Discontinuity, Nonlinearity, and Complexity

Volume 7 Issue 1 March 2018

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 University of Colorado, Boulder, and University of North Carolina, Chapel Hill, USA 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]

Dimitri Volchenkov Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA & Sichuan University of Science and Engineering, Sichuan, Zigong 643000, China Email: [email protected]

Associate Editors Marat Akhmet Department of Mathematics Middle East Technical University 06531 Ankara, Turkey Fax: +90 312 210 2972 Email: [email protected]

Ranis N. Ibragimov Department of Mathematics and Physics University of Wisconsin-Parkside 900 Wood Rd, Kenosha, WI 53144 Tel: 1(262) 595-2517 Email: [email protected]

J. A. Tenreiro Machado Institute of Engineering, Polytechnic of Porto, Dept. of Electrical Engineering, Rua Dr. Antonio Bernardino de Almeida, 431, 4249-015 Porto, Portugal Fax: 351-22-8321159 Email: [email protected]

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

Alexander N. Pisarchik Center for Biomedical Technology Technical University of Madrid Campus Montegancedo 28223 Pozuelo de Alarcon, Madrid, Spain E-mail: [email protected]

Josep J. Masdemont Department of Mathematics. Universitat Politecnica de Catalunya. Diagonal 647 (ETSEIB,UPC) Email: [email protected]

Marian Gidea Department of Mathematical Sciences Yeshiva University New York, NY 10016, USA Fax: +1 212 340 7788 Email: [email protected]

Gennady A. Leonov Department of Mathematics and Mechanics St-Petersburg State University 198504, Russia Email: [email protected]

Edgardo Ugalde Instituto de Fisica Universidad Autonoma de San Luis Potosi Av. Manuel Nava 6, Zona Universitaria San Luis Potosi SLP, CP 78290, Mexico Email: [email protected]

Juan Luis García Guirao Department of Applied Mathematics Technical University of Cartagena Hospital de Marina 30203-Cartagena, SPAIN Fax:+34 968 325694 Email: [email protected]

Elbert E.N. Macau Laboratory for Applied Mathematics and Computing, National Institute for Space Research, Av. dos Astronautas, 1758 C. Postal 515 12227-010 - Sao Jose dos Campos - SP, Brazil Email: [email protected], [email protected]

Michael A. Zaks Institut für Physik Humboldt Universität Berlin Newtonstr. 15, 12489 Berlin Email: [email protected]

Mokhtar Adda-Bedia Laboratoire de Physique Ecole Normale Supérieure de Lyon 46 Allée d’Italie, 69007 Lyon, France Email: [email protected]

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 the inside back cover

An Interdisciplinary Journal of Discontinuity, Nonlinearity, and Complexity Volume 7, Issue 1, March 2018

Editors Valentin Afraimovich Xavier Leoncini Lev Ostrovsky Dimitry Volchenkov

L&H Scientific Publishing, LLC, USA

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Discontinuity, Nonlinearity, and Complexity 7(1) (2018) 1-14

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

Existence and Uniqueness of Solutions for a Coupled System of Higher Order Fractional Differential Equations with Integral Boundary Conditions P. Duraisamy1 , T. Nandha Gopal2† 1 2

Department of Mathematics, Gobi Arts and Science College, Gobichettipalayam, Tamilnadu, India Department of Mathematics, Sri Ramakrishna Mission Vidyalaya College of Arts and Science, Coimbatore, Tamilnadu, India Submission Info Communicated by V. Afraimovich Received 25 December 2016 Accepted 2 March 2017 Available online 1 April 2018

Abstract In this article, we study the existence of solutions for a coupled system of higher order nonlinear fractional differential equations with non-local integral boundary condition by using Schaefer’s fixed point theorem and the uniqueness result is proved by the contraction mapping principle. Finally, examples are provided to the applicability our main results.

Keywords Coupled system Fractional differential equations Integral boundary conditions Schaefer’s fixed point theorem Contraction mapping principle

©2018 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Fractional differential equations have been of great interest recently and it arise in many fields of physics and engineering such as biophysics, electrodynamics of complex medium, signal processing etc., ([1–4]) and reference therein. Many physicists, mathematicians and engineers made remarkable contributions to the development of both theory and applications of fractional differential equations ([5–12]). Theory of coupled higher order fractional differential has been studied by few authors [13, 14]. Ahmad and Nieto [15] studied the existence of solution for a coupled fractional differential equation with three point boundary conditions whereas the study on coupled fractional differential equations with non-local integral boundary conditions is investigated in [16–18]. Recently, Ahmad and Ntouyas [16] studied the existence for a coupled fractional differential equation with integral boundary conditions of the type (c Dq + kc Dq−1 )x(t) = f (t, x(t), y(t)),t ∈ [0, 1], 2 < q ≤ 3, k > 0, (c D p + kc D p−1 )y(t) = g(t, x(t), y(t)),t ∈ [0, 1], 2 < p ≤ 3, k > 0, † Corresponding

author. Email address: [email protected] ISSN 2164-6376, eISSN 2164-6414/$-see front materials © 2018 L&H Scientific Publishing, LLC. All rights reserved. DOI:10.5890/DNC.2018.03.001

2

P. Duraisamy, T. Nandha Gopal / Discontinuity, Nonlinearity, and Complexity 7(1) (2018) 1–14

supplemented with coupled non-local integral boundary conditions ˆ η (η − s)β −1 x(s)ds, β > 0, 0 < η < ξ < 1 x(0) = 0, x (0) = 0, x(ξ ) = a Γ(β ) 0 ˆ θ (θ − s)γ −1 y(s)ds, γ > 0, 0 < θ < z < 1. y(0) = 0, y (0) = 0, y(z) = b Γ(γ ) 0 Integral boundary conditions are encountered in various applications such as population dynamics, blood flow models, chemical engineering and cellular systems. Moreover, boundary value problems with integral boundary conditions constitute a very interesting and important class of problems. The purpose of this paper is to prove the existence and uniqueness of solutions for a coupled system of higher order nonlinear fractional differential equation of the form ⎫ c α D u(t) = f (t, v(t),c D p v(t)),t ∈ [0, 1], ⎪ ⎪ ⎪ ⎪ c β c q ⎪ ⎪ D v(t) = g(t, u(t), D u(t)),t ∈ [0, 1], ⎪ ⎪ ⎬ ˆ η (m−2) (1) (0) = 0, u(1) = ζ u(s)ds⎪ u(0) = 0, u (0) = 0, u (0) = 0, ..., u ⎪ 0 ⎪ ⎪ ˆ δ ⎪ ⎪ ⎪ (m−2) ⎭ (0) = 0, v(1) = γ v(s)ds, ⎪ v(0) = 0, v (0) = 0, v (0) = 0, ..., v 0

where 0 < ζ , η < 1, m ∈ N, m ≥ 2, α , β ∈ (m − 1, m), 0 < p, q < 1 are real numbers, c Dα ,c Dβ ,c D p ,c Dq are the Caputo fractional derivatives and f , g : [0, 1] × R × R → R are continuous functions. The paper is organized as follows. In Section 2 we recall the definition of the Caputo fractional derivative and some basic lemma have been given. In Section 3, we study the existence and the uniqueness of solutions for the higher order nonlinear fractional differential equation (1). In section 4, we close the article with a specific example to illustrate the construction.

2 Preliminaries In this section, we recall some basic concepts of fractional calculus [19–22] and we prove a lemma before stating our main results. Let X = {u : u ∈ C[0, 1],c Dq u ∈ C[0, 1]}, we define ||u|| = max |u(t)|, t∈[0,1]

||u||X = max{||u||, ||c Dq u||};

where α ∈ (m − 1, m) and 0 < q < 1. Let Y = {v : v ∈ C[0, 1],c D p v ∈ C[0, 1]}, we define ||v|| = max |v(t)|, t∈[0,1]

||v||Y = max{||v||, ||c D p v||};

where β ∈ (m − 1, m) and 0 < p < 1. Thus, (X ×Y, ||.||X×Y ) is a Banach Space with the norm defined by ||(u, v)||X×Y = max{||u||X , ||v||Y } for (u, v) ∈ X ×Y. Definition 1. For a continuous function y : (0, ∞) −→ R, the Riemann-Liouville fractional integral of order α is defined as ˆ t (t − s)α −1 α y(s)ds, α > 0, I y(t) = Γ(α ) 0 where Γ is the gamma function.


