, Nonlinearity, and Complexity 5(4) (2016) 415–425 −(1−α )

where [0 Dt x(t)]t=0 = 0. Let β = 1 − α , α3 = αL3 , and b = a−1 , then the conclusion can be easily obtained by the Theorem 9. 2 In the following, we consider the Riemann-Liouvile fractional dynamic system as follows: RL α a Dt x(t)

= f (t, x(t))

(35)

with the initial condition x0 = x(a), where α ∈ (0, 1), f is piecewise continuous in t and locally Lipschitz in x. α ∗ ∗ ∗ If RL a Dt x = f (t, x ), the constant x is called the equilibrium point of Riemann-Liouvile fractional nonlinear system (35). Without loss generality, we suppose the equilibrium point be x = 0. If the equilibrium point is x∗ = 0, we consider the change of variable y(t) = x(t) − x∗ . Then the α th order derivative of y is given by RL α a Dt y(t)

α ∗ = RL a Dt (x(t) − x ) = f (t, x(t)) −

x∗ t−α = g(t, y(t)), Γ(1 − α )

where g(t, 0) = 0, and the system has equilibrium at the origin about the new variable y. Theorem 11. Let the assumptions in Theorem 9 be satisfied except replacing C0 Dtα by same conclusion ||x(t)|| ≤ (

V (0)KEβ

α1

1 a

α − aα3 t 2

) e

RL Dα , t 0

then we have the

, i.e. x = 0 is locally exponentially stable.

α C α Proof. It follows from V (t, x(t)) ≥ 0 and Lemma 1 that C0 Dtα V (t, x(t)) ≤ RL 0 Dt V (t, x(t)). Then 0 Dt V (t, x(t)) ≤ RL Dα V (t, x(t)) ≤ −α ||x(t)||. Therefore the conclusion can be obtained by the Theorem 9. 2 3 t 0

4 Three illustrative examples In this section three illustrative examples are used as proofs of the concept. Example 1. For a fractional order system RL α 0 Dt (x(t)sgn(x(t)))

= −x(t)sgn(x(t)),

(36)

where 0 < α < 1 and sgn(·) is the sign function. Choose the Lypunov function V (t, x(t)) = xsgn(x), then = −x(t)sgn(x(t)) ≤ 0. When selecting α1 = α2 = α3 = 1 and a = b = 1, it follows Theorem 11 that ||x(t)|| ≤ Me−t , i.e. x = 0 is locally exponentially stable. The numerical simulation of the fractional differential equations (36) is shown in Fig. 1, which demonstrates the efficiency and applicability of the proposed approach. RL Dα V (t, x(t)) t 0

However, if we apply the Laplace transform directly to properties of Riemann-Liouville definition [16] that

RL Dα (x(t)sgn(x(t))) t 0

= −x(t)sgn(x(t)), it follows

sα L{x(t)sgn(x(t))} − [0 Dtα −1 (x(t)sgn(x(t)))]t=0 = −L{x(t)sgn(x(t))}. Using the inverse of Laplace transform on above equation, we can get x(t)sgn(x(t)) = [0 Dtα −1 (x(t)sgn(x(t)))]t=0 Eα (−t α ). However [0 Dtα −1 (x(t)sgn(x(t)))]t=0 = 0 for any finite x(0), which implies that the stability of system (36) cannot be derive directly from solving equation (36). Example 2. We consider the fractional order autonomous system as follows: C α 0 Dt (x(t))

= f (x(t)),

(37)

f (x) x˙ ≤ −α1 ||x(t)||(α1 ≥ 0), and where 0 < α < 1. Let x = 0 be the equilibrium point of system (37), f (x) d dx ||x(t)||2 ≤ L|| f (x)||2 (L > 0), then the equilibrium x = 0 is stable.

Tianzeng Li, Yu Wang, Yong Yang / Discontinuity, Nonlinearity, and Complexity 5(4) (2016) 415–425

423

1 α=0.6 α=0.7 α=0.8 α=0.9

0.9 0.8 0.7

x(t)

0.6 0.5 0.4 0.3 0.2 0.1 0

0

10

20

30

40

50

time Fig. 1 Time waveforms of numerical solutions x(t) of the fractional order system (36) with α = 0.6, 0.7, 0.8, 0.9, respectively. (x) dV Proof. Choosing the Lypunov function V (t, x) = 12 f 2 (x) ≥ 0, we have dV ˙ = f (x) fdx x(t) ˙ ≤ −α1 ||x(t)||. dt = dx x(t) 2 2 2 2 2 It follows ||x(t)||2 ≤ L|| f (x)||2 that ||x(t)||2 ≤ L || f (x)||2 ≤ L V (t, x) ≤ L V (0, x(0)). Hence the conclusion is obtained, i.e. the equilibrium point x = 0 is stable. 2

Example 3. We consider the fractional order system as follows: C α 0 Dt (x(t))

= f (t, x(t)),

(38)

where 0 < α < 1. Let x = 0 be the equilibrium point of system (38) and f (t, x(t)) satisfies Lipschitz condition with Lipschitz constant L > 0. Assume that there is a Lypunov function V (t, x(t)) satisfying

α1 ||x(t)|| ≤ V (t, x(t)) ≤ α2 ||x(t)||, dV (t, x(t)) ≤ −α3 ||x(t)||, dt

(39) (40)

where αi > 0(i = 1, 2, 3). Then ||x(t)|| ≤ where KE1−α is obtained by Lemma 5.

V (0)KE1−α − Lαα3 t e 2 , α1

(41)

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Proof. By the properties of Caputo derivative and Lemma 7, we get C 1−α V (t, x(t)) 0 Dt

−(1−α )

where [0 Dt

= 0 Dt−α V˙ (t, x(t))

≤ −α30 Dt−α ||x(t)|| α3 ≤ − 0 Dt−α || f (t, x(t))|| L α3 ≤ − ||0 Dt−α f (t, x(t))|| L α3 ≤ − ||x(t)||, L

x(t)]t=0 = 0. Then, by the Theorem 10 we can obtain ||x(t)|| ≤

V (0)KE1−α − αα3 t α3 1−α V (0) E1−α (− t )≤ e 2 . α1 Lα2 α1

Therefore, the conclusion is obtained. 2 Remark 3. In this example, the conditions, which the Lypunov function satisfies, are the same as classical version of stability theorem for integer order derivative. 5 Conclusion Stability of the nonlinear dynamical systems is important for scientists and engineers. Fractional dynamic systems were used intensively during the last decade in order to describe the behaviors of complex systems in physical and engineering. In this manuscript the stability of nonlinear fractional order dynamic system is studied. We discussed the properties of the Caputo and Riemann-Liouville derivatives and proposed the comparison theorem. And by using the properties of Mittag-Leffler function and Laplace transform, we proposed the extending Lyapunov direct method which is the sufficient condition of stability for fractional order dynamic systems. This enriches the knowledge of both the system theory and the fractional calculus. We partly extended the application of Riemann-Liouville fractional order systems by using fractional comparison theorem and Caputo fractional order systems. Finally, Three illustrative examples were proposed to demonstrate the applicability of the proposed approach. Acknowledgements The work is supported by Found of Science & Technology Department of Sichuan Province (Grant No.2016JQ0046), Artificial Intelligence Key Laboratory of Sichuan Province (Grant No.2016RYJ06), Found of Sichuan University of Science and Engineering (Grant 2014PY06, 2015RC10), the Opening Project of Sichuan Province University Key Laborstory of Bridge Non-destruction Detecting and Engineering Computing (Grant No.2015QYJ02, 2014QZJ03), Opening Project of Key Laboratory of Higher Education of Sichuan Province for Enterprise Informationalization and Internet of Things (Grant No.2016WYJ04). References [1] Das, S. and Gupta, P. (2011), A mathematical model on fractional Lptka-Volterra equations, Joural of Theoretical Biology, 277, 1-6. [2] Naber, M. (2004), Time fractional Schr¨odinger equation, Journal of Mathematical Physics, 45, 3339–3352. [3] Burov, S. and Barkai, E. (2008), Fractional Langevin equation: overdammped, underdamped, and cirtical behaviors, Physical Review E, 78, 031112.

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[4] Ryabov, Y. and Puzenko, A. (2002), Damped oscillation in view of the fractional oscillator equation, Physical Review B, 66, 184–201. [5] Bonnet, C. and Partington, J.R. (2000), Coprime factorizations and stability fo fractional defferential systems, System & Control Letters 41, 167–174. [6] Deng, W.H., Li, C.P. and L¨u, J.H. (2007), Stability analysis of linear fractional dirrerential system with multiple timedelays, Nolinear Dynamics, 48, 409–416. [7] Kheirizad, I., Tavazoei, M.s. and Jalali, A.A. (2010), Stability criteria for a class of fractional order systems, Nonlinear Dynamics, 61, 153–161. [8] Zhang, F. and Li, C.P. (2011), Stability analysis of fractional differential systems with order lying in (1,2), Advances in Difference Equations, 2011, 213485. [9] Sadati, S.J., Baleanu, D., Ranjbar, A., Ghaderi, R. and Abdeljawad, T. (2010), Mittag-Leffler stability theorem for fractional nonlinear systems with delay, Abstract and Applied Analysis 2010, 108651. [10] Liu, L. and Zhong, S. (2011), Finite-time stability analysis of fractional-order with multi-state time delay, Word Academy of Science, Eniineering and Technology, 76, 874–877. [11] Li, Y., Chen, Y.Q. and Podlubny, I. (2009), Mittag-Leffler stability of fractional order nonlinear dynamic systems. Automatica 45, 1965–1969. [12] Li, Y., Chen, Y.Q. and Podlubny, I. (2010), Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability, Computers & Mathematics with Application, 59, 1810–1821. [13] Wang, Y. and Li, T.Z. (2014), Stability analysis of fractional-order nonlinera systems with delay. Mathematical Problems in Engineering 2014, 301235. [14] Chen, Y.Q. and Moore, K.L. (2002), Analytical stability bound for a class of delayed fractional order dynamic systems, Nonlenear Dynamics. 29, 191–200. [15] Li, T.Z., Wang, Y., Yang, Y. (2014), Designing synchronization schemes for fractional-order chaotic system via a single state fractional-order controller, Optik 125, 6700–6705. [16] Podlubny, I. (1999), Fractional Differential Equations, Academic Press, San Diego. [17] Li, T.Z., Wang, Y. and Luo, M.K. (2014), Control of fractional chaotic and hyperchaotic systems based on a fractional order controller, Chinese Physics B, 23, 080501. [18] Li, T.Z., Wang, Y. and Yang, Y. (2014), Synchronization of fractional-order hyperchaotic systems via fractional-order controllers, Discrete Dynamics in Nature and Society, 2014, 408972. [19] Sabatier, J., Agrawal, Q.P. and Machado, T.J.A. (2007), Advances in fractional calculus-theoretical developments and applications in physics and engineering, Springer. [20] De la Sen, M. (2011), About robust stability of Caputo linear fractional dynamic systems with time delays through fixed point theory, Fixed Point Theory and Applications, 2011, 867932. [21] Chen, L.P., He, Y.G., Chai, Y. and Wu, R.C. (2014), New results on stability and stabilization of a calss of nonlinear fractional-order systems, Nonlinear Dynamics., 75, 633–641. [22] Ye, H., Gao, J. and Ding, Y. (2007), A generalized Gronwall inequality and its application to a fractional differential equation, Journal of Mathematical Analysis and Applications, 328, 1075–1081.

Discontinuity, Nonlinearity, and Complexity 5(4) (2016) 427–446

Discontinuity, Nonlinearity, and Complexity https://lhscientificpublishing.com/Journals/DNC-Default.aspx

Robust Exponential Stability of Impulsive Stochastic Neural Networks with Markovian Switching and Mixed Time-varying Delays Haoru Li1 , Yang Fang2 , Kelin Li2† 1 School

of Automation and Electronic Information, Sichuan University of Science & Engineering, Sichuan 643000, P.R. China 2 School of Science, Sichuan University of Science & Engineering, Sichuan 643000, P.R. China Submission Info Communicated by A.C.J. Luo Received 21 January 2016 Accepted 26 April 2016 Available online 1 January 2017 Keywords Robust exponential stability impulsive stochastic neural networks Markovian switching Lyapunov-Krasovskii functional

Abstract This paper is concerned with the robust exponential stability problem for a class of impulsive stochastic neural networks with Markovian switching, mixed time-varying delays and parametric uncertainties. By construct a novel Lyapunov-Krasovskii functional, and using linear matrix inequality (LMI) technique, Jensen integral inequality and free-weight matrix method, several novel sufficient conditions in the form of LMIs are derived to ensure the robust exponential stability in mean square of the trivial solution of the considered system. The results obtained in this paper improve many known results, since the parametric uncertainties have been taken into account, and the derivatives of discrete and distributed time-varying delays need not to be 0 or smaller than 1. Finally, three illustrative examples are given to show the effectiveness of the proposed method. ©2016 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction In the past two decades, the stability analysis of neural networks has played an important role in modern cybernetic field, since most of the successfully applications of neural networks significantly depend on the stability of the equilibrium point of neural networks. Many papers related to this problem have been published in the literature, see [1] for a survey. During implementation of artificial neural networks, time-varying delays are unavoidable due to finite switching speeds of the amplifiers, as well as the neural signal propagation is often distributed in a certain time period with the presence of an amount of parallel pathways with a variety of axon sizes and lengths. Therefore, it is necessary to put mixed time-varying delays into the models. There are many works focusing on the mixed time-varying delays [2–8]. It is well known that the other three sources which may causing an instability and poor performances in neural networks are stochastic perturbation, impulsive perturbations and parametric uncertainties. Most of this viewpoint is attributable to the following three reasons: 1. A neural network can be stabilized or destabilized by † Corresponding

author. Email address: [email protected]

ISSN 2164 − 6376, eISSN 2164 − 6414/$-see front materials © 2016 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/DNC.2016.12.008

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certain stochastic inputs [9–11]; 2. In the real world, many evolutionary processes are characterized by abrupt changes at time. These changes are called impulsive phenomena, which have been found in various fields, such as physics, optimal control and biological mathematics, etc [12]; 3. The effects of parametric uncertainties cannot be ignored in many applications [13–15]. Hence, stochastic perturbation, impulsive perturbations and parametric uncertainties also should be taken into consideration when dealing with the stability issue of neural networks. On the other hand, Markovian jumping systems [16] can be seen as a special class of hybrid systems with two different states, which involve both time-evolving and event-driven mechanisms. So such systems would be used to model the abrupt phenomena such as random failures and repairs of the components, changes in the interconnections of subsystems, sudden environment changes, etc. Thus, many relevant analysis and synthesis results for Markovian jumping systems have been reported, see [17–26] and the references therein. Recently, various interesting works on impulsive stochastic neural networks with Markovian switching and mixed time delays have received considerable research attention, see [27–40] and the references therein. Rakkiyappan and Balasubramaniam [30] utilized the Lyapunov-Krasovskii functional and stochastic stability theory, delay-interval dependent stability criteria are obtained in terms of linear matrix inequalities for Markovian jumping impulsive stochastic Cohen-Grossberg neural networks with discrete interval and distributed delays. Zhang et al. [32] employed the Lyapunov-Krasovskii functional approach and linear matrix inequality (LMI) technique, delay-dependent sufficient condition for the stability problem of neutral-type impulsive neural networks with mixed time-varying delays and Markovian jumping is obtained. By using the concept of the minimum impulsive interval, Bao & Cao [37], Gao el al. [36] derived some sufficient conditions to ensure exponential stability for neutral-type delayed neural networks with impulsive perturbations and Markovian switching. In [38], Raja et al. employed a Lyapunov functional approach for the stability of a class of impulsive Hopfield neural networks with Markovian jumping parameters and time-varying delays, new delay-dependent stochastic stability criteria are obtained in terms of linear matrix inequalities. However, in [32, 36–38], the authors ignored parametric uncertainties. And in [28–32, 34], the derivatives of time-varying delays need to be zero or smaller than one. So far, there are few results on the study of robust exponential stability of impulsive stochastic neural networks with Markovian switching, mixed time-varying delays and parametric uncertainties. Motivated by above discussion, this paper investigates the robust exponential stability in mean square of impulsive stochastic neural networks with Markovian switching, mixed time-varying delays and parametric uncertainties. By employing Lyapunov-Krasovskii functional, linear matrix inequality (LMI) technique, Jensen integral inequality and free-weight matrix method, several novel sufficient conditions in terms of linear matrix inequalities (LMIs) are derived to guarantee the robust exponential stability in mean square of the trivial solution of the considered model. The proposed method in this paper is improve those given in [28–32, 34, 36–38], since the parametric uncertainties has been taken into account, and the restriction of the derivatives of discrete and distributed time-varying delays need to be 0 or smaller than 1 in [28–32, 34] is removed. The organization of this paper is as follows. In Section 2, the exponential stability problem of impulsive stochastic neural networks with Markovian switching, mixed time-varying delays and parametric uncertainties is described and some necessary definitions and lemmas are given. Some new robust exponential stability criteria are obtained in Section 3. In Section 4, three illustrative examples are given to show the effectiveness of the proposed method. Finally, conclusions are given in Section 5. Notation: Let R denotes the set of real numbers, R+ denotes the set of all nonnegative real numbers, Rn and Rn×m denote the n-dimensional and n × m dimensional real spaces equipped with the Euclidean norm,  ·  refers to the Euclidean vector norm and the induced matrix norm. N+ denotes the set of positive integers. For any matrix X ∈ Rn×n , X > 0 denotes that X is a symmetric and positive definite matrix. If X1 , X2 are symmetric matrices, then X1 ≤ X2 means that X1 − X2 is a negative semi-definite matrix. X T and X −1 mean the transpose of X and the inverse of a square matrix. I denotes the identity matrix with appropriate dimensions. Let τ > 0 and C([−τ , 0]; Rn ) denote the family of all continuous Rn -valued functions ξ (θ ) on [−τ , 0] with the norm |ξ | = sup ξ (θ ). Let ω (t) = [ω1 (t), ω2 (t), · · · , ωn (t)]T be an n-dimensional Brownian motion defined −τ ≤θ ≤0