3

Definition 2. The Caputo fractional derivative of order α for a continuous function y(t) is defined by ˆ t 1 c α D y(t) = (t − s)n−α −1y(n) (s)ds, Γ(n − α ) 0 where n − 1 < α < n, n = [α ] + 1, and [α ] denotes the integer part of number α . Lemma 1. [23] For q > 0, the general solution of the fractional differential equation c Dq x(t) = 0 is given by x(t) = c0 + c1t + c2t 2 + · · · + cn−1t n−1 , where ci ∈ R, i = 1, 2, . . . , n − 1, n = [q] + 1. From the above lemma, it follows that I q c Dq x(t) = x(t) + c0 + c1t + c2t 2 + · · · + cn−1t n−1 , for some ci ∈ R, i = 0, 1, 2, . . . , sn − 1, n = [q] + 1. Lemma 2. For σ ∈ C[0, 1], the fractional boundary value problem, c

Dα u(t) = σ (t), 0 < t < 1, α ∈ (m − 1, m],

(m−2)

u(0) = 0, u (0) = 0, u (0) = 0, . . . , u

ˆ

η

(0) = 0, u(1) = ζ

u(s)ds, 0

has a unique solution given by 1 u(t) = Γ(α )

ˆ

t

α −1

(t − s)

σ (s)ds −

0

t n−1

ˆ [

1

(1 − s)α −1 σ (s)ds Γ(α )

η (1 − ζ m ) 0 ˆ η ˆ s (s − t)α −1 ( σ (τ )d τ )ds]. −ζ Γ(α ) 0 0 m

Proof. From the above lemma, we have I α σ (t) = u(t) + c0 + c1t + c2t 2 + · · · + cm−1t m−1 u(t) = I α σ (t) − c0 − c1t − c2t 2 − · · · − cm−1t m−1 . Using definition 1, we get ˆ

t

u(t) = 0

(t − s)α −1 σ (s)ds − c0 − c1t − c2t 2 − · · · − cm−1t m−1 . Γ(α )

(2)

Then ˆ

t (t − s)α −2 σ (s)ds − c1 − 2c2t − · · · − (m − 1)cm−1t m−2 u (t) = Γ( α − 1) 0 ˆ t (t − s)α −3 u (t) = σ (s)ds − 2c2 − · · · − (m − 1)(m − 2)cm−1t m−3 , . . . , 0 Γ(α − 2)

Applying the boundary conditions, we have c0 = 0, c1 = 0, c2 = 0, . . . , cm−2 = 0.

(3)

4


Using (3) into (2), we get

ˆ

t

u(t) = 0

Also

ˆ

η

ζ

ˆ u(s)ds = ζ

0

ˆ

η 0

(t − s)α −1 σ (s)ds − cm−1t m−1 . Γ(α )

1 ( Γ(α )

ˆ

s

(s − t)α −1 σ (τ )d τ )ds −

0

(4)

ζ cm−1 η m . m

η

Since u(1) = ζ

u(s)ds, we get 0

cm−1 =

1 (1 −

ζ ηm m

)

ˆ [

1

0

(1 − s)α −1 σ (s)ds − Γ(α )

ˆ 0

η

ˆ (

s 0

(s − τ )α −1 σ (τ )d τ )ds]. Γ(α )

Substuting (5) into (4), we get u(t) =

ˆ 1 ˆ t (1 − s)α −1 t n−1 1 [ (t − s)α −1 σ (s)ds − σ (s)ds m η Γ(α ) 0 Γ(α ) (1 − ζ m ) 0 ˆ η ˆ s (s − t)α −1 ( σ (τ )d τ )ds]. −ζ Γ(α ) 0 0

This completes the proof.

3 Main results Let us define an operator F : X ×Y → X ×Y as F(u, v)(t) = (F1 v(t), F2 u(t)), where ˆ t 1 (t − s)α −1 f (s, v(s),c D p v(s))ds F1 v(t) = Γ(α ) 0 ˆ 1 1 t m−1 [ (1 − s)α −1 f (s, v(s),c D p v(s))ds − ζ η m Γ(α ) 0 (1 − m ) ˆ s ˆ η 1 ( (s − τ )α −1 f (τ , v(τ ),c D p v(τ ))d τ )ds], −ζ Γ(α ) 0 0 and 1 F2 u(t) = Γ(β )

ˆ

t

(t − s)β −1 f (s, u(s),c Dq u(s))ds

0 m−1 t

ˆ 1 1 (1 − s)β −1 f (s, u(s),c Dq u(s))ds − m [ (1 − γδm ) Γ(β ) 0 ˆ s ˆ δ 1 ( (s − τ )β −1 f (τ , u(τ ),c Dq u(τ ))d τ )ds]. −γ 0 Γ(β ) 0

To prove the existence and uniqueness of solutions, we need the following assumptions: (H1) There exists constants l1 , l2 > 0, such that | f (t, x1 , y1 ) − f (t, x2 , y2 )| ≤ l1 (|x1 − x2 | + |y1 − y2 |), |g(t, x1 , y1 ) − g(t, x2 , y2 )| ≤ l2 (|x1 − x2 | + |y1 − y2 |), for each t ∈ [0, 1] and all x1 , x2 , y1 , y2 ∈ R.

(5)


5

(H2)

θ1 = max{

1 2l1 [1 + m (1 + ζ η )], Γ(α + 1) |1 − ζ η | m

1 (m − 1) 2l1 (1 + ζ η )]}, [ + Γ(2 − q) Γ(α ) Γ(α + 1)|1 − ζ η m | m

and

θ2 = max{

1 2l2 [1 + m (1 + γδ )], Γ(β + 1) |1 − γδ | m

1 (m − 1) 2l2 [ + (1 + γδ )]}, Γ(2 − p) Γ(β ) Γ(β + 1)|1 − γδ m m

where θ = max{θ1 , θ2 } < 1. (H3) Set L1 = max | f (t, v(t),c D p v(t))| and t∈[0,1]

L2 = max |g(t, u(t),c Dq u(t))|. t∈[0,1]

(H4) Also set

μ1 =

1 L1 {1 + m (1 + |ζ |η )}, Γ(α + 1) |1 − ζ η | m

L1 Γ(m) L1 {1 + |ζ |η }, + μ2 = Γ(α − q + 1) Γ(α + 1)Γ(m − q)|1 − ζ η m | m

1 L2 {1 + μ3 = m (1 + |γ |δ )}, Γ(β + 1) |1 − γδ |

and

m

L2 Γ(m) L2 {1 + |γ |δ }. + μ4 = Γ(β − p + 1) Γ(β + 1)Γ(m − p)|1 − γδ m | m

3.1

Existence result via Schaefer’s fixed point theorem

Lemma 3. (Schaefer’s fixed point theorem). Let E be a normed linear space. Let F : E → E be a completely continuous operator, that is, it is continuous and the image of any bounded set is contained in a compact set and let ζ (F) = {x ∈ E; x = λ Fx for some 0 < λ < 1}. Then either ζ (F) is unbounded or F has a fixed point. Theorem 4. If f , g : [0, 1]× R× R → R are continuous functions and the assumptions (H3) and (H4) are satisfied, then the BVP (1) has a solution. Proof. Step:1 To show that F : E → E Let E = {(u(t), v(t)) : (u(t), v(t)) ∈ X ×Y, ||(u(t), v(t))|| ≤ r}, where r = max{μ1 , μ2 , μ3 , μ4 }. Now ˆ t 1 (t − s)α −1 | f (s, v(s),c D p v(s))|ds |F1 (v)(t)| ≤ Γ(α ) 0 ˆ 1 1 |t m−1 | [ (1 − s)α −1 | f (s, v(s),c D p v(s))|ds + η m Γ(α ) 0 |1 − ζ m | ˆ s ˆ η 1 ( (s − τ )α −1 | f (τ , v(τ ),c D p v(τ ))|d τ )ds]. +|ζ | Γ( α ) 0 0

6


Then ||F1 (v)|| = max |F1 (v)(t)| ≤ t∈[0,1]

1 L1 {1 + m (1 + |ζ |η )} = μ1 . Γ(α + 1) |1 − ζ η | m

Considering F1 v(t)

ˆ

1 = Γ(α − 1)

t

(t − s)α −2 f (s, v(s),c D p v(s))ds

0

ˆ 1 1 [ (1 − s)α −1 f (s, v(s),c D p v(s))ds − ηm (1 − ζ m ) Γ(α ) 0 ˆ η ˆ s 1 ( (s − τ )α −1 f (τ , v(τ ),c D p v(τ ))d τ )ds], −ζ Γ( α ) 0 0 ˆ t (t − s)−q sα −1 ds = t α −q B(α , 1 − q), (m − 1)t m−2

and

0

B(α , n − q) = we obtain c

Dq (F1 v)(t) =

1 Γ(1 − q) +

ˆ

t

(t − s)−q [

0

(m − 1)sm−2 ηm (1 − ζ m ) ˆ η

Γ(α )Γ(n − q) , (α − q + n − 1)Γ(α − q + n − 1)

ˆ s 1 (s − t)α −2 f (τ , v(τ ),c D p v(τ ))d τ Γ(α − 1) 0 ˆ 1 (1 − τ )α −1 f (τ , v(τ ),c D p v(τ ))d τ

1 Γ(α ) 0 ˆ τ (τ − δ )α −1 f (δ , v(δ ),c D p v(δ ))d δ )d τ }]