Haoru Li, Yang Fang, Kelin Li / Discontinuity, Nonlinearity, and Complexity 5(4) (2016) 427–446

429

on a complete probability space (Ω, F , P) with a natural filtration {Ft }t≥0 (i.e., Ft = σ {ω (s) : 0 ≤ s ≤ t}), p ([−τ , 0]; Rn )(t ≥ 0) denote the family of all Ft measurable which satisfies Ed ω (t) = 0 and E[d ω (t)]2 = dt. LF t ´0 bounded C([−τ , 0]; Rn )-valued random variables ξ = {ξ (θ ) : −τ ≤ θ ≤ 0} such that −τ E|ξ (s)| p ds < ∞, where E{·} stands for the correspondent expectation operator with respect to the given probability measure P. The notation  always denotes the symmetric block in one symmetric matrix. Matrix dimensions, if not explicitly stated, are assumed to be compatible for operations. 2 Model Description and Preliminaries Let {r(t),t ≥ 0} be a right continuous Markov chain in a complete probability space (Ω, F , P) taking values in a finite state space S = {1, 2, · · · , N} with generator Π = (πi j )N×N given by  πi j Δt + o(Δt), if i = j P{r(t + Δt) = j|r(t) = i} = 1 + πii Δt + o(Δt), if i = j where Δt > 0 and limΔt→0 (o(Δt/Δt) = 0. Here πi j ≥ 0(i = j) is the transition rate from mode i to mode j while πii = − ∑ j=i πi j is the transition rate from mode i to mode i. Considering a class of impulsive stochastic neural networks with Markovian jumping parameters, mixed time-varying delays and parametric uncertainties, which can be presented by the following impulsive integrodifferential equation: ⎧ dy(t) = [−C(r(t))y(t) + (A(r(t)) + ΔA(r(t)))g(y(t)) ⎪ ⎪ ⎪ ⎪ +(B(r(t)) + ΔB(r(t)))g(y(t ⎨ ´ t − τ1 (t))) +(D(r(t)) + ΔD(r(t))) t−τ2 (t) g(y(s))ds]dt (1) ⎪ ⎪ ⎪ σ (t, r(t), y(t), y(t − τ (t)), y(t − τ (t)))d ω (t), t =  t , + 1 2 k ⎪ ⎩ + t = tk , k ∈ N+ , y(tk ) = Wk (r(t))y(tk− ), for t > 0, where y(t) = (y1 (t), y2 (t), . . . , yn (t))T ∈ Rn is the state vector associated with n neurons at time t. In the continuous part of system (1), C(r(t)) = diag{c1 (r(t)), c2 (r(t)), · · · , cn (r(t))} is a diagonal matrix with positive entries ci (r(t)) > 0(i = 1, 2, · · · , n); the matrices A(r(t)) = (ai j (r(t)))n×n and B(r(t)) = (bi j (r(t)))n×n are the connection weight matrix, the discrete time-varying delay connection weight matrix and the distributeddelay connection weight matrix, respectively; ΔA(r(t)), ΔB(r(t)) and ΔD(r(t)) are the time-varying parametric uncertainties; g(y(t)) = (g1 (y1 (t)), g2 (y2 (t)), · · · , gn (yn (t)))T ∈ Rn is the nonlinear neuron activation function which describes the behavior in which the neurons respond to each other; τ1 (t) and τ2 (t) are namely the discrete and distributed time-varying delay; the noise perturbation (or the diffusion coefficient) σ (t, r(t), y(t), y(t − τ1 (t)), y(t − τ2 (t))) : R+ × S × Rn × Rn → Rn×n is a Borel measurable function. In the discrete part of system (1), y(tk ) = Wk (r(t))y(tk− ), k ∈ N+ is the impulse at the moment of time tk ; Wk (r(t)) ∈ Rn×n is the impulse gain matrix at the moment of time tk ; the discrete instant set {tk } satisfies 0 = t0 < t1 < t2 < · · · < tk < · · · , limk→∞ tk = ∞; y(tk− ) and y(tk+ ) are the left-hand and right-hand limits at tk , respectively; as usual, we always assume that y(tk+ ) = y(tk ). For convenience, we denote r(t) = i, i ∈ S, then the matrices C(r(t)), A(r(t)), B(r(t)), D(r(t)), ΔA(r(t)), ΔB(r(t)) and ΔD(r(t)) will be written as Ci , Ai , Bi , Di , ΔAi , ΔBi , ΔDi , respectively. Therefore, system (1) can be rewritten as follows: ⎧ dy(t) = [−Ci y(t) + (Ai + ΔAi)g(y(t)) ⎪ ⎪ ⎪ ⎪ +(Bi + ΔBi)g(y(t ⎨ ´ t − τ1 (t))) +(Di + ΔDi) t−τ2 (t) g(y(s))ds]dt (2) ⎪ ⎪ ⎪ σ (t, i, y(t), y(t − τ (t)), y(t − τ (t)))d ω (t), t =  t , + 1 2 k ⎪ ⎩ + t = tk , k ∈ N+ , y(tk ) = Wik y(tk− ),

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The initial condition of system (2) is give in the following form: y(s) = ϕ (s),

s ∈ [−τ , 0],

r(0) = i0 ,

for any ϕ (s) ∈ L2F0 ([−τ , 0]; Rn ), with τ > 0 being a constant to be determined later. Throughout this paper we assume that: (H1) The discrete time-varying delay τ1 (t) satisfies 0 ≤ τ1 (t) ≤ τ1 and τ˙1 (t) ≤ μ1 , the distributed time-varying delay τ2 (t) satisfies 0 ≤ τ2 (t) ≤ τ2 and τ˙2 (t) ≤ μ2 , where τ1 , τ2 , μ1 and μ2 are known positive constants. Moreover, τ = max{τ1 , τ2 }. + (H2) [51] Each neuron activation function g j is continuous, and there exist scalars l − j and l j such that

l− j ≤

g j (a) − g j (b) ≤ l+ j , a−b

(3)

− for any a, b ∈ R, a = b, j = 1, 2, · · · , n, where l + j and l j can be positive, negative or zero. And we set

L1 = diag(l1− , l2− , . . . , ln− ),

L2 = diag(l1+ , l2+ , · · · , ln+ ).

(H3) The noise matrix σ (t, i, ·, ·, ·) is local Lipschitz continuous and satisfies the linear growth condition as well, and σ (0, i, 0, 0, 0) = 0. Moreover, there exist positive definite matrices H1i , H2i and H3i (i ∈ S) such that trace[σ T (t, i, z1 , z2 , z3 )σ (t, i, z1 , z2 , z3 )] ≤ zT1 H1i z1 + zT2 H2i z2 + zT3 H3i z3 , for all z1 , z2 , z3 ∈ Rn , t ∈ R+ , and i ∈ S. (H4) The time-varying admissible parametric uncertainties ΔAi (t), ΔBi (t), ΔDi (t), i ∈ S are in terms of [ΔAi (t)

ΔBi(t)

ΔDi (t)] = Ei Fi (t)[Hi

Ji

Ki ],

where Ei , Hi , Ji and Ki are known real constant matrices with appropriate dimensions, Fi (t) is the uncertain time-varying matrix-valued function satisfying FiT (t)Fi (t) ≤ I,

∀t ≥ 0.

Next, Let y(t; ξ ) denote the state trajectory from the initial data y(θ ) = ξ (θ ) on −τ ≤ θ ≤ 0 in L2Ft ([−τ , 0]; Rn ). Based on Hypotheses (H2) and (H3), we know that g(0) = 0 and σ (0, i, 0, 0, 0) = 0, which means system (2) admits a trivial solution or zero solution y(t; 0) ≡ 0 corresponding to the initial condition ξ = 0. For simplicity, we write y(t; ξ ) = y(t). The following definition and lemmas are useful for developing our main results. Definition 1. [40] The trivial solution of system (2) is said to be exponentially stable in mean square if for every ξ ∈ L2F0 ([−τ , 0]; Rn ), there exist constants γ > 0 and M > 0 such that the following inequality holds: Ey(t; ξ )2 ≤ M e−γ t sup Eξ (θ )2 , −τ ≤θ ≤0

where γ is called the exponential convergence rate. Definition 2. (Yang [41]): The function V : [t0 , ∞) × Rn → R+ belong to class Ψ0 if: 1) the function V is continuous on each of the sets [tk−1 ,tk ) × Rn and for all t ≥ t0 , V (t, 0) ≡ 0;

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2) V (t, x) is locally Lipschitzian in x ∈ Rn ; 3) for each k = 1, 2, · · · , there exist finite limits lim

V (tk− , x),

lim

V (tk+ , x),

(t,z)→(tk− ,x) (t,z)→(tk+ ,x)

with V (tk+ , x) = V (tk , x) satisfied. Definition 3. Let C2,1 (R+ × S × Rn ; R) denote the family of all nonnegative functions V (t, i, x) ∈ Ψ0 on R+ × S × Rn , which are continuously twice differentiable in x and differentiable in t. If the function V ∈ C2,1 (R+ × S × Rn ; R), then an operator LV from R+ × S × Rn to R along the trajectory of system (2) is defined as: LV (t, i, y(t)) = Vt (t, i, y(t)) +Vy (t, i, y(t))[−Ci y(t) + (Ai + ΔAi)g(y(t)) ˆ t g(y(s))ds] +(Bi + ΔBi)g(y(t − τ1 (t))) + (Di + ΔDi ) t−τ2 (t)

1 + trace[σ T (t)Vyy (t, i, y(t))σ (t)], 2

t = tk ,

k ∈ N+ ,

(4)

where Vt (t, i, y(t)) =

∂ V (t, i, y(t)) ∂ V (t, i, y(t)) ∂ V (t, i, y(t)) ∂ 2V (t, i, y(t)) ,Vy (t, i, y(t)) = ( ,..., ),Vyy (t, i, y(t)) = ( )n×n , ∂t ∂ y1 ∂ yn ∂ yi ∂ y j σ (t) = σ (t, i, y(t), y(t − τ1 (t)), y(t − τ2 (t))).

Lemma 1. (Jensen integral inequality, see Gu [42]). For any constant matrix M > 0, any scalars s1 and s2 with s1 < s2 , and a vector function η (t) : [a, b] → R such that the integrals concerned are well defined, then the following inequality holds: ˆ s2 ˆ s2 ˆ s2 η (s)ds)T M( η (s)ds) ≤ (s2 − s1 ) η (s)M η (s)ds. ( s1

s1

s1

Lemma 2. (Wang et al. [43]) For given matrices E, F and G with F T F ≤ I and scalar ε > 0, the following inequality holds: GFE + E T F T GT ≤ ε GGT + ε −1 E T E. Remark 1. A series of inequalities are useful to derive less conservative conditions for the analysis and synthesis problems of time-delay systems, for example, Gronwall-Bellman inequality [44], Halanay inequality [45], Jensen integral inequality and Wirtinger integral [46], in which Jensen integral inequality is the most used. 3 Main Results In this section, the robust exponential stability in mean square of the trivial solution for the system (2) is studied under Hypotheses (H1) to (H4). Before deriving our main results, by using the model transformation technique, we rewritten system (2) as dy(t) = z(t)dt + σ (t)d ω (t),

t = tk ,

k ∈ N+ ,

where

(5) ˆ

z(t) = −Ci y(t) + (Ai + ΔAi)g(y(t)) + (Bi + ΔBi)g(y(t − τ1 (t))) + (Di + ΔDi)

t

t−τ2 (t)

g(y(s))ds.

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Theorem 3. Assume that Hypotheses (H1) − (H4) hold. Then the trivial solution of system (2) is robustly exponentially stable in mean square if there exist positive scalars λi (i ∈ S), γ , κ , positive definite matrices Pi (i ∈ S), Q1 , Q2 , Q3 , positive diagonal matrices Ri , Si (i ∈ S), any real matrices Nq (q = 1, 2, · · · , 6) of appropriate dimensions such that

Φi =

Pi T W PlWik   ik Φi Γi ∗ −κ I

≤ λi I, ≤ Pi

(6) [here

r(tk ) = l],

(7)

< 0,

(8)

where

)8×8 , Φ i = (φimn

m = 1, 2, . . . , 8,   Pi Ei , Γi = 07n×n

n = 1, 2, . . . , 8,

N

φi11 = −PiCi −CiT Pi + γ Pi + λi H1i + ∑ πi j Pj + e−γτ1 Q1 + e−γτ2 Q2 − 2L1 Ri L2 + N1 + N1T + N4 + N4T , j=1

φi12 = −N1 + N2T ,

φi13 = −N4 + N5T ,

φi16 = Pi Di ,

φi14 = Pi Ai + (L1 + L2 )Ri ,

φi17 = −N1 + N3T ,

φi15 = Pi Bi ,

φi18 = −N4 + N6T ,

φi22 = λi H2i − (1 − μ1 )h(μ1 )Q1 − 2L1 Si L2 − N2 − N2T ,

φi25 = (L1 + L2 )Si ,

φi27 = −N2 − N3T ,



φi33 = λi H3i − (1 − μ2 )h(μ2 )Q2 − N5 − N5T , φi38 = −N5 − N6T ,

τ



φi44 = eγτ − 1 Q3 − 2Ri + κ HiT Hi , φi55 = −2Si + κ JiT Ji , φi66 = −Q3 + κ KiT Ki , γ

φi77 = −N3 − N3T ,

φi88 = −N6 − N6T ,

the function h(u) ∈ R+ , u ∈ R is defined as  h(u) =

1, u > 1, e−2γτ , u ≤ 1.

and other elements of Φ i are all equal to 0. Proof. Construct a Lyapunov-Krasovskii functional in the following form: V (t, i, y(t)) = V1 (t, i, y(t)) +V2 (t, i, y(t)) +V3 (t, i, y(t)), where ˆ V2 (t, i, y(t)) =

t

t−τ1 (t)

V1 (t, i, y(t)) = eγ t yT (t)Pi y(t), ˆ t eγ (s−τ1 ) yT (s)Q1 y(s)ds + eγ (s−τ2 ) yT (s)Q2 y(s)ds, t−τ2 (t)

ˆ V3 (t, i, y(t)) = τ

0 ˆ

−τ

t

t+β

eγ (s−β ) gT (y(s))Q3 g(y(s))dsd β .