{

1 ( Γ( α) 0 0 ˆ t 1 (t − s)α −q−1 f (s, v(s),c D p v(s))ds = Γ(α − q) 0 ˆ 1 t m−q−1 Γ(m) { (1 − s)α −1 f (s, v(s),c D p v(s))ds + ηm (1 − ζ m )Γ(α )Γ(m − q) 0 ˆ η ˆ s ( (s − τ )α −1 f (τ , v(τ ),c D p v(τ ))d τ )ds}, +|ζ | +|ζ |

0

0

Thus ||c Dq (F1 v)|| =

1 Γ(α − q)

ˆ

t

(t − s)α −q−1| f (s, v(s),c D p v(s))|ds

0

ˆ 1 { (1 − s)α −1 | f (s, v(s),c D p v(s))|ds

t m−q−1Γ(m)

+

η (1 − ζ m )Γ(α )Γ(m − q) 0 ˆ η ˆ s ( (s − τ )α −1| f (τ , v(τ ),c D p v(τ ))|d τ )ds} +|ζ | m

0

0

L1 Γ(m) L1 {1 + |ζ |η } = μ2 , + ≤ m ζ η Γ(α − q + 1) |1 − |Γ(α + 1)Γ(m − q) m

we obtain ||F1 (v)||X = max {||F1 (v)||, ||c Dq F1 (v)||} ≤ r. t∈[0,1]


Similarly, we get ||F2 (u)|| = max |F2 (u)(t)| t∈[0,1]

≤

1 L2 {1 + m (1 + γ |δ |)} = μ3 , Γ(β + 1) |1 − γδ | m

||c D p (F2 u)|| =

1 Γ(β − p)

ˆ

t

(t − s)β −p−1 | f (s, u(s),c Dq u(s))|ds

0 m−p−1 Γ(m) t

+

ˆ 1 { (1 − s)β −1 | f (s, u(s),c Dq u(s))|ds

(1 − γδm )Γ(β )Γ(m − p) 0 ˆ δ ˆ s ( (s − τ )β −1 | f (τ , u(τ ),c Dq u(τ ))|d τ )ds} + |γ | m

0

0

L2 Γ(m) L2 + {1 + |γ |δ } = μ4 , ≤ m γδ Γ(β − p + 1) |1 − |Γ(β + 1)Γ(m − p) m

we obtain ||F2 (u)||Y = max {||F2 (u)||, ||c D p F2 (u)||} ≤ r. t∈[0,1]

Hence, we conclude that ||F(u, v)||X×Y ≤ r. From the expression of (F1 v)(t),c Dq (F1 v)(t), (F2 u)(t)andc D p (F2 u)(t), it is easy to see that (F1 v)(t),c Dq (F1 v)(t), (F2 u)(t)andc D p (F2 u)(t) ∈ C[0, 1]. Consequently F : E → E. Step:2 To show that F is uniformly bounded For each t ∈ [0, 1], we have ||F1 (v)|| ≤

1 L1 {1 + m (1 + |ζ |η )} = μ1 , Γ(α + 1) |1 − ζ η | m

L1 Γ(m) L1 {1 + |ζ |η } = μ2 , + || D (F1 v)|| ≤ Γ(α − q + 1) |1 − ζ η m |Γ(α + 1)Γ(m − q) c

q

m

1 L2 {1 + ||F2 (u)|| ≤ m (1 + γ |δ |)} = μ3 , Γ(β + 1) |1 − γδ | m

L2 Γ(m) L2 {1 + |γ |δ } = μ4 , + ||c D p (F2 u)|| ≤ m γδ Γ(β − p + 1) |1 − |Γ(β + 1)Γ(m − p) m

which shows that F is uniformly bounded. Step:3 To show that F is equicontinuous For each t1 ,t2 ∈ [0, 1],t1 < t2 , and it implies that

7

8


ˆ t1 1 (t2 − s)α −1 f (s, v(s),c D p v(s))ds Γ(α ) 0 ˆ t2 1 + (t2 − s)α −1 f (s, v(s),c D p v(s))ds Γ(α ) t1 ˆ t1 1 (t1 − s)α −1 f (s, v(s),c D p v(s))ds − Γ(α ) 0 ˆ 1 (t2m−1 − t1m−1 ) 1 { (1 − s)α −1 f (s, v(s),c D p v(s))ds + ζ ηm Γ( α ) 0 |1 − m | ˆ s ˆ η 1 ( (s − τ )α −1 f (τ , v(τ ),c D p v(τ ))d τ )ds}| +|ζ | Γ(α ) 0 0 L1 L1 (1 + |ζ |η )(t2m−1 − t1m−1 ), (t2α − t1α ) + ≤ m ζ η Γ(α + 1) |1 − |Γ(α + 1)

|F1 (v)(t2 ) − F1 (v)(t1 )| = |

(6)

m

and |c Dq F1 (v)(t2 ) −c Dq F1 (v)(t1 )| ˆ t2 1 (t2 − s)α −q−1 f (s, v(s),c D p v(s))ds ≤| Γ(α − q) 0 ˆ t1 1 (t1 − s)α −q−1 f (s, v(s),c D p v(s))ds| − Γ(α − q) 0 m−q−1 m−q−1 ˆ 1 − t1 ) Γ(m)(t2 { (1 − s)α −1 | f (s, v(s),c D p v(s))|ds + ζ ηm |1 − m |Γ(α )Γ(m − q) 0 ˆ η ˆ s ( (s − τ )α −1 | f (τ , v(τ ),c D p v(τ ))|d τ )ds} +|ζ | 0

≤

0

Γ(m)L1 (1 + |ζ |η ) L1 (t2m−q−1 − t1m−q−1). (t α −q − t1α −q ) + ζ ηm Γ(α − q + 1) 2 |1 − |Γ(α + 1)Γ(m − q)

(7)

m

Similarly, |F2 (u)(t2 ) − F2 (u)(t1 )| L2 L2 β β (1 + |γ |δ )(t2m−1 − t1m−1 ), (t2 − t1 ) + ≤ m γδ Γ(β + 1) |1 − |Γ(β + 1)

(8)

m

|c D p F2 (u)(t2 ) −c D p F2 (u)(t1 )| Γ(m)L2 (1 + |γ |δ ) L2 β −p β −p (t2m−p−1 − t1m−p−1 ). (t2 − t1 ) + ≤ γδ m Γ(β − p + 1) |1 − |Γ(β + 1)Γ(m − p)

(9)

m

The right-hand sides of equations (6), (7), (8) and (9) tend to zero when t1 → t2 . Therefore, the operator F is equicontinuous. As a consequence of the above proof together with the Ascoli Arzela theorem, we can conclude that F is completely continuous. Step:4 A priori bounds Ω = {(u, v) ∈ X ×Y, (u, v) = λ F(u, v), 0 < λ < 1}. To show that Ω is bounded. Let (u, v) ∈ Ω, then (u, v) = λ F(u, v), for some 0 < λ < 1.


9

For t ∈ [0, 1], we have u(t) = λ F1 (v)(t) and v(t) = λ F2 (u)(t). Using 1 λ L1 {1 + ||u|| ≤ λ ||F1 (v)|| = m (1 + |ζ |η )} = λ μ1 , Γ(α + 1) |1 − ζ η | m

1 λ L2 {1 + ||v|| ≤ λ ||F2 (u)|| = m (1 + |γ |δ )} = λ μ3 , Γ(β + 1) |1 − γδ | m

|| D v|| ≤ λ || D (F1 v)|| = λ μ2 , and c

q

c

q

||c D p u|| ≤ λ ||c D p (F2 u)|| = λ μ4 . Consequently, ||u||1 = max(||u||, ||c Dq u||) ≤ λ μ2 , ||v||1 = max(||v||, ||c D p v||) ≤ λ μ4 . Therefore ||(u, v)||1 ≤ λ max(μ2 , μ4 ).This shows that Ω is bounded. As a conclusion of Schaefer’s fixed point theorem, we deduce that F has at least one fixed point, which is a solution of (1). This completes the proof. 3.2

Uniqueness result via Banach’s fixed point theorem

Theorem 5. If the assumptions (H1) and (H2) holds, then the BVP (1) has a unique solution. Proof. Transform the BVP (1) into a fixed point problem. Let ˆ t 1 (t − s)α −1 f (s, v(s),c D p v(s))ds F1 v(t) = Γ(α ) 0 ˆ 1 1 t m−1 [ (1 − s)α −1 f (s, v(s),c D p v(s))ds − η m Γ(α ) 0 (1 − ζ m ) ˆ s ˆ η 1 ( (s − τ )α −1 f (τ , v(τ ),c D p v(τ ))d τ )ds]. −ζ Γ( α ) 0 0 Then |F1 (v1 )(t) − F1 (v2 )(t)| ˆ t 1 (t − s)α −1 | f (s, v1 (s),c D p v1 (s)) − f (s, v2 (s),c D p v2 (s))|ds ≤ Γ(α ) 0 ˆ 1 1 t m−1 [ (1 − s)α −1 | f (s, v1 (s),c D p v1 (s)) + ζ η m Γ(α ) 0 |1 − | m

− f (s, v2 (s),c D p v2 (s))|ds ˆ s ˆ η 1 ( (s − τ )α −1 | f (τ , v1 (τ ),c D p v1 (τ )) +|ζ | Γ( α ) 0 0 − f (τ , v2 (τ ),c D p v2 (τ ))|d τ )ds].