(9)

Haoru Li, Yang Fang, Kelin Li / Discontinuity, Nonlinearity, and Complexity 5(4) (2016) 427–446

433

For t ∈ [tk−1 ,tk ), k ∈ N+ , from (2), (9) and Definition 2, 3, we get LV1 (t, i, y(t))

= γ eγ t yT (t)Pi y(t) + 2eγ t yT (t)Pi −Ci y(t) + (Ai + ΔAi)g(y(t)) + (Bi + ΔBi)g(y(t − τ1 (t))) ˆ t N

g(y(s))ds + eγ t trace[σ T (t)Pi σ (t)] + eγ t yT (t) ∑ πi j Pj y(t), +(Di + ΔDi) t−τ2 (t)

(10)

j=1

LV2 (t, i, y(t)) = eγ (t−τ1 ) yT (t)Q1 y(t) − (1 − τ˙1 (t))eγ (t−τ1 (t)−τ1 ) yT (t − τ1 (t))Q1 y(t − τ1 (t)) +eγ (t−τ2 ) yT (t)Q2 y(t) − (1 − τ˙2 (t))eγ (t−τ2 (t)−τ2 ) yT (t − τ2 (t))Q2 y(t − τ2 (t)), LV3 (t, i, y(t)) ˆ 0 ˆ eγ (t−β ) gT (y(t))Q3 g(y(t))d β − τ = τ −τ

ˆ

0 −τ

eγ t gT (y(t + β ))Q3 g(y(t + β ))d β

(11)

(12)

ˆ t e−γ s ds − τ eγ t gT (y(s))Q3 g(y(s))ds −τ t−τ ˆ t

γt T τ γτ gT (y(s))Q3 g(y(s))ds. e − 1 e g (y(t))Q3 g(y(t)) − τ eγ t = γ t−τ = τ eγ t gT (y(t))Q3 g(y(t))

0

Combining Hypothesis (H3) and (6), we have trace[σ T (t)Pi σ (t)] ≤ λi trace[σ T (t)σ (t)] T

(13) T

T

≤ λi (y (t)H1i y(t) + y (t − τ1 (t))H2i y(t − τ1 (t)) + y (t − τ2 (t))H3i y(t − τ2 (t))). From Hypothesis (H1), (8) and (11), we obtain LV2 (t, i, y(t))

≤ eγ t (yT (t) e−γτ1 Q1 + e−γτ2 Q2 y(t) − (1 − μ1 )h(μ1 )yT (t − τ1 (t))Q1 y(t − τ1 (t))

(14)

T

−(1 − μ2 )h(μ2 )y (t − τ2 (t))Q2 y(t − τ2 (t))), Based on Hypothesis (H1), (12) and Lemma 1, it is easily to derive that LV3 (t, i, y(t)) ˆ t

T  ˆ t

γt T τ γτ γt g(y(s))ds Q3 g(y(s))ds] e − 1 e g (y(t))Q3 g(y(t)) − e [ ≤ γ t−τ t−τ ˆ t

T  ˆ t

γt T τ γτ γt g(y(s))ds Q3 g(y(s))ds]. e − 1 e g (y(t))Q3 g(y(t)) − e [ ≤ γ t−τ2 (t) t−τ2 (t)

(15)

On the other hand, by Hypothesis (H2), one can get that there exist positive diagonal matrices Ri = diag{r1i , r2i , . . . , rni }, Si = diag{s1i , s2i , . . . , sni }, i ∈ S such that the following inequalities hold 0 ≤ 2eγ t

n

∑ r ji (g j (y j (t)) − l −j y j (t))(l +j y j (t) − g j (y j (t)))

(16)

j=1 γt T

= 2e (y (t)(L1 + L2 )Ri g(y(t)) − yT (t)L1 Ri L2 y(t) − gT (y(t))Ri g(y(t))), 0 ≤ 2eγ t

n

∑ s ji (g j (y j (t − τ1(t))) − l −j y j (t − τ1(t)))(l +j y j (t − τ1(t)) − g j (y j (t − τ1(t))))

j=1 γt T

= 2e (y (t − τ1 (t))(L1 + L2 )Si g(y(t − τ1 (t))) − yT (t − τ1 (t))L1 Si L2 y(t − τ1 (t)) −gT (y(t − τ1 (t)))Si g(y(t − τ1 (t)))).

(17)

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Moreover, by utilizing the well-known Newton-Leibniz formulae and (5), it can be deduced that for any matrices Nq , q = 1, 2, · · · , 6 with appropriate dimensions, the following equalities also hold ˆ t γt T T z(s)ds)T N3 ] (18) 0 = 2e [y (t)N1 + y (t − τ1 (t))N2 + ( ˆ ×[y(t) − y(t − τ1 (t)) −

t−τ1 (t)

ˆ

t

t−τ1 (t)

z(s)ds −

ˆ 0 = 2e [y (t)N4 + y (t − τ2 (t))N5 + ( γt

T

T

ˆ ×[y(t) − y(t − τ2 (t)) −

t−τ1 (t)

t

t−τ2 (t)

ˆ

t

t−τ2 (t)

z(s)ds −

t

σ (s)d ω (s)],

z(s)ds)T N6 ] t

t−τ2 (t)

(19)

σ (s)d ω (s)].

Noting that LV (t, i, y(t)) = LV1 (t, i, y(t)) + LV2 (t, i, y(t)) + LV3 (t, i, y(t)).

(20)

Considering Hypothesis (H4), substituting (10)—(19) and Ed ω (t) = 0 into (20) yields that for t ∈ [tk−1 ,tk ), k ∈ N+ ELV (t, i, y(t)) ≤ eγ t E χ T (t)Φ

i χ (t),

(21)

where

χ (t) = [yT (t) yT (t − τ1 (t)) yT (t − τ2 (t)) gT (y(t)) gT (y(t − τ1 (t))) ˆ t ˆ t ˆ t T T g(y(s))ds) ( z(s)ds) ( z(s)ds)T ]T , ( t−τ2 (t)

t−τ1 (t)

t−τ2 (t)

⎤T ⎡ ⎤ ⎡ ⎡ ⎤ ⎤T Pi Ei 0 0 Pi Ei ⎢ 0 ⎥ ⎢ 0 ⎥ ⎢ 0 ⎥ ⎢ 0 ⎥ ⎥ ⎥ ⎢ ⎢ ⎢ ⎢ ⎥ ⎥ ⎢ 0 ⎥ ⎢ 0 ⎥ ⎢ 0 ⎥ ⎢ 0 ⎥ ⎢ ⎢ T⎥ ⎢ T⎥ ⎢ ⎥ ⎥ ⎢ 0 ⎥ ⎢H ⎥ ⎢H ⎥ T ⎢ 0 ⎥



i i ⎥ ⎥ ⎢ ⎢ ⎢ ⎢ ⎥ ⎥ Φi = Φi |κ =0 + ⎢ ⎥ Fi (t) ⎢ J T ⎥ + ⎢ J T ⎥ Fi (t) ⎢ 0 ⎥ . ⎢ 0 ⎥ ⎢ i ⎥ ⎢ i ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎢ KT ⎥ ⎢ KT ⎥ ⎢ 0 ⎥ ⎢ ⎢ i ⎥ ⎢ i ⎥ ⎢ ⎥ ⎥ ⎣ 0 ⎦ ⎣ 0 ⎦ ⎣ 0 ⎦ ⎣ 0 ⎦ 0 0 0 0 ⎡

(22)

Combining Lemma 2 and (22) together yields that there exists a positive scalar κ such that ⎤⎡ ⎤T PiEi Pi Ei ⎢ 0 ⎥⎢ 0 ⎥ ⎥⎢ ⎥ ⎢ ⎢ 0 ⎥⎢ 0 ⎥ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢



−1 ⎢ 0 ⎥ ⎢ 0 ⎥ Φi ≤ Ξi = Φi |κ >0 + κ ⎢ ⎥ ⎢ ⎥ . ⎢ 0 ⎥⎢ 0 ⎥ ⎢ 0 ⎥⎢ 0 ⎥ ⎥⎢ ⎥ ⎢ ⎣ 0 ⎦⎣ 0 ⎦ 0 0 ⎡

(23)

By applying the Schur complement equivalence [50] to (8) yields Ξi < 0. Therefore, Φ

i < 0, which means ELV (t, i, y(t)) ≤ 0,

t ∈ [tk−1 ,tk ),

k ∈ N+ .

(24)

Haoru Li, Yang Fang, Kelin Li / Discontinuity, Nonlinearity, and Complexity 5(4) (2016) 427–446

435

For t = tk , k ∈ N+ , according to (7), (9) and Ed ω (t) = 0, we have EV (tk , l, y(tk )) − EV (tk− , i, y(tk− )) = Eeγ tk yT (tk− )(WikT PlWik − Pi )y(tk− ) ≤ 0, which implies that EV (tk , l, y(tk )) ≤ EV (tk− , i, y(tk− )).

(25)

Based on the generalized Ito’s ˆ formula and inequalities (24), (25), by the similar proof and Mathematical induction of [39, 40, 48], it is true that for all i ∈ S, k ∈ N+ and t ≥ 0 EV (t, i, y(t)) ≤ EV (0, r(0), y(0)).

(26)

Before estimate the maximum convergence amplitude of the trivial solution of system (3), defined L = diag{l1 , l2 , · · · , ln } + within l j = max{|l − j |, |l j |}, j = 1, 2, · · · , n. Then EV (0, r(0), y(0)) T

= y (0)Pr(0) y(0) + ˆ 7+

0

−τ2 (0)

ˆ

0 −τ1 (0)

eγ (s−τ1 ) yT (s)Q1 y(s)ds

eγ (s−τ2 ) yT (s)Q2 y(s)ds + τ

ˆ

0 −τ

ˆ

0

β

(27)

eγ (s−β ) gT (y(s))Q3 g(y(s))dsd β

e−γτ1 ≤ max λi sup Eξ (θ )2 + λmax (Q1 ) (1 − e−γτ1 ) sup Eξ (θ )2 i∈S γ −τ ≤θ ≤0 −τ ≤θ ≤0 +λmax (Q2 )

e−γτ2 τ eγτ − 1 (1 − e−γτ2 ) sup Eξ (θ )2 + λmax (LT Q3 L) ( − τ ) sup Eξ (θ )2 γ γ γ −τ ≤θ ≤0 −τ ≤θ ≤0

= M1 sup Eξ (θ )2 , −τ ≤θ ≤0

where M1 = max λi + λmax (Q1 ) i∈S

e−γτ1 e−γτ2 τ eγτ − 1 (1 − e−γτ1 ) + λmax (Q2 ) (1 − e−γτ2 ) + λmax (LT Q3 L) ( − τ) γ γ γ γ

From (9), (26) and (27), we obtain eγ t λmin (Pi )Ey(t)2 ≤ M1 sup Eξ (θ )2 ,

(28)

Ey(t)2 ≤ M e−γ t sup Eξ (θ )2 ,

(29)

−τ ≤θ ≤0

Thus −τ ≤θ ≤0

where M = M1 / mini∈S λmin (Pi ). By Definition 2 and (29), it can be seen that the trivial solution of system (2) is robustly exponentially stable in mean square with exponential convergence rate γ . This completes the proof of Theorem 1. Remark 2. The constructed exponential-type Lyapunov-Krasovskii functional in Theorem 1 is dependent on the upper bounds of discrete and distributed time-varying delays, which makes our results be a explicit delaydependent stability criterion, and it is generally less conservative than delay-independent ones, especially when the size of the delay is small.

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Remark 3. For delayed Markovian systems with impulses, it is difficult to proof exponential stability in mean square without constructing a exponential-type Lyapunov-Krasovskii functional, that is, the Markovian jumping may occurs at the impulsive time instants. If system (2) without parametric uncertainties and stochastic perturbation, by constructing the same LyapunovKrasovskii functional, from Theorem 1, the following corollary can be deduced to guarantee the exponential stability in mean square of the trivial solution of system (2). Corollary 4. Assume that Hypotheses (H1) − (H2) hold. Then the trivial solution of system (2) is exponentially stable in mean square if there exist positive scalar γ , positive definite matrices Pi(i ∈ S), Q1 , Q2 , Q3 , positive diagonal matrices Ri , Si (i ∈ S), any real matrices Nq (q = 1, 2, · · · , 6) of appropriate dimensions such that WikT PlWik ≤ Pi

[here

r(tk ) = l],

(30)

Φ i < 0, where

)8×8 , Φ i = (φimn

(31)

m = 1, 2, . . . , 8,

n = 1, 2, . . . , 8,

N

φi11 = −PiCi −CiT Pi + γ Pi + ∑ πi j Pj + e−γτ1 Q1 + e−γτ2 Q2 − 2L1 Ri L2 + N1 + N1T + N4 + N4T , j=1

φi12 = −N1 + N2T ,

φi13 = −N4 + N5T ,

φi16 = Pi Di ,

φi14 = Pi Ai + (L1 + L2 )Ri ,

φi17 = −N1 + N3T ,

φi15 = Pi Bi ,

φi18 = −N4 + N6T ,

φi22 = −(1 − μ1 )h(μ1 )Q1 − 2L1 Si L2 − N2 − N2T ,

φi25 = (L1 + L2 )Si ,

φi27 = −N2 − N3T ,



φi33 = −(1 − μ2 )h(μ2 )Q2 − N5 − N5T , φi38 = −N5 − N6T ,

τ



φi44 = eγτ − 1 Q3 − 2Ri , φi55 = −2Si , φi66 = −Q3 , γ

φi77 = −N3 − N3T ,

φi88 = −N6 − N6T ,

the function h(u) ∈ R+ , u ∈ R is defined as  h(u) =

1, u > 1, e−2γτ , u ≤ 1.

and other elements of Φ i are all equal to 0. If system (2) only has a mode, i. e., S = {1}, by constructing the same Lyapunov-Krasovskii functional with deleting the subscript i in Pi , from Theorem 1, the following corollary can be deduced to guarantee the robust exponential stability in mean square of the trivial solution of system (2) with one mode. Corollary 5. Assume that Hypotheses (H1) − (H4) hold. Then the trivial solution of system (2) is robustly exponentially stable in mean square if there exist positive scalars λ , γ , κ , positive definite matrices P, Q1 , Q2 , Q3 , positive diagonal matrices R, S, any real matrices Nq (q = 1, 2, · · · , 6) of appropriate dimensions such that P ≤ λ I, WkT Wk

≤ qI,

(32) (33)

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437



 Φ Γ Φ= < 0, ∗ −κ I where

)8×8 , Φ = (φmn

m = 1, 2, . . . , 8,   PE , Γ= 07n×n

(34)

n = 1, 2, . . . , 8,

φ11 = −PC −CT P + γ P + λ H1 + e−γτ1 Q1 + e−γτ2 Q2 − 2L1 RL2 + N1 + N1T + N4 + N4T ,

φ12 = −N1 + N2T ,

φ13 = −N4 + N5T ,

φ16 = PD,

φ14 = PA + (L1 + L2 )R,

φ17 = −N1 + N3T ,

φ15 = PB,

φ18 = −N4 + N6T ,

φ22 = λ H2 − (1 − μ1 )h(μ1 )Q1 − 2L1 SL2 − N2 − N2T ,

φ25 = (L1 + L2 )S,

φ27 = −N2 − N3T ,



φ33 = λ H3 − (1 − μ2 )h(μ2 )Q2 − N5 − N5T , φ38 = −N5 − N6T ,

τ



φ44 = eγτ − 1 Q3 − 2R + κ H T H, φ55 = −2S + κ J T J, φ66 = −Q3 + κ K T K, γ

φ77 = −N3 − N3T ,

φ88 = −N6 − N6T ,

the function h(u) ∈ R+ , u ∈ R is defined as  h(u) =

1, u > 1, −2 γτ , u ≤ 1. e

and other elements of Φ are all equal to 0. If system (2) without distributed time-varying delay τ2 (t) and parametric uncertainties, by constructing the same Lyapunov-Krasovskii functional with Q2 = 0, Q3 = 0 and Q1 = Q, the following corollary can be deduced to guarantee the exponential stability in mean square of the trivial solution of system (2). Corollary 6. Assume that Hypotheses (H1) − (H3) hold. Then the trivial solution of system (2) is exponentially stable in mean square if there exist positive scalars λi (i ∈ S), γ , positive definite matrices Pi (i ∈ S), Q, positive diagonal matrices Ri , Si , (i ∈ S), any real matrices Nq (q = 1, 2, 3) of appropriate dimensions such that Pi ≤ λi I, WikT PlWik where

)5×5 , Φ i = (φimn

≤ Pi Φ i

[here

(35) r(tk ) = l],

(36)

< 0,

m = 1, 2, . . . , 5,

(37) n = 1, 2, . . . , 5,

N

φi11 = −PiCi −CiT Pi + γ Pi + λiH1i + ∑ πi j Pj + e−γτ1 Q − 2L1Ri L2 + N1 + N1T , j=1

φi12 = −N1 + N2T ,

φi13 = Pi Ai + (L1 + L2 )Ri ,

φi14 = Pi Bi ,

φi15 = −N1 + N3T ,

φi22 = λi H2i − (1 − μ1 )h(μ1 )Q1 − 2L1 Si L2 − N2 − N2T ,

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Haoru Li, Yang Fang, Kelin Li / Discontinuity, Nonlinearity, and Complexity 5(4) (2016) 427–446

φi24 = (L1 + L2 )Si ,

φi33 = −2Ri ,

φi25 = −N2 − N3T ,

φi44 = −2Si ,

φi55 = −N3 − N3T ,

the function h(u) ∈ R+ , u ∈ R is defined as  h(u) =

1, u > 1, e−2γτ , u ≤ 1.

and other elements of Φ i are all equal to 0. 4 Numerical results In this section, three numerical examples are given to demonstrate the effectiveness of our delay-dependent results. Example 1. [47] Let the state space of Markov chain {r(t),t ≥ 0} be S = {1, 2} with generator   −1.2 1.2 Π= . 0.5 −0.5 Consider a 2-D delayed impulsive stochastic neural networks (2) with Markovian switching and parametric uncertainties:         0.8 0 0.9 0 0.2 −0.2 0.3 0.2 , C2 = , A1 = , A2 = , C1 = 0 0.7 0 0.7 0.3 0.1 −0.1 0.2         0.1 0.2 0.2 0.2 0.2 0.2 0.1 0.1 , B2 = , D1 = , D2 = , B1 = −0.3 0.2 −0.3 0.1 −0.3 0.1 −0.1 0.2      0.1 0 0.3 0 0.01 tanh (y(t)), y(t) ≤ 0, + , W2k = , k ∈ N , g(y(t)) = W1k = 0 0.1 0 0.3 0.02y(t), y(t) > 0,

τ1 (t) = 0.4 cos t + 0.5, τ2 (t) = 0.3 sin t + 0.5,      0.5 1.2 sin(t) 0 0.7 0.4 , F1 (t) = , H1 = J1 = K1 = , E1 = −0.6 1.3 0 cos(t) −1.2 0.8       0.4 1.1 cos(t) 0 0.6 0.2 , F2 (t) = , H2 = J2 = K2 = , E2 = −0.4 1.2 0 sin(t) −1.1 0.7     0.3y2 (t) 0.3y1 (t) 0 0 + , σ (t, i, y(t), y(t − τ1 (t)), y(t − τ2 (t))) = 0 0.2y1 (t − τ1 (t)) 0 0.2y2 (t − τ2 (t)) 

i ∈ S.

Then system (3) satisfies Hypotheses (H1)-(H4) with

τ1 = 0.9, L1 = 0,

μ1 = 0.4, L2 = 0.02I,

H11 = H12 = 0.18I,

τ2 = 0.8, and

μ2 = 0.3,

L = 0.02I

H21 = H22 = 0.08I,

in

τ = 0.9, (27),

H31 = H32 = 0.08I.

We set tk = 0.4 + tk−1 , k ∈ N+ , Δt = 0.001. The 2-state Markov chain with r(0) = 1 is shown in Fig. 1, among which the right continuous Markov chain {r(t),t ≥ 0} is denoted by the solid blue line, and the Markov chain of the impulsive time instants {r(tk ), k ∈ N+ } is denoted by the red point, and the black point is used to judge whether the Markovian jumping occurs at the impulsive time instants, i. e. r(tk ) − r(tk − Δt). From Fig. 1,

Haoru Li, Yang Fang, Kelin Li / Discontinuity, Nonlinearity, and Complexity 5(4) (2016) 427–446

439

we can conclude that the Markovian jumping does not occurs at the impulsive time instants when tk = 0.4 +tk−1 , k ∈ N+ , Δt = 0.001. While the 2-state Markov chain with r(0) = 1, tk = 0.4 + tk−1 , k ∈ N+ and Δt = 0.05 is shown in Fig. 2, from which we can see that the Markovian jumping occurs at some points of the impulsive time instants when tk = 0.4 + tk−1 , k ∈ N+ , Δt = 0.05. However the authors in [36,37] ignored that the the Markovian jumping may occurs at the impulsive time instants, so we can say that the method proposed in this paper is better than that in [36, 37].

3 2.5

r(t),r(tk),r(tk)−r(tk−Δ t)

2 1.5 1 0.5 0 −0.5 r(t) r(t )

−1

k

−1.5 −2

r(t )−r(t −Δ t) k

0

5

10

15 t

20

25

k

30

Fig. 1 The 2-state Markov chain with tk = 0.4 + tk−1, k ∈ N+ , Δt = 0.001 in Example 1.

Let Δt = 0.001, by using the LMI toolbox in MATLAB, we search the maximum allowable exponential convergence rate subjects to LMIs (6)-(8) is 12.2126. Let γ = 2, we can obtain the following feasible solutions to LMIs (6)-(8) in Theorem 1:       0.0237 0.0018 0.0518 −0.0164 0.2181 −0.0019 , P2 = , Q1 = , P1 = 0.0018 0.0243 −0.0164 0.0442 −0.0019 0.2161       0.0983 −0.0016 0.1752 −0.0139 0.3573 0 , Q3 = , R1 = , Q2 = −0.0016 0.0967 −0.0139 0.1543 0 0.3573       0.3523 0 0.2595 0 0.2595 0 , S1 = , S2 = , R2 = 0 0.3523 0 0.2595 0 0.2595       −0.1385 −0.0009 0.1751 0.0013 0.1756 0.0015 , N2 = , N3 = , N1 = −0.0010 −0.1406 0.0015 0.1771 0.0013 0.1776       −0.1600 −0.0041 0.1883 0.0031 0.2110 0.0015 , N5 = , N6 = , N4 = −0.0041 −0.1652 0.0031 0.1921 0.0015 0.2128

λ1 = 0.0294,

λ2 = 0.1886,

κ = 0.0563.

Setting the simulation step size h = 0.001, and r(0) = 1, Δt = 0.001. The dynamic behavior of system (2) is presented in Fig. 3, with the initial condition y(s) = [0.5, −0.4]T , s ∈ [−0.9, 0]. Therefore, it can be verified that system (2) is robustly exponentially stable in mean square with a exponential convergence rate 2.

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3 2.5

r(t),r(tk),r(tk)−r(tk−Δ t)

2 1.5 1 0.5 0 −0.5 r(t) r(t )

−1

k

−1.5

r(t )−r(t −Δ t) k

−2

0

5

10

15 t

20

k

25

30

Fig. 2 The 2-state Markov chain with tk = 0.4 + tk−1, k ∈ N+ , Δt = 0.05 in Example 1.

0.8 0.6 0.4 0.2 y1(t)&y2(t)

0 −0.2 −0.4 y1(t)

−0.6

y (t) 2

−0.8

0

1

2

3

4 t

5

6

7

8

Fig. 3 The dynamic behavior of system (2) with the initial condition y(s) = [0.5, −0.4]T , s ∈ [−0.9, 0] in Example 1.

Example 2. [49] Let the state space of Markov chain {r(t),t ≥ 0} be S = {1, 2} with generator   −1 1 Π= . 0.5 −0.5 Consider a 2-D delayed impulsive stochastic neural networks (2) without distributed time-varying delay and

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parametric uncertainties:   70 , C1 = 04  B2 =

 −1 1 , 0 2

 C2 =

 30 , 08

 A1 =

 2 −1 , 0 −1 

D1 = 0,

D2 = 0,

W1k =

 A2 =

 0.5 0 , 0 0.5

 −1 0 , 3 2 

W2k =

 B1 =

 0.5 0 , 0 0.5

441

 12 , 30 k ∈ N+ ,

g(y(t)) = (|y(t) + 1| − |y(t) − 1|)/2, τ1 (t) = ε (ρ cos t + 0.8), , ε ≥ 0, ρ ≥ 0,     0.6y1 (t) 0.5y2 (t) 0 0 + , i ∈ S. σ (t, i, y(t), y(t − τ1 (t))) = 0 0.5y1 (t − τ1 (t)) 0 0.6y2 (t − τ1 (t)) Then system (3) satisfies Hypotheses (H1)-(H4) with

τ1 = ε (ρ + 0.8), L1 = 0,

μ1 = ερ ,

L2 = I,

and

H11 = H12 = 0.72I,

τ = ε (ρ + 0.8),

L=I

in

(27),

H21 = H22 = 0.72I,

We set tk = 0.5 + tk−1 , k ∈ N+ , Δt = 0.001. The 2-state Markov chain with r(0) = 1 is shown in Fig. 4, which has the same descriptions of Fig. 1. From Fig. 4, we can conclude that the Markovian jumping does not occurs at the impulsive time instants when tk = 0.5 + tk−1 , kinN+ , Δt = 0.001. According to Corollary 3, the maximum allowable exponential convergence rate γ subjects to LMIs (35)— (37) in Corollary 3 for different values of ε are listed in Table 1, and the maximum allowable exponential convergence rate γ subjects to LMIs (36)—(38) for different values of ρ are listed in Table 2. From Table 1 and Table 2, we find out that if the upper bound of differential of time delay μ1 ∈ [0, 0.99], when given the same value of μ1 , the maximum allowable exponential convergence rate γ decreases monotonically with increasing upper bound of time delay τ1 . And the maximum allowable exponential convergence rate γ also decreases monotonically with increasing μ1 . Table1. The Maximum allowable exponential convergence rate γ for different values of ε .

ρ = 0.1 Corollary 3

ε γ

1.1 11.1050

3.3 3.7000

6.6 1.7679

9.9 1.0005

Table 2. The Maximum allowable exponential convergence rate γ for different values of ρ .

ρ = 11 Corollary 3

3:

ε γ

0.01 99.0001

0.03 31.4000

0.06 15.1009

0.09 8.6000

Let γ = 0.5, ε = 3.3, ρ = 0.1, we can obtain the following feasible solutions to LMIs (36)-(38) in Corollary       0.4070 −0.0036 0.0652 0.0013 0.2750 −0.0061 , P2 = , Q= , P1 = 10−3 × −0.0036 0.4014 0.0013 0.0569 −0.0061 0.2782       0.0882 0 0.1705 0 0.3855 0 −3 , R2 = , S1 = 10 × , R1 = 0 0.0882 0 0.1705 0 0.3855       0.3153 0 −0.3445 0.0031 0.3228 0.0013 , N1 = , N2 = , S2 = 0 0.3153 0.0009 −0.3375 0.0012 0.3205   0.3226 −0.0001 , λ1 = 6.5647 × 10−4 , λ2 = 0.1508. N3 = 0 0.3208

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Setting the simulation step size h = 0.001. The dynamic behaviors of systems (2) are presented in Fig. 5, with the initial conditions are uniformly randomly selected from [−1, 1]T , s ∈ [−1.8, 0]. From which we can see that exponential convergence rate γ significantly dependents on time delay τ1 (t). Therefore, it can be verified that system (2) is exponentially stable in mean square with a exponential convergence rate 0.03.

3 2.5

r(t),r(tk),r(tk)−r(tk−Δ t)

2 1.5 1 0.5 0 −0.5 r(t) r(t )

−1

k

−1.5 −2

r(t )−r(t −Δ t) k

0

5

10

15 t

20

k

25

30

Fig. 4 The 2-state Markov chain with tk = 0.5 + tk−1, k ∈ N+ , Δt = 0.001 in Example 2.

Remark 4. For Example 2, the conditions of [49] fails to verify the stability since only Markovian switching has been taken into account. Therefore, we can say that for this system of Example 2, the results in this paper are much effective and less conservative than that in [49]. Example 3. [38] Let the state space of Markov chain {r(t),t ≥ 0} be S = {1, 2} with generator   −1 1 Π= . 2 −2 Consider a 2-D delayed Markovian switching impulsive neural networks (2) without parametric uncertainties and stochastic perturbation:     1.4576 0 1.7631 0 , C2 = , A1 = 0, A2 = 0, C1 = 0 1.3680 0 0.0253         −0.9220 −1.7676 −2.8996 0.4938 0.5 −0.5 0.3 0.2 , B2 = , D1 = , D2 = , B1 = −0.6831 −2.0429 −0.6736 −1.0183 0.2 0.7 −0.5 0.4      0.1 0 0.3 0 (0.2 tanh (x1 (t)), 0.3 tanh (x2 (t)))T , mode1, , W2k = , g(x(t)) = W1k = 0 0.1 0 0.3 (0.4 tanh (x1 (t)), 0.6 tanh (x2 (t)))T , mode 2,

τ1 (t) = τ2 (t) = τ . Then system (2) satisfies Hypotheses (H1)—(H2) with h1 = 0,

h2 = τ ,

τ2 = τ ,

μ1 = 0,

μ2 = 0,

τ = τ ,

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443

1 0.8 0.6 0.4 0.2 y1(t)&y2(t)

0 −0.2 −0.4 −0.6 −0.8 −1 0

2

4

6

8

10

12

14

t

Fig. 5 The dynamic behaviors of systems (2) with the initial conditions which are uniformly randomly selected from [−1, 1]T , s ∈ [−2, 0] in Example 2.



L1 = 0, L1 = 0,

L2 = diag(0.2, 0.3), L2 = diag(0.4, 0.6),

L = diag(0.2, 0.3), mode 1, L = diag(0.4, 0.6), mode 2,

We set tk = 1 + tk−1 , k ∈ N+ , Δt = 0.001. The 2-state Markov chain with r(0) = 1 is shown in Fig. 6, which has the same descriptions of Fig. 1. From Fig. 6, we can conclude that the Markovian jumping does not occurs at the impulsive time instants when tk = 1 + tk−1 , k ∈ N+ , Δt = 0.001. Let γ = 1, by combining the LMI toolbox in MATLAB and Corollary 3, we search the maximum allowable delay bound (MADB) is 11.1700, which is larger than 6.7568 that given in Example 1 of [38]. Hence, we can say for this system of Example 3, the results in this paper are much effective and less conservative than that in [38]. Setting the simulation step size h = 0.01, and τ = 8. The dynamic behavior of system (2) in Example 3 is presented in Fig. 7, with the initial condition of every state is uniformly randomly selected from [−1, 1]T , s ∈ [−8, 0]. Therefore, it can be verified that system (2) in Example 3 is exponentially stable in mean square with a exponential convergence rate 1. 5 Conclusion This paper has investigated the problem of robust exponential stability in mean square for a class of impulsive stochastic neural networks with Markovian switching, mixed time-varying delays and parametric uncertainties. Based on a novel exponential-type Lyapunov-Krasovskii functional, the delay-dependent sufficient conditions for the robust stability analysis problem have been presented in terms of LMIs. These conditions are improve some existing ones in the literature. That is, parametric uncertainties have been taken into account, and the derivatives of discrete and distributed time-varying delays are not necessarily zero or smaller than 1. Finally, three numerical examples have been provided to illustrate the proposed method.

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3 2.5 2

k

1

k

0.5

k

r(t),r(t ),r(t )−r(t −Δ t)

1.5

0 −0.5 r(t) r(tk)

−1 −1.5 −2

r(tk)−r(tk−Δ t) 0

5

10

15 t

20

25

30

Fig. 6 The 2-state Markov chain with tk = 1 + tk−1, k ∈ N+ , Δt = 0.001 in Example 3.

1 0.8 0.6 0.4 0.2 y (t)&y (t) 1

2

0 −0.2 −0.4 −0.6 y1(t)

−0.8

y2(t)

−1 0

1

2

3

4 t

5

6

7

8

Fig. 7 The dynamic behaviors of systems (2) with the initial conditions which are uniformly randomly selected from [−1, 1]T , s ∈ [−8, 0] in Example 3.

Acknowledgements This work was supported by the Opening Project of Sichuan Province University Key Laboratory of Bridge Non-destruction Detecting and Engineering Computing under Grants No. 2014QZJ01 and No. 2015QYJ01, National Natural Science Foundation of China under Grant 61573010.