10


Using | f (s, v1 (s),c D p v1 (s)) − f (s, v2 (s),c D p v2 (s))| ≤ 2l1 ||v1 − v2 ||X , we have ||F1 (v1 )(t) − F1 (v2 )(t)|| ≤

1 Γ(α )

ˆ

t

(t − s)α −1 2l1 ||v1 − v2 ||X ds

0

ˆ 1 1 (1 − s)α −1 2l1 ||v1 − v2 ||X ds ζ η m Γ(α ) 0 |1 − m | ˆ s ˆ η 1 +|ζ | ( (s − τ )α −1 2l1 ||v1 − v2 ||X d τ )ds] Γ(α ) 0 0 1 2l1 [1 + ≤ m (1 + ζ η )]||v1 − v2 ||X , Γ(α + 1) |1 − ζ η | +

1

[

m

As |(F1 v1 ) (t) − (F1 v2 ) (t)| ˆ t 1 (t − s)α −2 | f (s, v1 (s),c D p v1 (s)) − f (s, v2 (s),c D p v2 (s))|ds ≤ Γ(α − 1) 0 ˆ 1 (m − 1)t m−2 1 [ (1 − s)α −1 | f (s, v1 (s),c D p v1 (s)) + ζ ηm Γ( α ) 0 |1 − m | c p − f (s, v2 (s), D v2 (s))|ds ˆ s ˆ η 1 ( (s − τ )α −1| f (τ , v1 (τ ),c D p v1 (τ )) +|ζ | Γ( α ) 0 0 − f (τ , v2 (τ ),c D p v2 (τ ))|d τ )ds] (m − 1) 1 (1 + ζ η )]||v1 − v2 ||X , + ≤ 2l1 [ Γ(α ) Γ(α + 1)|1 − ζ η m | m

we obtain |c Dq (F1 v1 )(t) −c Dq (F1 v2 )(t)| ˆ t 1 (t − s)−q |(F1 v1 ) (s) − (F2 v2 ) (s)|ds ≤ Γ(1 − q) 0 1 (m − 1) 2l1 (1 + ζ η )]||v1 − v2 ||X . [ + ≤ Γ(2 − q) Γ(α ) Γ(α + 1)|1 − ζ η m | m

Consequently ||F1 v||X ≤ max{||F1 (v1 ) − F1 (v2 )||, ||c Dq (F1 v1 ) −c Dq (F1 v2 )||} ≤ θ1 ||v1 − v2 ||X . Similarly, 1 F2 u(t) = Γ(β )

ˆ

t

(t − s)β −1 f (s, u(s),c Dq u(s))ds

0 m−1 t

ˆ 1 1 (1 − s)β −1 f (s, u(s),c Dq u(s))ds γδ m Γ(β ) 0 (1 − m ) ˆ s ˆ δ 1 ( (s − τ )β −1 f (τ , u(τ ),c Dq u(τ ))d τ )ds]. −γ Γ( β ) 0 0 −

[


Then |F2 (u1 )(t) − F2 (u2 )(t)| ˆ t 1 (t − s)β −1 | f (s, u1 (s),c Dq u1 (s)) − f (s, u2 (s),c Dq u2 (s))|ds ≤ Γ(β ) 0 ˆ 1 t m−1 1 + (1 − s)β −1 | f (s, u1 (s),c Dq u1 (s)) m [ |1 − γδ | Γ(β ) 0 m

− f (s, u2 (s),c Dq u2 (s))|ds ˆ s ˆ δ 1 ( (s − τ )β −1 | f (τ , u1 (τ ),c Dq u1 (τ )) +|γ | Γ( β ) 0 0 − f (τ , u2 (τ ),c Dq u2 (τ ))|d τ )ds]. Using | f (s, u1 (s),c Dq u1 (s)) − f (s, u2 (s),c Dq u2 (s))| ≤ 2l2 ||u1 − u2 ||Y , we have ||F2 (u1 )(t) − F2 (u2 )(t)|| ≤

1 Γ(β )

ˆ

t

(t − s)β −1 2l2 ||u1 − u2 ||Y ds

0

ˆ 1 1 (1 − s)β −1 2l2 ||u1 − u2 ||Y ds γδ m Γ(β ) 0 |1 − m | ˆ s ˆ δ 1 ( (s − τ )β −1 2l2 ||u1 − u2 ||Y d τ )ds] +|γ | Γ( β ) 0 0 1 2l1 [1 + ≤ m (1 + γδ )]||u1 − u2 ||Y . Γ(β + 1) |1 − γδ | +

1

[

m

As |(F2 u1 ) (t) − (F2 u2 ) (t)| ˆ t 1 (t − s)β −2 | f (s, u1 (s),c Dq u1 (s)) − f (s, u2 (s),c Dq u2 (s))|ds ≤ Γ(β − 1) 0 ˆ 1 (m − 1)t m−2 1 [ (1 − s)β −1 | f (s, u1 (s),c Dq u1 (s)) + m |1 − γδ | Γ(β ) 0 m

− f (s, u2 (s),c Dq u2 (s))|ds ˆ s ˆ δ 1 ( (s − τ )β −1 | f (τ , u1 (τ ),c Dq u1 (τ )) +|γ | Γ( β ) 0 0 − f (τ , u2 (τ ),c Dq u2 (τ ))|d τ )ds] (m − 1) 1 (1 + γδ )]||v1 − v2 ||Y , + ≤ 2l2 [ Γ(β ) Γ(β + 1)|1 − γδ m | m

we obtain |c D p (F2 u1 )(t) −c D p (F2 u2 )(t)| ˆ t 1 (t − s)−p |(F2 u1 ) (s) − (F2 u2 ) (s)|ds ≤ Γ(1 − p) 0 1 (m − 1) 2l2 (1 + γδ )]||u1 − u2 ||Y . [ + ≤ Γ(2 − p) Γ(β ) Γ(β + 1)|1 − γδ m | m

11

12


Consequently ||F2 u||Y ≤ max{||F2 (u1 ) − F2 (u2 )||, ||c D p (F2 u1 ) −c D p (F2 u2 )||} ≤ θ2 ||u1 − u2 ||Y . Hence we conclude that, ||F(u, v)||X×Y ≤ θ ||(u1 − u2 ), (v1 − v2 )||X×Y . Since θ < 1, we say that F is a contraction. As a consequence of the Banach fixed point theorem, we deduce that F has a unique fixed point which is the solution of the problem. This complets the proof.

4 Example Example 4.1 Consider the following coupled system of fractional differential equations with integral boundary conditions: ⎫ 1 c 32 ⎪ D u(t) = 10t + 5t 2 + c D 3 t 2 ⎪ ⎪ ⎪ 5 2 ⎪ c 3 2 2t c 3 2t ⎪ ⎪ D v(t) = 2t + te + D e ⎪ ⎪ ⎬ ˆ 3 5 1 (10) u(0) = 0, u (0) = 0, u (0) = 0, u (0) = 0, u(1) = u(s)ds⎪ ⎪ 3 0 ⎪ ⎪ ⎪ ˆ 1 ⎪ ⎪ 2 ⎪ 1 ⎭ v(s)ds. ⎪ v(0) = 0, v (0) = 0, v (0) = 0, v (0) = 0, v(1) = 5 0 Here α = 52 , β = 73 , p = 13 , q = 23 , m = 5, ζ = 14 , η = 23 , γ = 15 , δ = 17 . Now max f (t, v(t), c D p v(t)) = 16.33

t∈[0,1]

and

max f (t, u(t), c Dq u(t)) = 23.0597

t∈[0,1]

Hence (H3) holds with L1 = 16.33 and L2 = 23.0597. From (H4), we can find 1 L1 {1 + m (1 + |ζ |η )} = 10.6847 = μ1 , Γ(α + 1) |1 − ζ η | m

L1 Γ(m) L1 {1 + |ζ |η } = 24.4149 = μ2 , + Γ(α − q + 1) Γ(α + 1)Γ(m − q)|1 − ζ η m | m

1 L2 {1 + m (1 + |γ |δ )} = 16.8378 = μ3 , Γ(β + 1) |1 − γδ | m

L2 Γ(m) L2 {1 + |γ |δ } = 25.4582 = μ4 . + Γ(β − p + 1) Γ(β + 1)Γ(m − p)|1 − γδ m | m

Also r = max{μ1 , μ2 , μ3 , μ4 } = 25.4582.


13

Thus all the conditions of the Theorem 4 are satisfied. Hence the system (10) has a solution. Example 4.2 Consider the following fractional differential system with integral boundary conditions: ⎫ 1 1 1 c 75 ⎪ ⎪ D u(t) = tv(t) + t 2c D 4 v(t) + t, ⎪ ⎪ 15 15 ⎪ ⎪ ⎪ 5 5 1 1 ⎪ c 3 2c 7 ⎪ ⎪ D v(t) = tu(t) + t D u(t) + t, ⎪ ⎬ 12 12 ˆ 1 1 3 ⎪ u(s)ds,⎪ u(0) = 0, u (0) = 0, u (0) = 0, u(1) = ⎪ ⎪ 2 0 ⎪ ⎪ ⎪ ˆ 1 ⎪ ⎪ 3 ⎪ 1 ⎭ v(s)ds. ⎪ v(0) = 0, v (0) = 0, v (0) = 0, v(1) = 5 0

(11)

Here α = 75 , β = 53 , p = 14 , q = 57 , m = 4, ζ = 12 , η = 13 , γ = 15 , δ = 14 . 7 1 1 tv(t) + t 2c D 4 v(t) + t and 15 15 5 1 1 g(t, x, y) = tu(t) + t 2c D 7 u(t) + t. 12 12

f (t, x, y) =

Note that 1 (|x1 − x2 | + |y1 − y2 |) and 15 1 |g(t, x1 , y1 ) − g(t, x2 , y2 )| ≤ (|x1 − x2 | + |y1 − y2 |). 12

| f (t, x1 , y1 ) − f (t, x2 , y2 )| ≤

1 and l2 = Hence (H1) holds with l1 = 15 From (H2), we can show that

1 12 .

1 2l1 [1 + m (1 + ζ η )] = 0.2327 < 1, Γ(α + 1) |1 − ζ η | m

1 (m − 1) 2l1 (1 + ζ η )] = 0.7424 < 1, [ + Γ(2 − q) Γ(α ) Γ(α + 1)|1 − ζ η m | m

1 2l2 [1 + m (1 + γδ )] = 0.2268 < 1, Γ(β + 1) |1 − γδ | m

1 (m − 1) 2l2 (1 + γδ )] = 0.1472 < 1. [ + Γ(2 − p) Γ(β ) Γ(β + 1)|1 − γδ m | m

Thus, under the appropriate conditions on the functions f and g satisfies the hypotheses (H1) and (H2). All the conditions of the Theorem 3.3 are satisfied, therefore the system (11) has a unique solution.

5 Conclusion We have established the existence and uniqueness results for the Boundary value problem of fractional coupled system with integral boundary conditions. Existence result of the problem is based on Schaefer’s fixed point theorem, while the uniqueness of the solutions is proved by the application of contraction mapping principle. Also examples are illustrated in the applicability of our main results.

14


References [1] Abdellaoui, M., Dahmani, M.Z., and Bedjaoui, N. (2015), New existence results for a coupled system of nonlinear differential equations of arbitrary order, International Journal of Nonlinear Analysis and Applications, 6, 65-75. [2] Bai, C.Z. and Fang, J.X. (2004), The existence of a positive solution for a singular coupled system of nonlinear fractional differential equations, Applied Mathematics and Computation, 150, 611-621. [3] Bengrine, M.E. and Dahmani, Z. (2012), Boundary value problems for fractional differential equations, International Journal of Open Problems in Computer Science and Mathematics, 5, 7-15. [4] Gaber, M. and Brikaa, M.G. (2012), Existence results for a coupled system of nonlinear fractional differential equation with three point boundary conditions, Journal of Fractional Calculus and Applications, 3, 1-10. [5] Bakkyaraj, T. and Sahadevan, R. (2014), An approximate solution to some classes of fractional nonlinear partial difference equation using Adomian decomposition method, Journal of Fractional Calculus and Applications, 5, 37-52. [6] Bakkyaraj, T. and Sahadevan, R. (2016), Appriximate analytical solution of two coupled time fractional nonlinear Schrodinger equations, International Journal of Applied and Computational Mathematics, 2, 113-135. [7] Bakkyaraj, T. and Sahadevan, R. (2014), On solutions of two coupled fractional time derivative Hirota equations, Nonlinear Dynamics, 77, 1309-1322. [8] Chen, Y., Chen, D., and Lv, Z. (2012), The existence results for a coupled system of nonlinear fractional differential equations with multi-point boundary conditions, Bulletin of the Iarnian Mathematical Society, 38, 607-624. [9] Hu, Z., Liu, W., and Rui, W. (2014), Existence of solutions for a coupled system of Fractional Differential Equations, Bulletien of the Malasian Mathematical Science Society, 37, 1111-1121. [10] Sahadevan, R. and Prakash, P. (2016), Exact solution of certain time fractional nonlinear partial differential equations, Nonlinear Dynamics, DOI:10.1007/s 11071-016-2714-4, 15 pages. [11] Su, X. (2009), Boundary value problem for a coupled system of nonlinear fractional differential equations, Applied Mathematics Letters, 22, 64-69. [12] Wang, J., Xiang, H., and Liu, Z. (2010), Positive solution to nonzero Boundary Values Problem for a Coupled System of Nonlinear Fractional Differential Equations, International Journal of Differential Equations, Artical ID:186928, 12 pages. [13] Ahmad, B. and Alsaedi, A. (2010), Existence and uniqueness of solutions for coupled systems of higher-order nonlinear fractional differential equations, Fixed Point Theory and Applications, Article ID:364560, 17 pages. [14] Ahmad, B. and Nieto, J.J. (2009), Existence of solutions for nonlocal boundary value problems of higher order nonlinear fractional differential equations, Abstract and Applied Analysis, Article ID:494720, 9 pages. [15] Ahmad, B. and Nieto, J.J. (2009), Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions, Computers and Mathematics with Applications, 58, 1838-1843. [16] Ahmad, B. and Ntouyas, S.K. (2015), Existence results for a coupled system of caputo type sequential fractional differential equations with nonlocal integral boundary conditions, Applied Mathematics and Computation, 266, 615622. [17] Alsaedi, A., Ntouyas, S.K., Agarwal, R., and Ahmad, B. (2015), On caputo type sequential fractional differential equations with nonlocal integral boundary conditions, Advances in Difference Equations, 33, DOI:10.1186/s 13662015-0379-9, 12 pages. [18] Ntouyas, S.K. and Obaid, M. (2012), A coupled system of fractional differential equations with nonlocal integral boundary conditions, Advances in Difference Equations, DOI:10.1186/1687-1847-2012-130, 8 pages. [19] Granas, A. and Dugundji, J. (2003), Fixed Point Theory, Springer-Verlag, New York. [20] Kai, D. (2004), The Analysis of Fractional Differential Equations, Springer. [21] Kilbas, A.A, Srivastava, H.M., and Trujillo, J.J. (2006), Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam. [22] Podlubny, (1999), Fractional Differential Equations, Academic Press, New York. [23] Lakshmikantham, V., Leela, S., and Devi, J.V. (2009), Theory of Fractional Dynamic Systems, Cambridge Academic, Cambridge, UK.



Almost Periodicity in Chaos Marat Akhmet1†, Mehmet Onur Fen2 1 2

Department of Mathematics, Middle East Technical University, 06800 Ankara, Turkey Department of Mathematics, TED University, 06420 Ankara, Turkey Submission Info Communicated by V. Afraimovich Received 27 December 2016 Accepted 18 March 2017 Available online 1 April 2018 Keywords Chaos with almost periodic motions Li-Yorke chaos Cascade of almost periodic solutions Control of chaos Stabilization of tori

Abstract Periodicity plays a significant role in the chaos theory from the beginning since the skeleton of chaos can consist of infinitely many unstable periodic motions. This is true for chaos in the sense of Devaney [1], Li-Yorke [2] and the one obtained through period-doubling cascade [3]. Countable number of periodic orbits exist in any neighborhood of a structurally stable Poincaré homoclinic orbit, which can be considered as a criterion for the presence of complex dynamics [4–6]. It was certified by Shilnikov [7] and Seifert [8] that it is possible to replace periodic solutions by Poisson stable or almost periodic motions in a chaotic attractor. Despite the fact that the idea of replacing periodic solutions by other types of regular motions is attractive, very few results have been obtained on the subject. The present study contributes to the chaos theory in that direction. In this paper, we take into account chaos both through a cascade of almost periodic solutions and in the sense of Li-Yorke such that the original Li-Yorke definition is modified by replacing infinitely many periodic motions with almost periodic ones, which are separated from the motions of the scrambled set. The theoretical results are valid for systems with arbitrary high dimensions. Formation of the chaos is exemplified by means of unidirectionally coupled Duffing oscillators. The controllability of the extended chaos is demonstrated numerically by means of the Ott-Grebogi-Yorke [9] control technique. In particular, the stabilization of tori is illustrated. ©2018 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction The transition between regular and chaotic motions has been a field of interest among scientists since the second half of the twentieth century. Investigations of chaos for continuous-time dynamics started with the studies of Poincaré [10], Cartwright and Littlewood [11], Levinson [12], Lorenz [13], and Ueda [14]. Chaos theory has many applications in various disciplines such as neuroscience, economics, mechanics, electronics, meteorology, and medicine [15]. The first mathematical definition of chaos was introduced in the paper [2] for discrete-time systems. According to Li and Yorke [2], a continuous map on an interval has points of all periods provided that it has a point of period three, and there exists an uncountable scrambled subset [16] of the interval. The part of the Li-Yorke † Corresponding