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Discontinuity, Nonlinearity, and Complexity 5(4) (2016) 447–455

Discontinuity, Nonlinearity, and Complexity https://lhscientificpublishing.com/Journals/DNC-Default.aspx

Slowing Down of So-called Chaotic States: “Freezing” the Initial State M. Belger1 , S. De Nigris†2 , X. Leoncini‡1,3 1

Aix Marseille Univ, Univ Toulon, CNRS, CPT, Marseille, France Department of Mathematics and Namur Center for Complex Systems-naXys, University of Namur, 8 rempart de la Vierge, 5000 Namur, Belgium 3 Center for Nonlinear Theory and Applications, Shenyang Aerospace University, Shenyang 110136, China 2

Submission Info Communicated by Valentin Afraimovich Received 31 March 2016 Accepted 8 June 2016 Available online 1 January 2017 Keywords Macroscopic Chaos Hamiltonian Systems Networks Long-Range systems

Abstract The so-called chaotic states that emerge on the model of XY interacting on regular critical range networks are analyzed. Typical time scales are extracted from the time series analysis of the global magnetization. The large spectrum confirms the chaotic nature of the observable, anyhow different peaks in the spectrum allows for typical characteristic time-scales to emerge. We find that these time scales τ (N) display a critical slowing down, i.e they diverge as N → ∞. The scaling √ law is analyzed for different energy densities and the behavior τ (N) ∼ N is exhibited. This behavior is furthermore explained analytically using the formalism of thermodynamicequations of the motion and analyzing the eigenvalues of the adjacency matrix. ©2016 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Macroscopic chaotic behavior is often linked to out-of-equilibrium states, one of the most prominent display of such phenomenon is most certainly turbulence. The resulting chaotic or turbulent states result from various macroscopic instabilities and bifurcations, and their persistence is usually driven by strong gradients or energy fluxes. When considering isolated systems with many degrees of freedom, some similar behavior can be found, but typically it is a transient during which, starting from a given initial condition, the system relaxes to some thermodynamical equilibrium [1]. Microscopic “molecular” chaos plays there an important role for relaxation; however, in the equilibrium state, macroscopic variables are at rest, despite the microscopic chaos. It is nevertheless possible to extend this transient state: indeed in recent years there has been an extensive study of the so-called quasi-stationary states (QSSs), that emerge after a violent relaxation in systems with long-range interactions [2–5]. These states have the peculiarity that their lifetime diverges with the number of constituents, so that the limits N → ∞ and t → ∞ do not commute. In fact it has been shown that some of these states are nonstationary but can display regular oscillations and, therefore, they represent a different kind of steady state [6–9]. Moreover, as can be observed in [10], both the lifetime of the state and the “transient” relaxation time from the †

email address: [email protected] author, email address: [email protected]

‡ Corresponding

ISSN 2164 − 6376, eISSN 2164 − 6414/$-see front materials © 2016 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/DNC.2016.12.009

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QSS to the equilibrium diverge with system size. During these relaxation periods we can expect to observe some long lived but transient chaotic-like features in these isolated states [11]. As mentioned,these transient periods correspond to some kind of relaxation, nevertheless, more recently, persistent chaotic macroscopic behavior in a isolated system has been exhibited . These states occur over a wide range of energy. They were first spotted on systems of rotators evolving on a regular lattice, with a critical range of interaction and number of neighbors [12]. Further studies have shown that this behavior occurred as well on so-called lace networks, when the effective network dimension was around d = 2 [13]. Studying these systems for different number of constituents N and a fixed density of energy ε , it was discovered that the chaotic behavior of the order parameter was persistent and that the width of the fluctuations around its mean value was not changing with N, implying an infinite susceptibility over a given range of values of ε . However it was evident, at least qualitatively, that some changes in the characteristic time scales of the fluctuations were present and depended on the system’s sizes. In this paper we focus on√ this dependence of the fluctuations time scales with system size, we shall show that the observed scaling τ (N) ∼ N is different than the typical relaxation time scales observed in QSS, and provide a theoretical explanation of these time scales in the low energy range. The paper is organized as follows: in the first part we describe the considered model and remind the reader of the previously obtained results. We then move on to a a numerical study of the characteristic time scales of the fluctuations √ by analyzing the frequency spectrum of the measured order parameter, where a scaling behavior τ (N) ∼ N is clearly exhibited. The presence of a large and broad spectrum allows us to infer that the signal is indeed chaotic. We then perform an analytical study of the thermodynamical wave spectrum at low energies and we indeed confirm the numerically exhibited scaling. This evidence confirms that these chaotic states are not QSS’s and that the chaotic behavior can be expected to be an actual permanent feature or characteristic of these “equilibrium” states. 2 Description of the model Originally the model we shall consider was tailored in order to uncover the threshold of a long range interacting system. As such it was inspired from the fact that the so-called α −HMF model (see [14, 15]) displayed similar thermodynamical properties as the mean field model (for α < 1), also dubbed the HMF model, which over the years has become de facto the paradigmatic model to study and test new ideas when studying long range system. In the α −HMF rotators are located on a one-dimensional lattice, and the coupling constant Ji j between the spins decreases according to a power-law with the distance between the rotators Ji j ∼ |i − j|−α , so that all rotators are coupled. The initial idea of the proposed model was to consider a range r of neighbors who equally interact with a sharp edge, meaning that Ji j ∼ Cst if |i − j| < r and 0 if |i − j| ≥ r. We set up a window function, but what is important here is that we allow r to be a function of the total number of spins N. The range is parametrized using a characteristic exponent 1 ≤ γ ≤ 2, which measures as well the total number of links (interactions) being present in the system. When γ = 1, we are on a one-dimensional chain with a short range interactions (in our case with just nearest neighbors interactions), while when γ = 2, we retrieve the mean field model, with all rotators equally interacting with each other. To get more specific we now present the details of the rotators model placed on a one-dimensional lattice with periodic boundary conditions. The Hamiltonian of the considered system writes N

1 N p2i + ∑ εi, j (1 − cos(qi − q j )) , 2k i, j i=1 2

H=∑

(1)

where k is the constant number of links (connections) per rotator which scales with γ as k≡

22−γ (N − 1)γ 1 , εi, j = ∑ N i> j N

and is related to the range by the simple relation k = 2r. The matrix εi, j is the adjacency matrix, defined as

(2)

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449

Fig. 1 Magnetization versus time, for a fixed density of energy ε ≈ 0.4 and different values of γ . The size of the system is N = 213 . For γ = 1.25 there is no magnetization (the residual magnetization is due to finite size effects see for instance ), for γ = 1.75 we observed a finite almost constant magnetization, while for γ = 1.5 large fluctuations of order one are observed. Simulations have been performed using a time step δ t = 10−3.

 1 if i − j  r εi, j = , 0 otherwise

(3)

where i − j stands for the smallest distance between two site on the one dimensional lattice with periodic boundary conditions. From the Hamiltonian we directly get the equations of the motion of the rotators. q˙i = pi p˙i = −

(4) N

1 εi, j sin(qi − q j ) . k ,∑ j=1

(5)

A full study of the equilibrium properties of this model has been made in [12, 16]. The order parameter that we monitored is the total magnetization of the system M, defined as   cos ϕ Mx = N1 ∑Ni=1 cos qi =M . M= 1 N My = N ∑i=1 sin qi sin ϕ The results are as follows: • For γ < 1.5 the system behaves as a short range model, meaning that no order parameter emerges in the thermodynamic limit and no phase transition exists. For the short range case (γ = 1), this result is consistent with the predictions of the Mermin-Wagner theorem, which predicts that no order parameter can exist for systems with dimensions d ≤ 2, due to the existence of a continuous symmetry group (here the global translation/rotation symmetry qi → qi + θ ). • For γ > 1.5 the system behaves like the mean field model, meaning that a second order transition at a critical density of energy of εc = 0.75, is observed. All curves Mγ (ε ) appear as independent of γ and fall on the mean field one.

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Fig. 2 Time evolution of the magnetization of the system with ε ≈ 0.4. The system size is here N = 29 . The final time and data sampling of the simulation is identical to the one performed in Fig. 2. We notice that the fluctuations are indeed of the same order, however we can notice that the typical time scale of the fluctuations appear to be faster than in Fig. 2.

• For γ = 1.5 for a range of temperatures below the critical energy one, a chaotic state is observed. The magnetization displays steady and large incoherent fluctuations, which do not appear to be dependent on system size, implying an infinite susceptibility. The time dependence of these fluctuations is the subject of this paper. Note also that the transition of the Berezinsky-Kosterlitz-Thouless type has not been detected (see for details [12]). To illustrate the phenomena described, we have plotted in Fig. 2 the evolution of the order parameter at a fixed density of energy ε for three different values of γ and a system size of N = 213 . Simulations have been performed using the optimal fifth order symplectic integrator described in [17], and the fast-Fourier transform made use of the FFTW package. We can notice the peculiar regime that appears for γ = 1.5 where the magnetization displays what looks like a macroscopic chaotic behavior. In the next section we shall study in more detail the temporal behavior of the order parameter in these chaotic states. 3 Critical slowing down 3.1

Numerical study

In this section we study numerically the behavior of the order parameter for different values of ε , γ = 1.5 and different system sizes with the aim of uncovering the timescales characterizing the fluctuations. Indeed we can notice in Fig. 2 that the typical time scale of the fluctuations appears to depend on the system size, as the magnetization fluctuations are much faster for N = 512 (Fig. 2) than for N = 8192 (Fig. 2). Also, even though the signal plotted in Fig. 2 looks turbulent, it may just be the consequence of the presence of a few unrelated modes. In order to confirm the chaotic nature of the signal, we decided to analyze its Fourier spectrum. An example of such spectrum is displayed in Fig. 3. We can notice that the spectrum is continuous, differently from the one given by a quasi-periodic signal, so it is definitely of the chaotic (turbulent) type. However we can

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Fig. 3 Fourier spectrum of a “chaotic” signal of the order parameter. The considered system size is N = 214 . We can notice that we obtain a continuous spectrum with some broad peaks, and associated harmonics. The time dependence is typically chaotic, and does not correspond to a quasi-periodic signal.

notice as well some broad peaks which are associated to the decreasing harmonics in this signal. Indeed, we can relate these peaks to the typical scale of fluctuations that visually appeared in the figures 2 and 2. In order to determine the scaling with system size, we performed a sequence of numerical simulations, with a fixed density of energy, fixed total time and different system sizes. In these simulations, the initial condition is extracted from a Gaussian distribution for both the pi ’s and qi ’s. The signal analysis is performed over the data that has been averaged during the second half of the total simulation’s time. The results are displayed in Fig. 4, where the locations of the three first peaks displayed in Fig. 3 are represented versus system size in a log-log plot. One notices a universal scaling of the typical fluctuation time scale, with all peaks having a frequency that decreases as f ∼ N −1/2 . This scaling was initially not anticipated as one would naturally expect a behavior similar to what has been observed for QSS’s, with a typical lifetime scaling τ ∼ N α , with α = 1 or higher values. The observed scaling τ ∼ N 1/2 is another confirmation that these chaotic states do not correspond to transient regimes, but are “steady”. We had already run very large time simulations in without noticing any visible change in the dynamics of the order parameter, but a transient with a large value of α could still have been possible. We now move on to a theoretical hint at the observed scaling law, and the confirmation as well that these are not transient states. 3.2

Theoretical analysis

In order to perform our analysis we carry out a similar calculation as the one performed in [12], that had allowed us to prove that γ = 1.5 was a threshold between the short range and the long range behavior. The method was proposed in a general context and explicitly developed for lattice’s system in [18]. In order to be more self-consistent we review the method from the start, and apply it to the considered system (1). As already stated we consider a lattice (in dimension D = 1 for our system) of N sites with coordinates xi = 1, . . . , N. At each site i we have a particle, coupled to some neighbors, each having a momentum pi and conjugate coordinate qi . We recall that we shall consider thermodynamical equilibrium properties (even though we are looking at some dynamical properties) so the units are such that the lattice spacing, the Boltzmann constant, and the mass are equal to one. Also from the form of the Hamiltonian (1), a calculation within the

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Fig. 4 Scaling of the localization of the frequency peaks versus system size. The locations of the first three peaks (harmonics) are represented. One notices a global uniform scaling of the different peaks, and√a slowing down of the typical fluctuation time. The scaling law shows a decrease of the peak frequencies as f ∼ 1/ N.

canonical ensemble will imply that the pi are distributed according to a Gaussian distribution. Since we are working on a lattice, with periodic boundary conditions, we can represent the momentum as a superposition of Fourier modes: pi =

Nk0

∑ α˙ k cos(kxi + φk ) ,

(6)

k=0

where the wavenumber k is in the reciprocal lattice (an integer multiple of k0 = 2π /N (1/D) ), the wave amplitude is α˙ k , and since we want the momenta to be Gaussian distributed variables in the thermodynamic limit, we consider that the random phase φk is uniformly distributed on the circle. Therefore, we should, given some conditions on the amplitude, be able to apply the central limit theorem. The momentum set is labeled, using (6), with the set of phases  ≡ {φk }. Note also that this equation can also be interpreted as a change of variables, from p to α , with constant Jacobian (the change is linear and we chose an equal number of modes and particles). Before proceeding, we would like to make some remarks. First, it is clear from the Hamiltonian (1) that we have a translational invariance, which implies that the total momentum of the system is conserved. Since physics should not change we make a simple Galilean transform in order to choose a reference frame where the total momentum is zero. The total momentum is directly linked to the zero mode, so this choice implies thus that we have to take α˙ 0 = 0. Second, since we know that in the canonical ensemble the variance of pi is fixed and equal to the temperature of the system, we shall assume that the α˙ k are all of the same order (we need a large number of relevant modes for the center-limit theorem to apply). Given these assumptions and using the relation p2i = ∑ α˙ k2 /2 (we average over the random phases), we write p2i ≈ T and obtain α˙ k2 ≈ O[(T /N)] (we call this relation the Jeans condition [19]). We now move on to the associated conjugated variable of pi , since we have q˙i = pi , we write it as qi = α0 +

Nk0

∑ αk cos(kxi + φk ) ,

(7)

k=k0

where α0 is a constant since α˙ 0 = 0, corresponding to the constant average of the qi ’s. In order to proceed, since we are below the mean-field critical temperature, we make a low temperature hypothesis: thus, we can

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assume that neighboring qi ’s are not too different although no global long range order exists. . Assuming that the difference qi − q j when the rotators interact is small, we expand the Hamiltonian and obtain: H=∑ i

1 p2i + ∑ εi j (qi − q j )2 . 2 4k i, j

Using the previous expressions derived for qi and pi and averaging over the random phases we end up with an effective Hamiltonian

H 1 N 2 = ∑ α˙ l + αl2 (1 − λl ), (8) N 2 l=1 where

2 k/2 2π ml ) λl = ∑ cos( k m=1 N

(9)

are the eigenvalues of the adjacency matrix. We can extract from this a dispersion relation, indeed we have d ∂ H ∂ H ( )=− dt ∂ α˙l ∂ αl α¨l = −ωl2 αl

(10)

As mentioned this computation was already used in in order to show that the critical threshold between short range and long range behavior was γ = 1.5; we used this formalism in order to compute analytically the value of the magnetization in the thermodynamic limit. In the present case, we stress the fact that the dispersion relation (10) embeds also some dynamical informations since we have access to the typical frequencies that we can expect to find in the system. This dynamical information was not used in previous papers levering this formalism, but nevertheless the understanding of the observed scaling law could provide new avenues for this approach. We can now use this dynamical feature in order to explain the critical slowing down by monitoring√how the spectrum behaves as we change the size of the system, for the specific situation with γ = 1.5, i.e k ∼ N. For this purpose, we consider a specific mode l; we have

ωl2 = (1 − λl ) = 1−



(k+1)l π N 1 sin  lπ  [ k sin N

(11)

 − 1] .