16

Marat Akhmet, Mehmet Onur Fen / Discontinuity, Nonlinearity, and Complexity 7(1) (2018) 15–29

Theorem concerning periodic motions is a specific version of the Sharkovskii Theorem [17]. The presence of the scrambled set is the feature that distinguishes Li-Yorke chaos from other types [16]. Besides, a possible way for systems to achieve chaos is the period-doubling cascade, which was first observed in quadratic maps by Myrberg [18–20]. In this phenomenon, as some experimental parameter of the considered system varies, a motion with a fundamental period changes to a periodic motion with twice the period of the original oscillation, and as the parameter is changed further, the same procedure occurs. The period-doubling bifurcation values of the parameter accumulates at a finite value, after which chaos is observable [3, 21, 22]. The period-doubling onset of chaos exhibits universal behavior [3], and it can be observed in many nonlinear systems, particularly in neural networks [23], ecological models [24], nonlinear circuits [25], and semiconductor lasers [26]. Throughout the paper, R, N and Z will denote the sets of real numbers, natural numbers and integers, respectively. The main object of the present investigation is the dynamics of the following system, y = Ay + G(t, y) + H(x(t)),

(1)

where the function G : R × Rn → Rn is continuous in both of its arguments and it is almost periodic in t uniformly for y ∈ Rn , the function H : Rm → Rn is continuous, and A is a constant n× n matrix with real entries all of whose eigenvalues have negative real parts. Equation (1) is obtained by perturbing the system u = Au + G(t, u),

(2)

with the term H(x(t)). The perturbation function x(t) will be explained in a detailed form in Section 3 and Section 4. Our purpose is to prove rigorously that if the perturbation is chaotic, then system (1) possesses chaos with infinitely many almost periodic motions. We will make use of coupled Duffing oscillators to illustrate the formation of the chaos. Moreover, the controllability of the generated chaos will be demonstrated numerically. In the present study, two different types of chaos formation will be presented. In the first one, solutions of a system of differential equations that possesses a period-doubling cascade will be utilized as the perturbation function x(t) in (1). Secondly, we will make use of chaotic sets of functions as the source for the perturbation x(t) to obtain Li-Yorke chaos with infinitely many almost periodic motions, which are separated from the scrambled set. The main difficulty of indicating almost periodicity in chaos lies in the fact that systems which generate chaos do this with periodical scenario. Being based on special input-output mechanisms, the chaos replication method introduced and developed in our studies [27–38] allows to superpose solutions with incommensurate periods and to apply chaotic perturbation to the system with purely almost periodic motion to create infinitely many unstable almost periodic motions embedded in the chaotic attractor. The paper [28] deals with the general technique of dynamical synthesis, which was developed in [39–42]. The study [32] was concerned with the extension of chaos in the sense of Devaney [1], Li-Yorke [2] and the one obtained through period-doubling cascade [3, 21, 22] for unidirectionally coupled systems. The appearance of infinitely many quasi-periodic motions in a chaotic attractor was revealed in [32]. For that purpose, a system with an asymptotically stable equilibrium point was influenced by two chaotic systems that possess infinitely many periodic motions with incommensurate periods giving rise to quasi-periodicity. However, in the present study, we make use of only a single source of perturbations to obtain chaos with infinitely many almost periodic motions, and the perturbed system has a different structure. The idea that unstable periodic motions embedded in a chaotic attractor can be replaced by more general types of regular motions stems from the investigations of Poincaré and Birkhoff [43]. In any neighborhood of a structurally stable Poincaré homoclinic orbit, there exist nontrivial hyperbolic sets containing a countable number of saddle periodic orbits and continuum of non-periodic Poisson stable orbits [4–6]. According to Seifert [8], there exists discrete chaos with infinitely many almost periodic motions. It was certified in the paper [7] that, in general, in place of a countable set of periodic solutions to form chaos, one can take an uncountable collection of Poisson stable motions which are dense in a quasiminimal set. This can also be


17

observed in the Horseshoe attractor [44]. Motivated by these studies, continuous chaotic attractors with infinitely many almost periodic motions are considered in the present paper. The concept of snap-back repellers for high dimensional maps was introduced in [45]. According to Marotto [45], if a multidimensional continuously differentiable map has a snap-back repeller, then it is Li-Yorke chaotic. Marotto’s Theorem was used in [46] to prove the existence of Li-Yorke chaos in a spatiotemporal chaotic system. Li-Yorke sensitivity, which links the Li-Yorke chaos with the notion of sensitivity, was studied in [47]. Moreover, generalizations of Li-Yorke chaos to mappings in Banach spaces and complete metric spaces were provided in [48–50]. In the present paper, we develop the concept of Li-Yorke chaos with infinitely many almost periodic motions, which are separated from the motions of a scrambled set, and provide a method for its formation. Almost periodic and in particular quasi-periodic motions have an important place in the theory of neural networks. In the book [51], the dynamics of the brain activity is considered as a system of many coupled oscillators with different incommensurable periods. According to Pasemann et al. [52], quasi-periodic solutions are noteworthy in biological and artificial systems since they are associated with various kinds of central pattern generators. Watanabe et al. [53] demonstrated that chaotic dynamics works as means to learn new patterns and increases the memory capacity of neural networks. The consideration of infinitely many almost periodic motions instead of periodic ones provides dynamics with a higher complexity. Therefore, our results may be useful for neural networks to obtain a memory with a larger capacity than that with periodic motions. The extension of chaos through unidirectional couplings has been considered in the theory of synchronization [54–62]. In our study, we do not consider the chaos synchronization problem, but we say that chaos with infinitely many almost periodic motions is generated. The asymptotic proximity of the solutions underlies the presence of synchronization. However, we do not take into account the solutions from the asymptotic point of view. Furthermore, in the synchronization of chaotic systems, one of the assumptions is that the response system is chaotic in the absence of the driving. On the contrary, in our case, the unperturbed system (2) possesses an asymptotically stable almost periodic solution, and therefore, it is non-chaotic. The remaining parts of the paper is organized as follows. In Section 2, we investigate the bounded solutions of (1) and their attractiveness property. Section 3 is devoted for chaos through a cascade of almost periodic solutions. In Section 4, we describe the Li-Yorke chaos with infinitely many almost periodic motions and deal with the theoretical results concerned with its appearance in system (1). An example is provided in Section 5 by means of unidirectionally coupled Duffing oscillators. In Section 6, the control of the extended chaos as well as the stabilization of tori embedded in the chaotic attractor of the coupled Duffing oscillators are numerically demonstrated. Finally, some concluding remarks are given in Section 7.

2 Preliminaries In the remaining parts of the paper, we will make use of the usual Euclidean norm for vectors and the norm induced by the Euclidean norm for square matrices. Since the matrix A in system (1)has eigenvalues all with negative real parts, one can verify that there exist positive numbers N and ω such that eAt ≤ Ne−ω t for all t ≥ 0. The following conditions are required. (C1) There exists a positive number L1 < y2 ∈ Rn ;

ω N

such that G(t, y1 ) − G(t, y2 ) ≤ L1 y1 − y2 for all t ∈ R, y1 ,

(C2) There exists a positive number MG such that supt∈R, y∈Rn G(t, y) ≤ MG . Utilizing the theory of quasilinear equations [63], it can be verified under the conditions (C1) and (C2) that for any given bounded function x(t), there exists a unique bounded on R solution φx(t) (t) of system (1), which satisfies the following relation,

18


ˆ

φx(t) (t) =

t −∞

eA(t−s) [G(s, φx(t) (s)) + H(x(s))]ds.

(3)

Moreover, for a fixed x(t), if y(t) is a solution of (1) with y(0) = y0 ∈ Rn , then the inequality φx(t) (t) − y(t) ≤ Nφx(t) (0) − y0 e(NL1 −ω )t holds for t ≥ 0 so that φx(t) (t) − y(t) → 0 as t → ∞. In other words, the bounded on R solution φx(t) (t) attracts all other solutions of (1). 3 Chaos through a cascade of almost periodic solutions In this section, we will deal with the presence of chaos in the dynamics of (1) through a cascade of almost periodic solutions. For that purpose, we will take advantage of a period-doubling cascade [3, 21, 22, 64, 65] of a system which will be used as the source for the perturbation function x(t) in (1). We will understand chaos in system (1) as the presence of sensitivity, which is the main ingredient of chaos [1, 13, 66], and the existence of infinitely many unstable almost periodic solutions in a bounded region. Let us consider the system x = F(t, x, μ ),

(4)

where μ is a parameter and the function F : R × Rm × R → Rm is continuous in all of its arguments. Suppose that there exists a positive number T such that F(t + T, x, μ ) = F(t, x, μ ) for all t ∈ R, x ∈ Rm and μ ∈ R. We assume that system (4) admits a period-doubling cascade [3, 21, 22], that is, there exist a finite number μ∞ and a sequence of period-doubling bifurcation values μ j , j ∈ N, satisfying μ j → μ∞ as j → ∞ such that for each j as the parameter μ increases or decreases through μ j , system (4) undergoes a period-doubling bifurcation, which gives rise to the formation of a new periodic solution with a twice period of the former one. At the parameter value μ = μ∞ , there exist infinitely many unstable periodic solutions of (4) all lying in a bounded region. The perturbation function x(t) in (1) will be provided by system (4) with μ = μ∞ , that is, it will be a solution of the system x = F(t, x, μ∞ ).

(5)

It is worth noting that the results of the present section are valid even if we replace the non-autonomous system (5) with the autonomous equation x = F(x, μ∞ ),

(6)

where F : Rm × R → Rm is a continuous function in all of its arguments. We suppose that system (5) ((6)) admits a chaotic attractor, let us say a set in Rm for (6). Fix x0 from the attractor, and take a solution x(t) of (6) with x(0) = x0 . Since x0 is a member of the chaotic attractor, we will call x(t) a chaotic solution [1, 3, 13, 66]. There exists a compact set Λ ⊂ Rm such that the trajectories of the chaotic solutions of (5) ((6)) lie inside Λ for all t. If we denote MH = maxx∈Λ H(x), then one can verify that supt∈R φx(t) (t) ≤ N(MGω+MH ) for each chaotic solution x(t) of (5) ((6)). The following conditions are needed throughout the section. (C3) There exists a positive number L2 such that H(x1 ) − H(x2 ) ≤ L2 x1 − x2 for all x1 , x2 ∈ Λ; (C4) There exists a positive number L3 such that H(x1 ) − H(x2 ) ≥ L3 x1 − x2 for all x1 , x2 ∈ Λ; (C5) There exists a positive number L4 such that F(t, x1 , μ∞ ) − F(t, x2 , μ∞ ) ≤ L4 x1 − x2 for all t ∈ R, x1 , x2 ∈ Λ. The next subsection is concerned with the extension of sensitivity.


3.1

19

Extension of sensitivity

Let us describe the sensitivity feature for system (5) as well as its replication by system (1). System (5) is called sensitive if there exist positive numbers ε0 and Δ such that for an arbitrary positive number δ0 and for each chaotic solution x(t) of (5), there exist a chaotic solution x(t) of the same system, t0 ∈ R and an interval J ⊂ [t0 , ∞) with a length no less than Δ such that x(t0 ) − x(t0 ) < δ0 and x(t) − x(t) > ε0 for all t ∈ J. We say that system (1) replicates the sensitivity of (5) if there exist positive numbers ε1 and Δ such that for an arbitrary positive number δ1 and for each chaotic solution x(t) of (5), there exist a chaotic solution x(t) of (5), t0 ∈ R and an interval J 1 ⊂ [t0 , ∞) with a length no less than Δ such that φx(t) (t0 ) − φx(t) (t0 ) < δ1 and φx(t) (t) − φx(t) (t) > ε1 for all t ∈ J 1 . The next assertion is about the sensitivity feature of system (1). Lemma 1. Under the conditions (C1) − (C5), system (1) replicates the sensitivity of (5). We omit the proof of Lemma 1 since it can be proved in a very similar way to Lemma 5.1 [32]. We will handle the presence of chaos through a cascade of almost periodic motions in system (1) in the following subsection. 3.2

Cascade of almost periodic solutions

We say that system (1) is chaotic through a cascade of almost periodic solutions if for each periodic solution x(t) of (5), system (1) possesses an almost periodic solution. One can verify by using the results of [67] that if x(t) is a periodic solution of (5), then the function G(t, y) + H(x(t)) in (1) is almost periodic in t uniformly for y ∈ Rn and the unique bounded on R solution φx(t) (t) of (1) is almost periodic. On the other hand, if x1 (t) and x2 (t) are two different periodic solutions of (5), then the corresponding almost periodic solutions φx1 (t) (t) and φx2 (t) (t) are different from each other under the condition (C4). Because system (5) is chaotic through period-doubling cascade, there exists a cascade of almost periodic solutions in the dynamics of (1). Consequently, system (1) possesses infinitely many almost periodic solutions in a bounded region. The instability of the existing almost periodic motions is ensured by Lemma 1. This result is mentioned in the following theorem. Theorem 2. Under the conditions (C1) − (C5), system (1) is chaotic through a cascade of almost periodic solutions. A corollary of Theorem 2 is as follows. Corollary 3. Under the conditions (C1) − (C5), the coupled system (5) + (1) is chaotic through a cascade of almost periodic solutions. In the next section, we will consider the formation of Li-Yorke chaos with infinitely many almost periodic motions.

4 Li-Yorke Chaos with Infinitely Many Almost Periodic Motions In the original paper [2], chaos with infinitely many periodic solutions, which are separated from the elements of a scrambled set, was introduced. We modify the Li-Yorke definition of chaos by replacing periodic motions by almost periodic ones, and prove its presence in system (1) rigorously. In opposition to the descriptions of regular motions such as periodic, quasi-periodic and almost periodic motions, one encounters with interaction of motions in order to describe the Li-Yorke chaos. Therefore, we need to introduce the concept of chaotic sets of functions.

20


Let Λ be a compact subset of Rm , and consider the set of uniformly bounded functions A whose elements are of the form x(t) : R → Λ. We suppose that A is an equicontinuous family on R. In this section, the perturbation function x(t) in system (1) will be provided from the elements of the collection A . A couple of functions (x(t), x(t)) ∈ A × A is called proximal if for an arbitrary small number ε > 0 and an arbitrary large number E > 0 there exists an interval J ⊂ R with a length no less than E such that x(t) − x(t) < ε for all t ∈ J. Besides, a couple of functions (x(t), x(t)) ∈ A × A is frequently (ε0 , Δ)−separated if there exist numbers ε0 > 0, Δ > 0 and infinitely many disjoint intervals each with a length no less than Δ such that x(t) − x(t) > ε0 for each t from these intervals. It is worth noting that the numbers ε0 and Δ depend on the functions x(t) and x(t). We say that a couple of functions (x(t), x(t)) ∈ A × A is a Li-Yorke pair if they are proximal and frequently (ε0 , Δ)−separated for some positive numbers ε0 and Δ. The definition of a Li-Yorke chaotic set with infinitely many almost periodic motions is as follows [68]. Definition 1. A is called a Li-Yorke chaotic set with infinitely many almost periodic motions if: i. There exists a countably infinite set R ⊂ A of almost periodic functions; ii. There exists an uncountable set D ⊂ A , the scrambled set, such that the intersection of D and R is empty and each couple of different functions inside D × D is a Li-Yorke pair; iii. For any function x(t) ∈ D and any almost periodic function x(t) ∈ R, the couple (x(t), x(t)) is frequently (ε0 , Δ)−separated for some positive numbers ε0 and Δ. To provide a rigorous study of the subject, we introduce the following set of functions, B = φx(t) (t) | x(t) ∈ A .

(7)

The following assertions, which are about the proximality and frequent separation features for system (1), can be proved in a similar way to Lemma 6.1 and Lemma 6.2 [32]. Lemma 4. Under the conditions (C1) − (C3),if a couple of functions (x(t), x(t)) ∈ A × A is proximal, then the same is true for the couple φx(t) (t), φx(t) (t) ∈ B × B. Lemma 5. Assume that the conditions (C1), (C2) and (C4) hold. If a couple of functions (x(t), x(t)) ∈ (ε0 , Δ)−separated for some positive numbers ε0 and Δ, then the couple of functions A × A is frequently φx(t) (t), φx(t) (t) ∈ B × B is frequently (ε1 , Δ)−separated for some positive numbers ε1 and Δ. The main result of the present section is mentioned in the following theorem. Theorem 6. Suppose that the conditions (C1) − (C4) are fulfilled. If the set A is chaotic in the sense of Definition 1, then the same is true for the set B. Proof. Let R be the set of almost periodic functions inside A . If x(t) belongs to R, then the function G(t, y) + H(x(t)) in (1) is almost periodic in t uniformly for y ∈ Rn [67]. In this case, the bounded solution φx(t) (t) of (1) is almost periodic [67]. Consider the set R = φx(t) (t) | x(t) ∈ R . According to condition (C4) there is a Since the set R is countably infinite, the set R is also countably one-to-one correspondence between R and R. infinite. Suppose that D is a scrambled set inside A . Define the set D = φx(t) (t) | x(t) ∈ D . is uncountable since the same is true for D. Moreover, the intersection of D and R is empty since the The set D intersection of D and R is empty.