(12)

√ In order to proceed we shall consider that N → ∞, thus N N , i.e N k and that l is fixed, we can then perform an expansion of the expression (11), and in order to avoid the first order ωl2 = 0 result, we shall expand it to third order using sin(x) = x − x3 /6 + o(x3 ). We then obtain (omitting the o(x3 ) notation)

ωl2

3 3 3 (k+1)l π l π − (k+1) N 6N 3 lπ l3 π 3 N − 6N 3 2 l2 π 2 1 − (k+1) k+1 6N 2 [1 − ] 2 2 k 1 − l6Nπ2 k + 1 (k + 1)2 l 2 π 2 l 2 π 2

k+1 1 − ≈ k k ≈ ≈

k

[

6N 2



6N 2

]

1 k2 ∼ . 2 N N We recover analytically the critical slowing down exhibited numerically √ √ in Fig. 4 and confirm that the scaling law leads to ω ∼ 1/ N, and thus characteristic time scales of order N. ∼

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4 Concluding remarks In this paper we have analyzed the typical time scale τ (N) over which the chaotic behavior (fluctuations) of the order parameter evolves as a function of system size. First after a numerical study, we have exhibited √ by showing that each of that τ (N) ∼ N. Then this behavior has been afterwards confirmed theoretically, √ the frequencies, associated to modes of the dual lattice, scaled as ωk ∼ 1/ N with system size. The direct consequences of these results go in two directions. First we confirmed the chaotic states observed and discussed in [12, 13, 16] indeed are not a transient state like a QSS and, because of the presence of a large continuous spectrum, we can as well confirm the chaotic nature of the macroscopic behavior in these states, much like a turbulent one. Second, when performing our theoretical analysis using the formalism developed in [18], we were able to show for the first time that it is possible to uncover some dynamical information from this formalism, and the successful prediction of the scaling law shows that the formalism is adequate to predict some finite size dynamical features of systems with many degrees of freedom with underlying Hamiltonian microscopic dynamics. As a whole the typical decay of the characteristic time scale has another important consequence: indeed should we consider an N → ∞ limit, the fluctuations should stop and the system will end up frozen in its initial magnetic state. It is important to comment that still the infinite susceptibility would remain, so the system should remain extremely sensitive to any external perturbation. This critical slowing down with system size has been observed in other types of networks with different structure. Thus, beside confirming the same behavior arises considering lace networks as a substrate, an interesting perspective would be to check if there are any similarities to what has been already reported, and if this phenomenon could be of practical use, like for instance to slow down the waves propagation in some localized regions. Acknowledgements S.D.N and X.L. would like to thank S. Ogawa for fruitful discussions and remarks during the preparation of this manuscript. References [1] Fermi, E., Pasta, J., and Ulam, S. (1955), Los Alamos Reports, (LA-1940). [2] Dauxois, T., Ruffo, S., Arimondo, E., and Wilkens, M., editors. (2002), Dynamics and Thermodynamics of Systems with Long Range Interactions, volume 602 of Lect. Not. Phys., Springer-Verlag, Berlin. [3] Campa, A., Dauxois, T., and Ruffo, S. (2009), Statistical mechanics and dynamics of solvable models with long-range interactions, Phys. Rep., 480, 57–159. [4] Campa, A.,Dauxois, T.,Fanelli, D., and Ruffo, S. (2014), Physics of Long-Range Interacting Systems, Oxford University Press. [5] Levin, Y., Pakter, R., Rizzato, F.B., Teles, T.N., and C. Benetti, F.P. (2014), Nonequilibrium statistical mechanics of systems with long-range interactions, Phys. Rep., 535, 1–60. [6] Holloway, J.P. and Dorning, J.J. (1991), Phys. Rev. A, 44, 3856. [7] Van den Berg, T.L., Fanelli, D., and Leoncini, X. (2010), Stationary states and fractional dynamics in systems with long range interactions. EPL, 89, 50010. [8] Yamaguchi, Y.Y. (2011), Phys. Rev. E, 84, 016211. [9] Ogawa, S., Barre, J.,Morita, H., and Yamaguchi, Y.Y. (2014), Phys Rev. E, 89, 063007. [10] Turchi, A., Fanelli, D., and Leoncini, X. (2011), Existence of Quasi-stationary states at the Long Range threshold, Commun. Nonlinear. Sci. Numer. Simulat., 16(12), 4718–4724. [11] Antunes, F.L., Benetti, F.P.C., Pakter, R., and Levin, Y. (2015), Chaos and relaxation to equilibrium in systems with long-range interactions. Phys. Rev. E, 92, 052123. [12] De Nigris, S. and Leoncini, X. (2013), Emergence of a non trivial fluctuating phase in the XY model on regular networks. EPL, 101, 10002. [13] De Nigris, S. and Leoncini, X. (2015), Crafting networks to achieve, or not achieve, chaotic states. Phys. Rev. E, 91,

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042809. [14] Anteneodo, C. and Tsallis, C. (1998), Breakdown of exponential sensitivity to initial conditions: Role of the range of interactions. Phys. Rev. Lett., 80, 5313–5316. [15] Campa, A., Giansanti, A., Moroni, D., and Tsallis, C. (2001), Long-range interacting classical systems: universality in mixing weakening. Phys. Lett. A, 286, 251. [16] De Nigris, S. and Leoncini, X. (2013), Critical behaviour of the XY -rotors model on regular and small world networks , Phys. Rev. E, 88(1-2), 012131. [17] McLachlan, R.I. and Atela, P. (1992), The accuracy of symplectic integrators, Nonlinearity, 5, 541–562. [18] Leoncini, X. and Verga, A. (2001), Dynamical approach to the microcanonical ensemble, Phys. Rev. E, 64(6), 066101. [19] Jeans, J.H. (1916), The Dynamical Theory of Gases, Cambridge Univ. Press, Cambridge.

Discontinuity, Nonlinearity, and Complexity 5(4) (2016) 457–474

Discontinuity, Nonlinearity, and Complexity https://lhscientificpublishing.com/Journals/DNC-Default.aspx

Relaxation Oscillations and Chaos in a Duffing Type Equation: A Case Study L. Lerman1†, A. Kazakov2,1 , N. Kulagin3 1 Institute

of Information Technology, Mathematics & Mechanics, Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, 603950, Russia 2 National Research University Higher School of Economics, Nizhny Novgorod, 603155, Russia 3 Moscow Aviation Institute (MAI), Moscow, Russia Submission Info Communicated by A.C.J. Luo Received 1 April 2016 Accepted 15 June 2016 Available online 1 January 2017 Keywords System with slow varying parameter Adiabatic invariant Chaos Relaxation oscillation Reversibility Symmetric orbit

Abstract Results of numerical simulations of a Duffing type Hamiltonian system with a slow periodically varying parameter are presented. Using theory of adiabatic invariants, reversibility of the system and theory of symplectic maps, along with thorough numerical experiments, we present many details of the orbit behavior for the system. In particular, we found many symmetric mixed mode periodic orbits, both being hyperbolic and elliptic, the regions with a perpetual adiabatic invariant and chaotic regions. For the latter region we present details of chaotic behavior: calculation of homoclinic tangles and Lyapunov exponents.

©2016 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Slow-fast systems model different phenomena in many branches of science and their study is a rather developed part of the theory of dynamical systems and asymptotic analysis [1–3]. Here many tools collaborate to get a more or less detailed picture of dynamics. One of the first and most elaborated theory when applying to Hamiltonian systems is the adiabatic theory [3–5] which gives an approximate description of the orbit behavior in large regions of the phase space. For the dissipative systems the so-called geometric theory of slow-fast systems initiated by the work of Fenichel [6] is important. This technique is mainly applicable when somebody is interested in the orbit behavior near the sets made up of the hyperbolic equilibria or periodic orbits of the fast systems generated by a slow-fast system at some its limit. But when this set contains nonhyperbolic equilibria then other tools should be applied. As such, the blow-up methods are used here [7–9]. Also, many efforts were spent to study using other tools and numerically the chaotic orbit behavior in the stochastic regions near separatrix sets (see, for instance, [10–16]). † Corresponding

author. Email address: [email protected] ISSN 2164 − 6376, eISSN 2164 − 6414/$-see front materials © 2016 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/DNC.2016.12.010

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In recent paper [17] it was shown that blow-up methods can also be applied for studying slow fast Hamiltonian systems. In particular, it was shown there that for the case of one slow and one fast degrees of freedom the orbit behavior near a generic point on a singular curve of the slow manifold can be reduced in the principal approximation to the study of Painlev´e-I (for the case of a fold) or to the Painlev´e-II equation (for the case of a cusp). For particular cases it was known before [18–20]. An interesting problem is to understand the orbit behavior of a slow fast Hamiltonian system in a vicinity of such point and to connect this with the observed relaxation behavior. The most simple situation is met here when studying a nonautonomous Hamiltonian system with slow varying parameters to which a system with two degrees of freedom, one fast and one slow ones, can be often reduced. In this paper we study, as a representative example, a 2π -periodic nonautonomous differential system of the Duffing type in the phase space R2 × S1 = {(x, y, θ )}, θ (mod 2π ) x˙ = y = Hy , y˙ = − sin θ − x cos θ − x3 = −Hx , θ˙ = ε .

(1)

First two equations at ε > 0 and θ = ε t + θ0 give a periodic nonautonomous Hamiltonian system of period 2π /ε with the Hamiltonian x2 y2 x4 (2) H = + + cos(ε t + θ0 ) + x sin(ε t + θ0 ). 2 4 2 When the parameter ε is small, this system is slow fast with the slow varying variable θ and two fast variables (x, y). In a sense, it is a prototype of any Hamiltonian system in one degree of freedom with slow varying parameters that were by the subject of many investigations [4, 5, 12, 21–23]. We have deliberately chosen a system which on the one hand is very simple from the point of view of its fast dynamics, but from the other hand it does change its phase portrait passing through generic possible codimension-1 bifurcations. Nonetheless, the system is not chosen by chance, it appears in a slow fast Hamiltonian system with one fast and one slow degrees of freedom, when its 2-dimensional slow manifold has a cusp point w.r.t the projection of the slow manifold onto the space of slow variables. The fast systems near this point depends on two parameters (= slow variables) and on the corresponding leaf of the fast variables the fast system has a degenerate equilibrium of the type degenerate saddle or degenerate center. This equilibrium just corresponds to the cusp point on the slow manifold. Such equilibria are of codimension 2 generically. If one goes slowly around this specific point in the parameter plane (slow variables) in time, then one gets in the main approximation a system coinciding with that with Hamiltonian (2). 2 The model pecularities Sometimes it is convenient to consider this system as autonomous one. System (1) is reversible w.r.t. involution L of the phase space acting as L(x, y, θ ) = (−x, y, 2π − θ ). This means that if (x(t), y(t), θ (t)) is its solution, then (x1 (t), y1 (t), θ1 (t)) = (−x(−t), y(−t), 2π − θ (−t)) is the solution as well. The set of fixed points of the involution Fix(L) consists of two disjoint lines: x = 0, θ = 0 and x = 0, θ = π . As is known [24, 25], any orbit of a reversible system that intersect Fix(L) at exactly two points is symmetric periodic. We use this property to search for symmetric periodic orbits geometrically and numerically. For our case symmetric periodic orbits can be of three types: 1. orbits that intersect at one of its point the line x = 0, θ = 0 (mod 2π ) and at another point the line x = 0, θ = π (mod 2π ); using the symmetry L we conclude that such orbits go around the circle S1 odd times before their closing, in particular, one-round symmetric periodic orbits belong to this type; 2. orbits that intersect at both points the line x = 0, θ = 0 (mod 2π ), such orbits go around the circle even times before their closing;

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3. orbits that intersect at both points the line x = 0, θ = π (mod 2π ), such orbits also go around the circle even times before their closing. Generically, these symmetric periodic orbits differ from each other. System (1) for ε > 0 has a global cross-section, as such any plane θ = θ0 can be taken. Thus the Poincar´e map P is well defined on such a plane, this map is symplectic (area-preserving w.r.t. 2-form dx ∧ dy), since this map is generated by the nonautonomous periodic Hamiltonian system in its period 2π /ε . The map depends on the parameter ε , Pε , but the limit ε → +0 is singular, since the transition time 2π /ε goes to infinity and it is unclear which dynamical structures of Pε survive at this limit. For small positive ε the system is a slow fast one. For such a system it is useful to investigate the dynamics near its slow manifold (if it exists). This manifold is defined as the set of all equilibria for fast systems for all θ . Recall that the fast systems are obtained at the limit ε = 0 in (1). In fact, it is a one parameter family of Hamiltonian systems in one degree of freedom, the individual system is given, if one fixes a parameter θ = θ0 . In the phase space of the full system R2 × S1 the slow curve is made of these equilibria when parameter θ0 varies on the circle θ0 ∈ [0, 2π ]. For the system under consideration the slow curve is a smooth closed curve given by solutions of equations y = 0, x3 + x cos θ + sin θ = 0. In dependence on θ0 , solutions of this system consist generically either of three points or one point with two intermediate sections at angles θ = θ∗ , θ = 2π − θ∗ , where there are two equilibria. Here the angle π /2 < θ∗ < π is defined as follows. Double roots (in x) of the cubic equation x3 + x cos θ + sin θ = 0 arise when the derivative in x is also vanishes: 3x2 + cos θ = 0. From these two equations one can exclude θ since from two equations we derive sin θ = 2x3 , − cos θ = 3x2 > 0, and come to the equation 4σ 3 + 9σ 2 − 1 = 0, σ = x2 . The root under search should satisfy inequality 0 < σ < 1/3, this gives a unique root σ∗ ∼ 0.312. Thus, we have cos θ∗ = −3σ∗ , π /2 < θ∗ < π and the related x∗ > 0. The second related pair (−x∗ , 2π − θ∗ ) is given by symmetry. The specific section θ = θ∗ contains a disruption point (x∗ , θ∗ ) on the slow curve where two of three intersection points existing for θ∗ < θ < 2π − θ∗ coalesce at one point when decreasing θ . The similar situation for other two intersection points occurs near the second disruption point by the symmetry when increasing θ near 2π − θ∗ . These two specific sections θ = θ∗ and 2π − θ∗ divide the closed slow curve into four segments being each the graph of a function x = xi (θ ), y = 0, i = 1 − 4. Near the disruption point on section θ = θ∗ the slow curve has a representation y = 0, θ − θ∗ = a(x − x∗ )2 + · · · , a > 0. Indeed, at the point (x∗ , θ∗ ) the derivative in θ of the cubic function is −2σ∗2 − 3σ < 0, thus its solution near this point is given as θ − θ∗ = r(x − x∗ ), r(0) = 0, r (0) = 0, a = r (0) = 6x∗ /(x∗ sin θ∗ − cos θ∗ ) > 0. For the second disruption point we have similar representation, but the second derivative is negative, since −x∗ < 0. The whole picture of the fast phase portraits is presented schematically in Fig. 1, the phase portrait depends on the section θ = θ0 chosen. There are three significantly different types of phase portraits for such a system. One of them is the phase portrait of a nonlinear oscillator. Such a system has a unique equilibrium, a center, enclosed by periodic orbits of different periods. This orbit behavior takes place for |θ | < θ∗ (mod 2π ). The second type system occurs on intermediate sections |θ | = θ∗ , or what is the same, on sections θ = θ∗ and θ = 2π − θ∗ . Here one more equilibrium appears (disappears) on the x-axis. This additional equilibrium is parabolic with the double zero eigenvalue and 2-dimensional Jordan box of the linearized system at the equilibrium. The parabolic equilibrium possesses a unique symmetric (w.r.t. the symmetry (x, y) → (x, −y)) homoclinic orbit, orbits inside of the homoclinic orbit are periodic ones and they shrink to the center equilibrium as a value of the Hamiltonian changes in one direction but they expand to the homoclinic loop as the value of the Hamiltonian changes in another direction. All orbits outside of the loop are also periodic and tend to infinity as the value of the Hamiltonian for this θ increases to infinity. The representation for the homoclinic solution of the fast Hamiltonian system on the section θ = θ∗ has the form x(t) = x∗

1 2x2∗ t 2 − 3 16x2∗ t , y(t) = x , x2∗ = − cos θ∗ ∗ 2 2 2 2 2 2x∗ t + 1 (2x∗ t + 1) 3

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Fig. 1 Fast systems for different θ .

it is symmetric w.r.t. the involution (x, y) → (x, −y). The action I∗ corresponding to this solution (= the area inside it divided at 2π ) is equal to 4x3∗ . This will be useful for the further purposes. Also we present expressions for periodic orbits of the fast system on the section θ0 = π . Periodic solutions inside the negative loop (x < 0) are given as follows [26]: x(τ ) = −x1 dn(Kτ /π ), τ = ω t, ω = √ 2 1 + 4C 2 √ , k = 1 + 1 + 4C

π x1 √ , 2K

x1 =



√ 1 + 1 + 4C, y(τ ) =

k2 x21 √ sn(Kτ /π )cn(Kτ /π ), 2

here K is the complete elliptic integral of the first kind with parameter k. This solution is defined by an elliptic integral which is derived from the first equation using the Hamiltonian at θ = π ˆx  −x1

ds (x21 − s2 )(s2 − x22 )

√ , x22 = 1 − 1 + 4C, −x1 ≤ s ≤ −x2 .

In addition, we present the expressions for homoclinic solutions on the sections θ = θ0 for 2π − θ∗ < θ0 < θ∗ . Between two specific sections θ∗ < θ0 < 2π − θ∗ fast systems have three equilibria, a saddle with two homoclinic loops and two centers inside of the each loop, other orbits are periodic. Denote the equilibria as (xe , y), (xs , 0), (xe , y), xe < xs < xe . For Hamiltonian (2) let us denote Vθ (x) the potential, Vθ = x4 /4 + x2 cos θ /2 + x sin θ . Then the polynomial Vθ (x) −Vθ (xs ) has the double root xs , since V  (xs ) = Hx (xs , 0) = 0. Expressing y from the equation H = H(xs , 0) = V (xs ) and using the first equation in (1), we get a differential equation  1 x˙ = √ (x − xs ) (x − x1 )(x2 − x), 2

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where x1 , x2 are two remaining simple roots for the polynomial Vθ (x) − Vθ (xs ). Integration of this differential equation gives the following representations for its solutions, the left loop and right loops xs − x1 1 − tanh2 (α t/2) , r= , 2 x2 − xs 1 + r tanh (α t/2)  1 − tanh2 (α t/2) , α = (xs − x1 )(x2 − xs )/2. x(t) = xs + (x2 − xs ) 1 + r−1 tanh2 (α t/2)

x(t) = xs + (x1 − xs )

(3)

The area inside of the left homoclinic loop is monotonically increases from zero till the value 8π x3∗ , when θ0 increases from θ∗ till 2π − θ∗ . The area inside of the right homoclinic loop decreases from 8π x3∗ till zero on the same segment of θ . Saddle equilibria of the fast systems make up the middle piece of the slow curve. Thus, it is a hyperbolic invariant curve of the system at ε = 0. When approaching the specific section θ = 2π − θ∗ as θ increases, two equilibria of the fast system, the saddle and the left center, coalesce and then disappear through a parabolic equilibrium. For the system with small positive ε the orbits which start inside the small loop close to the left center move slowly in θ -direction. This is accompanied when crossing the section θ = 2π − θ∗ by the sharp transition from small amplitude fast oscillations near the piece of the slow curve to the fast oscillations of the large amplitude connected with going around near the former degenerate homoclinic orbit of the fast system. After that these fast large-amplitude oscillations are continued along some tube composed from periodic orbits of the fast systems due to an approximate preservation of a related adiabatic invariant [3] between sections θ = −θ∗ and θ = θ∗ . This tube is the surface of the constant value 4x3∗ of this adiabatic invariant. Numerical simulations with this system show several characteristic features in the orbit behavior and will be presented in the next sections. 3 Known results To substantiate further simulations recall some relevant known rigorous results. For the case of one fast and any number of slow degrees of freedom a slow fast Hamiltonian system can have a slow manifold which is generically filled with either center equilibria or saddle equilibria. For the former case the related part of the slow manifold was called in [27] (see also [28]) that near an almost elliptic slow manifold of an analytic slowfast Hamiltonian system with one fast and k slow degrees of freedom the Hamiltonian of the system can be transformed by an analytic symplectic ε -dependent transformation to the form where fast variables (x, y) enter to the transformed Hamiltonian only in the combination I = (x2 + y2 )/2 up to an exponentially small error in ε . For our case this theorem reads as follows: for those pieces of the slow curve where variables x, y can be expressed from the equations Hy = 0, Hx = 0 as functions of θ : x = f (θ ), y = g(θ ), and related equilibria of the fast system are centers, the Hamiltonian can be transformed by an analytic symplectic coordinate change Φ to the form (we preserve the same notation for new coordinates) H ◦ Φ = h(I, ε t) + R(x, y, ε t), I =

x2 + y2 , |R| ≤ C exp[−B/ε ]. 2

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The needed transformation is given first by the shift X = x − f (θ ), Y = y − g(θ ), and after that using the procedure developed by Neishtadt [29]. For our case the study is reduced to the theorem in [27], if one introduces a new Hamiltonian Hˆ = ρ + H(x, y, θ ), considering (x, y) and (θ , ρ ) as conjugated variables w.r.t the singular symplectic 2-form dy ∧ dx + ε −1 d ρ ∧ d θ . Then the system is reduced to the autonomous slow-fast system with two degrees of freedom, and results of [27] on the existence of almost invariant elliptic slow manifold are applicable.

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a) hyperbolic orbit

b)its projection on (x, y)-plane

Fig. 2 Relaxation symmetric periodic orbits (RSPO).

Fig. 3 The image and preimage of symmetry line x = 0, θ = 0 on θ = π in T = ±π /ε

This theorem says that the motion near the related pieces of the slow curve looks as fast rotations with small amplitudes around the curve. This indeed can be seen on Fig. 2 below. Another relevant result is due to Fenichel [30, 31]. It describes the behavior near that hyperbolic piece of the slow curve for which fast systems have saddle equilibrium points. For small ε > 0 near this piece there exists a true invariant smooth slow curve being for our case an orbit segment in θ : |θ − π | ≤ T1 < π − θ∗ of the flow with a hyperbolic nearby behavior. The drawback of this result is in the fact that many such orbits exist, since only finite segments of the orbits stay in the neighborhood of the slow curve: they leave the neighborhood in both directions in time through the incoming and outcoming boundary parts of the neighborhood. 4 Symmetric periodic orbits and relaxation symmetric periodic orbits In this section we present a method for finding symmetric periodic orbits. This method allows one to search not only elliptic or hyperbolic orbits but also parabolic periodic orbits from which, through a bifurcation, one can

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localize elliptic periodic orbits with very small regions of their stability (elliptic islands for the related Poincar´e map). The method was first presented in [32] but we give here its description, for completeness. We consider only the case of S-reversible area-preserving maps, f ◦ S = S ◦ f −1 for which the involution S has a smooth line of fixed points, Fix(S). For our case these maps are Poincar´e maps on the sections θ = 0 or θ = π . The map is reversible w.r.t. the involution S that is inherited by the involution L of our system on the sections. For the diffeomorphisms we consider these lines are given as x = 0 on the related section. Theorem 1. Suppose that f is a C2 -smooth area-preserving diffeomorphism that is reversible w.r.t. a smooth involution S, and the fixed points set Fix(S) of the involution is a smooth curve. If ξ = Fix(S) ∩ f p (Fix(S)) is a point of transversal intersection of these two curves, then ξ is a point on either an elliptic or a hyperbolic period-2p orbit, while if ξ is a point of quadratic tangency, it is a parabolic period-2p orbit. Proof. Since ξ ∈ Fix(S) ∩ f p (Fix(S)), then ξ = S(ξ ) and there is a point η ∈ Fix(S) such that f p (η ) = ξ . Consider first p = 1. Then we have f 2 (η ) = f ( f (η )) = f (ξ ) = f (S(ξ )) = S( f −1 (ξ )) = S(η ) = η . Similarly, one has f 2 (ξ ) = ξ . By induction, the same is true for any p ∈ Z. Below we work with p = 1 to facilitate calculations. According to the Bochner-Montgomery theorem [33] we can take two symplectic charts: V near η with coordinates (x, y) and U near ξ with coordinates (u, v) such that in V the involution S becomes S(x, y) = (x, −y), and similarly in U it becomes S(u, v) = (u, −v). Moreover, f |V = f1 : V → U is written as follows (we assume with no loss of generality that ξ and η have zero coordinates in the related charts)       u x F1 (x, y) =A + G1 (x, y) v y where A is a constant matrix and F1 and G1 are O2 (x, y). Similarly f |U = f2 : U → V has the form       x u F2 (u, v) . =B + G2 (u, v) y v Note that in both cases, du ∧ dv = dx ∧ dy by the area preservation. If ξ is the point of transverse intersection of f1 (Fix(S)) and Fix(S), then two vectors (a11 , a21 ) and (1, 0) are transverse, i.e., a21 = 0. In this case, when −1 < a12 a21 < 0, the point η is elliptic (its eigenvalues satisfy |λ1,2 | = 1), while if a12 a21 > 0 it is an orientable saddle, and if a12 a21 < −1 it is a non-orientable saddle. The tangency of D f1 (FixS) and FixS at ξ implies a21 = 0 and area preservation gives a22 = a−1 11 . The reversibility written in both coordinate charts provides the following relations for direct and inverse maps f1 ◦S = S ◦ f2−1 , f2 ◦ S = S ◦ f1−1 , or in coordinate form:        x a22 −a12 u F2 (u, −v) −1 , = + f1 : 0 a11 y −G2 (u, −v) v and f2−1 :

       u a a x F (x, y) , = 11 12 + 2 a21 a22 v G2 (x, y) y

from where we get relations: a11 = b22 , a12 = b12 , a22 = b11 , b21 = 0, U2 (x, y) = F1 (x, −y), V2 (x, y) = −G1 (x, −y), U1 (u, v) = F2 (u, −v), V1 (u, v) = −G2 (u, −v), here U1 ,V1 , U2 ,V2 are nonlinear terms of the inverse maps f1−1 , f2−1 . Denote below for brevity a11 = α , a12 = β , then a22 = α −1 . The quadratic tangency of f1 (FixS) and FixS at ξ implies ∂ 2 G1 /∂ x2 = 0 at (0, 0). The map f 2 near a 2-periodic point η has the form f2 ◦ f1 . Hence, the linear part of this map has the matrix   1 2β /α , γ = 2β /α = 0. 0 1

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Let us notice that for the map f 2 near the point η to guarantee its fixed point be parabolic (not more higher degenerate) we need only to check that in the local coordinates x1 = x + γ y + p(x, y), y1 = y + q(x, y), dx1 ∧ dy1 = dx ∧ dy the inequality ∂ 2 q/∂ x2 = 0 at the fixed point holds. For our case this quantity is the following 2 ∂ 2 G1 ∂ 2q 2 ∂ V1 (0, 0) = α (0, 0) − α (0, 0). ∂ x2 ∂ x2 ∂ u2

From identities derived from the representation for f1 and f2 = S ◦ f1−1 ◦ S we get 1 ∂ 2 G1 ∂ 2V (0, 0) = − (0, 0), ∂ u2 α ∂ x2 therefore we come to 2 ∂ 2 G1 ∂ 2 G1 ∂ 2q 2 ∂ V1 (0, 0) = α (0, 0) − α (0, 0) = 2 α (0, 0) = 0 ∂ x2 ∂ x2 ∂ u2 ∂ x2

due to the quadratic tangency of Fix(S) and f (Fix(S)) at ξ . In order to use this tool for finding periodic orbits, we need to search intersection points of symmetry line x = 0, θ = π with the image of the symmetry line x = 0, θ = 0 under the flow map in the passage time T = π /ε . These points give traces of symmetric periodic orbits of the type 1 mentioned above, they go around the circle one time. If we search for the intersection of the symmetry line x = 0, θ = 0 with its image in time T = 2π /ε , then we get type 2 symmetric periodic orbits, they go around the circle 2 times before closing. The same will occur, if one search the intersection points of the symmetry line x = 0, θ = π with its image in time T = 2π /ε . In fact, there are many such symmetric periodic orbits. The related results obtained by the numerical calculation of the flow orbits in time T = π /ε or T = 2π /ε are shown on Fig. 5. As we shall see, these orbits are closely connected with the dynamics in the chaotic region. One type of symmetric periodic orbits (SPO) is those which will be called relaxation symmetric periodic orbits (RSPO). They are similar to mixed mode oscillation orbits found in dissipative systems [2]. These are SPO which have on its period both the segments of small oscillations near an elliptic part of slow curve and segments of fast oscillations with large amplitudes. Such orbits can be seen on Fig. 2 and Fig. 8(c), one of which is hyperbolic and another one is elliptic. For example, at ε = 0.0499542 an elliptic RSPO cuts the section θ = 0 at the point (0, 0.000039). Its unfolding is plotted on Fig. 2. The reason of their existence is very transparent. Indeed, take a small segment of the symmetry line x = 0, θ = 0 |y| ≤ δ , and iterate it till the section θ = π . We will get a curvilinear segment of an almost same length, due to preservation of adiabatic invariant I near the related piece of slow curve between points (0, 0, 0) and (−1, 0, π ). The central point (0, 0, 0) of the segment is mapped to a point near (−1, 0, π ) (see blue line on Fig. 4). Let us iterate this curve further till it returns to the section θ = 2π . The curve extends around former separatrix of the parabolic point after passing the disruption point on the section 2π − θ∗ and we get as a result a curve on the section θ = 2π which make one and a half rounds in the polar angle ϕ (see Fig. 4). Thus this curve intersects symmetry line at least two times but in fact this curve acquires several folds. Hence, varying ε one can achieve the tangency of the curve and symmetry line. This guarantees the existence of elliptic periodic orbits by a small variation of ε . The smaller ε is the more long curve becomes and it makes more revolutions along ϕ -coordinate and simultaneously acquires the more and more folds. This gives a mechanism of the multiplication of symmetric periodic orbits. Moreover, they approach close to the origin on the section θ = 0 (see, Fig. 5b).

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Fig. 4 Intersections of a small symmetry line segment x = θ = 0 (blue line is its trace on the section θ = π .)

a)

b)

Fig. 5 Intersections of symmetry lines: (a) 0 → π and (b) π → 2π .

5 Three regions with the different behavior Based on the previous studies [3, 4, 10–14, 14, 15, 21, 22], we will distinguish three regions in the phase space R2 × S1 which we call the adiabatic region, the chaotic region and the transition region. Under the adiabatic region we will understand such that the system (1) possesses a perpetual adiabatic invariant. Recall some relevant results. As is known, the following theorem holds [3]. Theorem 2. For a smooth Hamiltonian system with one degree of freedom slow periodically varying in time the action I of the fast system is the perpetual adiabatic invariant in the region where all orbits of fast systems are periodic, if some nondegeneracy condition holds. For the system we study the adiabatic region is distinguished by the condition that for all θ ∈ S1 we choose for fast systems on the related plane the regions being out of separatrices of the saddle and parabolic equilibrium points. This is done for values |θ | ≥ θ∗ . For values of θ where |θ | ≤ θ∗ the fast system has the only equilibrium,

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the center, and we distinguish the region where the value of action I is greater than I∗ = 4x3∗ (see above). In such a region we introduce the action-angle variables (I, ϕ ) for any θ . This is done in the usual way [34] by means of the equations (1). The curve H = C on the θ -plane consists of one oval due to our assumption about the region out of separatrices. It is a periodic orbit of the related fast system. In virtue of the reversibility of a fast system w.r.t. the involution (x, y) → (x, −y), this oval is a symmetric curve relative to x-axis and intersects it at two points x1 (C, θ ) < x2 (C, θ ), being the roots of the polynomial H(x, 0) −C. To construct the action-angle variables we search according to [34], a canonical transformation (x, y) → (I, ϕ ) that satisfies two conditions ˛ 1)I = I(H);

2)

d ϕ = 2π ,

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Mh

here Mh is the curve H(x, y, θ ) = C on the related θ -plane. The change of variables has the form

ϕ=

∂S ∂S ∂S , y= , H(x, ) = h(I), ∂I ∂x ∂x

with the generating function S(I, x) of the canonical transformation (we omit here for brevity the dependence on θ ). If after the transformation Hamiltonian depends only on I and function h(I) has the inverse one (for instance, if h (I) = 0), then for a fixed I we get a closed curve and differential dS = SI dI + Sx dx of the function S equals dSI=const = ydx. Integrating dS along the curve gives in a neighborhood of x0 the generating function ´x S(I, x) = ydx. The complete variation of S x0

˛ ΔS =

ydx Mh

when going around the curve equals to the area bounded by the curve Mh , thus this function is multi-valued. But its derivative in x is the single-valued function though the function ϕ = ∂∂ SI has an increment by dΔS/dI when a complete route around the curve is done. In order this increment would be 2π one needs the equality 2π = dΔS/dI to hold, from which one gets ΔS = 2π I. Therefore, the action I has to be equal to the area bounded by the curve divided at 2π . The fast system after the transformation casts ⎧ ∂H ⎪ ⎪ ⎨ I˙ = 0 = ∂ϕ (6) ⎪ ∂ H ⎪ ⎩ ϕ˙ = ω (I) = − ∂I ´x2 In accordance to [34], the action variable is sought as I(x) = π1 ydx. From the Hamiltonian (2) we express x1

1  4C − x4 − 2x2 cos θ − 4x sin θ , y = ±√ 2 then one has 1 I(x; θ ,C) = √ π 2

x2ˆ(θ ,C)



4C − x4 − 2x2 cos θ − 4x sin θ dx.

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x1 (θ ,C)

This integral is elliptic, it can be transformed to the normal form by some transformation [35]. Denote P4 the polynomial in x under the square root P4 (x; θ ,C) = −x4 − 2x2 cos θ − 4x sin θ + 4C.

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P4 has two real and two imaginary roots, since we assumed that in the region where we work the level H = C consists of the only closed curve. Such closed curve is symmetric w.r.t. x-axis, hence P4 indeed has two real roots and can be represented as P4 (x; θ ,C) = −(x − x1 )(x − x2 )(x2 + ax + b),

where a2 − 4b < 0.

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The values of roots x1 (θ ,C) < x2 (θ ,C) depend on a θ -section chosen and a value of C. The elliptic integral can be transformed in such a way that the polynomial P4 will acquire the Legendre form 2 (α − s2 )(β 2 + s2 ) that is attained by the substitution x = (ps + q)/(s + 1), where p(θ ,C), q(θ ,C), p > q, are two real roots of the quadratic polynomial z2 − (p + q)z + pq with positive discriminant [36]: p+q = 2

ax1 x2 + b(x1 + x2 ) x1 x2 − b , pq = − . a + x1 + x2 a + x1 + x2

For P4 we have the equality x1 + x2 = a, since coefficient before x3 vanishes. Thus these formulas can be simplified x1 x2 + b x1 x2 − b . , pq = − p+q = x1 + x2 2 In case if x1 + x2 = 0, then a = 0 and P4 already has the needed form. After the change of variables the integral takes the form p−q I=A √ π 2

ˆα 

−α

(α 2 − s2 )(β 2 + s2 ) ds. (s + 1)4

Here constants α , β , A are the following



 

(q − x1 )(x2 − q) 2 q2 + aq + b 2 2 + ap + b (p − x )(p − x ).

, β =

, A = α =

p 1 2 (x1 − p)(x2 − p)

p2 + ap + b The elliptic integral J is calculated as ˆα J= −α

where J1 = 0, J2 = − J3 = − R=

(α 2 − s2 )(β 2 + s2 )) ds = J1 + J2 + J3 + R(G1 + G2 ), (s + 1)4

4α 4 β 4 − 4α 2 β 4 + 4α 4 β 2 + 2α 2 β 2 + 3β 4 + 3α 4 K(iα /β ), 3β (1 − α 2 )2 (1 + β 2 )2

β 4 + 10α 2 β 2 − 2β 2 + α 4 + 2α 2 + 2α 2 β 4 − 2α 4 β 2 β [K(iα /β ) − E(iα /β )], 3(1 − α 2 )2 (1 + β 2 )2

(1 + α 2 β 2 )(α 2 + β 2 )2 , 2(1 − α 2 )2 (1 + β 2 )2

G1 = 0, G2 = (2/β )Π(α 2 , iα /β ), here K, E, Π are complete elliptic integrals of the first, second, and third kinds, see [35–37]. In the action-angle variables the Hamiltonian takes the form H(I, θ ) with parameter θ , it does not depend on the angle variable ϕ . In order the theorem on the perpetual adiabatic invariant would be valid, the following conditions of “nonlinearity” have to be satisfied [3]. To express it, consider an analytic Hamiltonian H(I, θ ),

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θ = ε t, written in the action-angle variables (I, ϕ ). Suppose the frequency ω (I, θ ) = HI (I, θ ) = 0 in some domain, and its mean value in θ , ˆ 2π 1 ω¯ = ω (I, θ )d θ 2π 0 satisfies the inequality ω¯ (I) ≡ 0. Then I is the perpetual adiabatic invariant. For the case under study after introducing the action variable I = I(C, θ ) = I(H, θ ) the Hamiltonian H(I, θ ) is the inverse function of I. Thus, we need to require that IC = 1/HI = 0 in the region under consideration. Then the needed condition of nonlinearity casts as follows d dI

ˆ2π 0



= 0. IC (C, θ )

We checked this condition numerically and almost everywhere it is satisfied. In the adiabatic region the dynamics of the system is KAM-like type: the related Poincar´e map on the cross-section θ = 0 in any compact invariant subregion possesses an almost full measure set filled with invariant KAM curves interspersed with thin stochastic regions near resonant periodic orbits existing due to the resonances between frequencies of the integrable adiabatic system and the fast frequency 2π /ε . The picture reminds a usual behavior in the KAM region presented in many papers. The chaotic region is that where a stochastic orbit behavior was observed. It contains the slow curve of the system (1) and captures some its neighborhood. The behavior in this region will be discussed below. 6 System for large actions The system under interest for large values of variables x, y is more hard for simulations. To facilitate them one may use the following considerations. The system admits the transformation to the action-angle variables (I, ϕ ) where the action I is the perpetual adiabatic invariant. To this end one needs to introduce these coordinates for a fast Hamiltonian system where θ is a parameter as we did in the previous section. In these coordinates the fast system takes the standard form I˙ = 0, ϕ˙ = −Hˆ I (I, θ ) = 0. But this change of variable is rather hard implement. Therefore we may use the idea proposed by A.M. Lyapunov ˙ To display in [38] when he studied a stability of degenerate equilibrium for the equation x¨ + x2n−1 = X (x, x). this more precisely, let us introduce the generalized polar coordinates. The related coordinate transformation is as follows: x = rC(ϕ ), y = r2 S(ϕ ). 3 functions C, S of√ϕ are in fact√the elliptic Jacobi functions with the For the case of the √ √ √ nonlinearity x periodic modulus k = 1/ 2: C(ϕ ) = cn(ϕ ; 1/ 2), S(ϕ ) = sn(ϕ ; 1/ 2)dn(ϕ ; 1/ 2), of the period 4K( 2/2) with K being the complete elliptic integral of the first kind [35, 37]. We omit writing k further. Using the standard formulae for elliptic functions (see, for instance, [35, 37]): cn4 ϕ + 2sn2 ϕ dn2 ϕ ≡ 1, cn ϕ = −snϕ dnϕ , sn ϕ = cnϕ dnϕ , dn ϕ = −(snϕ cnϕ )/2, we come to the following system

(sin θ + r cos θ cnϕ )snϕ dnϕ , r (sin θ + r cos θ cnϕ )cnϕ ϕ˙ = −r− r2 r˙ = −

One may also use a symplectic transformation with similar properties: x = (3r)1/3C(ϕ ), y = (3r)2/3 S(ϕ ), dx ∧ dy = dr ∧ d ϕ .

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Then the equations cast as follows r˙ = −(3r)1/3 snϕ dnϕ sin θ + (3r)1/3 cnϕ cos θ ,

ϕ˙ = −

cnϕ sin θ + (3r)1/3 cn2 ϕ cos θ + 3r . (3r)2/3

We used system (10) for simulations at large values r ≥ 5. 7 Blow-up and Painlev´e-I equation The fast system has two specific θ -sections which contain each a parabolic point of the fast system along with its homoclinic orbit. We would like to investigate the full system near these layers for small nonzero ε > 0. The first problem here is to describe the transition of orbits for the system with small ε > 0 through a small neighborhood of a former parabolic point. To that end, let us consider this problem separately for any slow varying Hamiltonian H(x, y, s) (11) x˙ = Hy , y˙ = −Hx, s˙ = ε which has at ε = 0 a parabolic point x = y = 0. Since the study is local, we do not require here H to be periodic in s. Asymptotic expansions for such transition solutions were presented in [39] for the so-called primary parametric resonance equation ε iU  + (|U |2 − t)U = 1, ε  1, where U is a complex-valued function of t. This equation can be written in a Hamiltonian form w.r.t. real variables (u, v), U = u + iv and fast time t/ε = τ : 1 dv λ (u2 + v2 ) du + u − (u2 + v2 )2 , = λ v − v(u2 + v2 ) = Hv , = −1 − λ u + u(u2 + v2 ) = −Hu , H = dτ dτ 2 4 if λ = ετ = t considers as a parameter. The same Hamiltonian arises when studying a pendulum with a small slow varying periodic force near its 1:1 resonance of the center equilibrium [14]. After a passage to new variables (action-angle ones or symplectic polar coordinates) the same system appears in the first nonlinear approximation. The difference with the presented Hamiltonian is a small additional parameter in front of the linear √ term in u. For equation (11) the parabolic point for the frozen system (λ is a parameter) arises at λ∗ = 3 3 2/2. This parabolic point has a homoclinic loop enclosing a center equilibrium. The parabolic point breaks up into saddle and center for λ > λ∗ and disappears for λ < λ∗ . The center equilibrium inside of the former loop persists. Thus adding the equation λ˙ = ε we come to the same form of the Hamiltonian system. Now we add one more equation ε˙ = 0 to the system (11), then the extended system will have an equilibrium at the point (x, y, s, ε ) = (0, 0, 0, 0) (we preserve the old notations for variables to avoid extra letters). The linearization of the system at this equilibrium has a matrix being nothing else as 4-dimensional Jordan box. To study the solutions of this system near this equilibrium we, following the idea in [7, 8] (see also a close situation in [20]), blow up a neighborhood of this point by means of the coordinate change x = r2 X , y = r3Y, s = r4 Z, ε = r5 E.

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After blowing up we get five variables instead of former four. So we can take different charts in dependence of what four variables are assumed to be independent in the related chart. In fact, the blowing-up means passing to the space S3 × R instead of a neighborhood of R4 , thus the equilibrium at the origin is blown up to a unit sphere (X ,Y, Z, E) ∈ S3 and r ≥ 0. Since we consider ε > 0, then E is non-negative E ≥ 0, hence (X ,Y, Z, E) vary on the half sphere being the 3-ball D3 . In fact, it is not convenient to work near the sphere but it is better tackled in affine coordinates on the related tangent planes. This will be present elsewhere.

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Let us carry out the blow-up transformation for the initial system after the shift its disruption point to the origin (ξ = x − x∗ , y, u = θ − θ∗ , ε ) = (0, 0, 0, 0). The system casts in the form

ξ˙ = y, y˙ = a(u) + b(u)ξ − γξ 2 − ξ 3 , u˙ = ε , ε˙ = 0.

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For the system under consideration the coefficients are the following: a(u) = (3x2∗ + 2x4∗ ) sin u − x3∗ u2 /2 + O(u4 ), b(u) = 2x3∗ sin u − 3x2∗ (u2 /2 + O(u4 )), γ = 3x∗ > 0, for the point x = x∗ > 0, y = 0, θ = θ∗ . We denote below a0 = a (0) = 3x2∗ + 2x4∗ > 0, b0 = b (0) = 2x3∗ > 0. For the second disruption point x = −x∗ , y = 0, θ = 2π − θ∗ they are γ = −3x∗ < 0, −b0 . At the beginning we shall do the blow up near the point x = x∗ > 0, y = 0, θ = θ∗ , here we take γ > 0, b0 > 0, and after that shall do the same near the second point x = −x∗ , y = 0, θ = 2π − θ∗ where we set −γ < 0 instead of γ and b0 < 0. We shall work only in a chart which is generated on the 4-plane E = 1 being tangent to the sphere at the point (0, 0, 0, 1), then one obtains x = r2 X , y = r3Y, s = r4 Z, ε = r5 ,

(14)

or, since ε˙ = 0 we consider r = ε 1/5 as a small parameter. The system in these variables takes the form X˙ = rY, Y˙ = r(a0 Z − γ X 2 + O(r2 )), Z˙ = r. After re-scaling time rt = τ , denoting  = d/d τ , setting r = 0 we get X  = Y, Y  = a0 Z − γ X 2, Z  = 1 > 0. This system describes the behavior of the blown-up system inside of the ball D3 . The system is equivalent to the well known Painlev´e-I equation X  = a0 τ − γ X 2 [40–42]. The standard form of the Painlev´e-I equation is d 2W = 6W 2 − z, dz2 to which our equation can be transformed by a scaling of X and τ . When studying the system near the second disruption point (−x∗ , 0, 2π − θ∗ ) we need to change γ to −γ . Thus, Painlev´e-I equation describes approximately the behavior of solutions of our system near the disruption point (x∗ , 0, θ∗ ). Hence, some known solutions of Painlev´e-I equation have to play an essential role in the description of solutions of our system. Among them there is the so-called tritronqu´ee solution found first by Boutroux [43] (see details in [44, 45]). This solution is characterized by the property that it is the only real solution of the Painlev´e-I equation that is monotone in all its existence interval (it has a unique pole on the real line). For our case for system (1) at small ε > 0 this corresponds to its solution which passes near elliptic part of the slow curve and in the backward time direction it follows the stable separatrix of the former parabolic point at the distance O(ε 4/5 ) as ε → +0. The topological limit of this solution as ε → +0 is the curve made up of the elliptic part of the slow curve and the stable separatrices of the parabolic point. The role of this analog of the tritronqu´ee solution is that it is just the orbit around which all close solutions make fast rotations when passing near a related piece of slow curve (the instant center of rotations). In fact, all four known types of solutions of the Painlev´e-I equation [45] have analogs in the slow fast system near its disruption point. All this true in a neighborhood of the disruption point and will be presented elsewhere. 8 Stochastic region The simulations showed the existence of a stochastic region in the phase space. On the cross-section θ = 0 this region has the form of a disk filled with iterations of one orbit. The topological explanation of such the behavior

L. Lerman, A. Kazakov, N. Kulagin / Discontinuity, Nonlinearity, and Complexity 5(4) (2016) 457–474

a)

471

b)

Fig. 6 (a) Homoclinic tangle in the stochastic region; (b) Stochastic region for the Poincar´e map on θ = 0. Red points are iterations of multi-round elliptic island.

Fig. 7 The chart of Lyapunov’s exponents

is the presence of a number of saddle periodic orbits that exist in this region. Their separatrices intersect each other forming a tangle leading to the possibility a transition from a neighborhood of one saddle periodic orbit to another one. This is clear seen on Fig. 6. The existence of symmetric saddle fixed and periodic orbits can be explained by the reversibility of the flow. The related results were presented above. The complicated homoclinic tangle cannot explain the chaotic behavior of the system from the ergodic point of view: this set could be of a measure zero. Moreover, as was mentioned above, there are many elliptic orbits inside this stochastic region. So, what prevails is a very interesting and hard question [46, 47]. To give some insight, we performed a calculations of Lyapunov’s exponents. They appeared positive, see Fig. 7. Despite the slow fast character of the system under consideration, its behavior is similar to what was observed

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in other numerical simulations of area preserving maps starting since the standard map see, for instance, recent [15, 32]). All these simulations show the presence of elliptic islands around elliptic periodic orbits within the mess of chaotic orbits. Though here it is more subtle task to find such orbits due to a relaxation nature of the system, we found such orbits using the technique exploiting the reversibility. The related orbits are shown on Fig. 8.

a)

b)

c)

Fig. 8 (a) Image of the fixed point line, near a tangency. Intersection with x = 0 corresponds to symmetric PO; (b) near a tangency of fixed point lines; (c) graph of the elliptic periodic orbit

9 Conclusions We study the model Duffing-like system being slow fast with the periodically slow varying parameter. The combination of rigorous methods along with the accurate numerical simulations allowed us to find some new periodic orbits (relaxation symmetric periodic orbits), to find regions in the phase space where the dynamics is of KAM type (where there exists a perpetual adiabatic invariant) and a region with the clearly observed stochastic behavior. We present some explanations of this behavior using the features of the system, in particular, its reversibility. Acknowledgements Authors thank A.I. Neishtadt and P. Clarkson for useful discussions and explanations, and A. Gonchenko for a help in preparing figures. The research for this paper was supported by the following grants: the research of Sections 1 – 4 were supported by the Russian Foundation for Basic Research under the grant 14-01-00344 (N.K. and A.K.), the results from Sections 5 – 8 were supported by the Russian Science Foundation under the grant No. 14-41-00044. Also the results of L.L. were supported by the Russian Ministry of Science and Education (project 1.1410.2014/K, target part), results of A.K. were supported by the Basic Research Program at the National Research University Higher School of Economics (project 98) in 2016 and (partially) by the Dynasty Foundation. Numerical experiments were conducted using software package Computer Dynamics: Chaos. References [1] Mischenko, E.F. and Rozov, N.Kh. (1980), Differential Equations with Small Parameters and Relaxation Oscillations, Plenum Press, New York and London.

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An Interdisciplinary Journal of Discontinuity, Nonlinearity, and Complexity Volume 5, Issue 4

December 2016

Contents Further Results on the Stability of Neural Network for Solving Variational Inequalities Mi Zhou, Xiaolan Liu............................................................................................................

341-353

How the Minimal Poincar´e Return Time Depends On the Size of a Return Region in a Linear Circle Map N. Semenova, E. Rybalova, V. Anishchenko……..………….….…………………………..

355-364

Reversible Mixed Dynamics: A Concept and Examples S.V. Gonchenko......................................................................................................................

365-374

We Speak Up the Time, and Time Bespeaks Us Dimitri Volchenkov, Anna Cabigiosu, Massimo Warglien………………………….……...

375-395

On Quasi-periodic Perturbations of Duffing Equation A.D. Morozov†, T.N. Dragunov……………………..……......…..............…....…………..

397-406

A Study of the Dynamics of the Family f  ,   sin z 



z  k

where  ,    \ {0} and

k   \ {0} Patricia Domınguez, Josue Vazquez, Marco A. Montes de Oca……………….…...…..….

407-414

New Results on Exponential Stability of Fractional Order Nonlinear Dynamic Systems Tianzeng Li, Yu Wang, Yong Yang....…...…………………….…………..……….…….....

415-425

Robust Exponential Stability of Impulsive Stochastic Neural Networks with Markovian Switching and Mixed Time-varying Delays Haoru Li, Yang Fang, Kelin Li…………………………………………....…………….....

427-446

Slowing Down of So-called Chaotic States: “Freezing” the Initial State M. Belger, S. De Nigris, X. Leoncini………………….……………..…………….…….....

447-455

Relaxation Oscillations and Chaos in a Duffing Type Equation: A Case Study L. Lerman, A. Kazakov, N.Kulagin………………….……………………....…….…….....

457-474

Available online at https://lhscientificpublishing.com/Journals/DNC-Download.aspx

Printed in USA