21

Because each couple of different functions inside D × D is proximal, Lemma 4 implies the same feature × D. On the other hand, since each couple of different functions for each couple of different functions inside D (x(t), x(t)) ∈ D × D is frequently (ε0 , Δ)−separated for some positive numbers ε0 and Δ, Lemma 5 implies that ×D is frequently (ε1 , Δ)−separated for some positive numbers each couple of different functions (y(t), y(t)) ∈ D Therefore, the set ε1 and Δ. Additionally, the frequent separation feature holds also for each pair inside D × R. D is a scrambled set inside B. Consequently, B is a Li-Yorke chaotic set in accordance with Definition 1. Using the technique of the proof of Theorem 6, one can confirm that if the function G(t, y), which is used in the right hand side of system (1), is quasi-periodic (periodic) in t uniformly for y ∈ Rn , then under the conditions of Theorem 6, the set B is chaotic in the sense of Li-Yorke with infinitely many quasi-periodic (periodic) motions provided that the same is true for A . In the following section, the theoretical results of Section 4 will be supported by simulations. We will show how Theorem 6 can be utilized to obtain a Li-Yorke chaotic Duffing oscillator with infinitely many almost periodic motions. In order to construct the Li-Yorke chaotic set A , we will take into account the bounded solutions of a Duffing oscillator perturbed with a relay function whose switching moments are changing chaotically.

5 An example Let us consider the forced Duffing oscillator x + x +

11 x + 0.0003x3 = sin(2π t) + ν (t, θ0 ), 2

(8)

where ν (t, θ0 ) is the relay function defined as

ν (t, θ0 ) =

3.6, if θ2i < t ≤ θ2i+1 , 1.3, if θ2i−1 < t ≤ θ2i .

(9)

In the relay function (9), the sequence {θi } , i ∈ Z, of switching moments is defined through the equation θi = i + κi , in which the sequence {κi } , κ0 ∈ [0, 1], is generated by the logistic map such that

κi+1 = λ κi (1 − κi ),

(10)

where λ is a parameter. The interval [0, 1] is invariant under the iterations of (10) for the values of λ between 0 and 4 [69]. For λ = 3.9, the map (10) is Li-Yorke chaotic [2], and the family {ν (t, θ0 )} , θ0 ∈ [0, 1], is chaotic in the sense of Li-Yorke with infinitely many periodic motions [29]. Using the variables x1 = x and x2 = x , equation (8) can be written as a system in the form x1 = x2 , 11 x2 = − x1 − x2 − 0.0003x31 + sin(2π t) + ν (t, θ0 ). 2

(11)

According to the results of the paper [29], the set A consisting of the bounded on R solutions of (11) with λ = 3.9 is Li-Yorke chaotic such that it possesses infinitely many periodic solutions with periods 2p, p ∈ N, which are separated from the motions of the scrambled set. The chaoticity of the switching moments {θi } gives rise to the presence of chaos in system (11). The reader is referred to [28–31, 33, 34, 38] for further information about the dynamics of relay systems. In what follows, we will take into account system (11) with λ = 3.9. To illustrate the chaotic dynamics of (11), let us consider the solution of the system with the initial data x1 (t0 ) = 0.24, x2 (t0 ) = 0.16, where t0 = 0.61. The simulation results are shown in Figure 1, where one can observe the chaotic behavior of (11). The sequence {θi } of switching moments is used with θ0 = 0.61 in the simulation. It is worth noting that thebounded on R solutions of (11) lie inside the compact region Λ = (x1 , x2 ) ∈ R2 : 0 ≤ x1 ≤ 0.9, −1 ≤ x2 ≤ 1.3 .

22


x

1

1 0.5 0 0

10

20

30

40

50

60

70

80

0

10

20

30

40

50

60

70

80

t

x2

1 0 Ŧ1

t

Fig. 1 Li-Yorke chaos in system (11).

Next, let us consider the quasi-periodically forced Duffing oscillator 1 t 3 3 u + u + 2u + 0.0001u3 = sin( √ ) + sin(π t). 2 2 2 2

(12)

By means of the variables u1 = u and u2 = u one can reduce (12) to the system u1 = u2 ,

1 t 3 3 u2 = −2u1 − u2 − 0.0001u31 + sin( √ ) + sin(π t). 2 2 2 2

(13)

We perturb (13) with the solutions of (11), and set up the following system, y1 = y2 + 6 arctan(x1 (t)) 1 t 3 3 y2 = −2y1 − y2 − 0.0001y31 + sin( √ ) + sin(π t) + 2x2 (t). 2 2 2 2 System (14) is in the form of (1) with

⎛ A=⎝

0

−2 −

⎛ ⎜ G(t, y1 , y2 ) = ⎝

3 2

⎞ ⎠, ⎞

0

−0.0001y31 +

and H(x1 , x2 ) = The eigenvalues of the matrix A are − 34 ± i (y1 , y2 ) ∈ R2 . One can confirm that

1

√ 23 4 ,

⎟ ⎠, 1 t 3 sin( √ ) + sin(π t) 2 2 2

6 arctan(x1 ) 2x2

.

and the function G(t, y1 , y2 ) is quasi-periodic in t uniformly for √

√ ⎞ 23 23 cos( t) − sin( t) ⎟ −1 ⎜ 4 4 ⎟P , eAt = e−3t/4 P ⎜ √ √ ⎠ ⎝ 23 23 t) cos( t) sin( 4 4 ⎛

(14)


23

u1

1 0 Ŧ1

0

50

0

50

t

100

150

100

150

u2

1 0 Ŧ1

t

Fig. 2 The quasi-periodic behavior of system (13).

⎞ 0 1 ⎟ ⎜ P = ⎝ √23 3 ⎠ . − 4 4 Therefore, eAt ≤ Ne−ω t , t ≥ 0, where N = PP−1 ≈ 2.0029 and ω = 3/4. In order to show that system (13) possesses an asymptotically stable quasi-periodic solution, let us consider the solution of (13) with initial data u1 (0) = 0.1, u2 (0) = 0.2. The u1 and u2 coordinates of the solution are depicted in Figure 2, where one can observe that the represented solution approaches to the asymptotically stable quasi-periodic solution such that the system does not possess chaos. It can be 2numerically verified that the bounded on R solutions of (14) lie inside the compact region R = (y1 , y2 ) ∈ R : −1.1 ≤ y1 ≤ 4.7, −5.3 ≤ y2 ≤ 0.5 . Therefore, it is reasonable to consider the dynamics of (14) √ conditions (C1) − (C4) are valid for system (14) with MG = 2.0104, L1 = 0.006627, L2 = √ inside R, and 6 2, and L3 = 2. According to the theoretical results of Section 4, the set B consisting of the bounded on R solutions of (14) for which (x1 (t), x2 (t)) belongs to A is Li-Yorke chaotic with infinitely many quasiperiodic solutions, which are separated from the motions of the scrambled set. That is, the applied perturbation H(x1 (t), x2 (t)) effects (13) in such a way that Li-Yorke chaos takes place in the dynamics of (14). In system (14) we use the solution of (11) that is depicted in Figure 1, and we represent in Figure 3 the solution of (14) corresponding to the initial data y1 (t0 ) = 1.32, y2 (t0 ) = −2.25, where t0 = 0.61. Figure 3 supports the result of Theorem 6 such that system (14) possesses chaotic motions. Now, we will demonstrate that the chaos formation mechanism can be proceeded further. For that purpose, we take into account the Duffing equation ⎛

where

v + 2v + 4v + 0.0002v3 = 0,

(15)

which is equivalent to the system v1 = v2 , v2 = −4v1 − 2v2 − 0.0002v31 ,

(16)

in view of the variables v1 = v and v2 = v . One can verify that (16) admits an asymptotically stable equilibrium point. We perturb system (16) with the solutions of (14) to constitute the system 1 z1 = z2 + y1 (t) + sin(y1 (t)), 2 1 1 z2 = −4z1 − 2z2 − 0.0002z31 + y2 (t) + y32 (t). 5 10

(17)

24


y1

4 2 0 0

10

20

30

40

50

60

70

80

90

100

0

10

20

30

40

50

60

70

80

90

100

t

y2

0 Ŧ2 Ŧ4

t

Fig. 3 The graphs of the y1 and y2 coordinates of system (14).

z1

2 0

Ŧ2

0

10

20

30

40

50

60

70

80

90

100

0

10

20

30

40

50

60

70

80

90

100

t

z2

2 0 Ŧ2 Ŧ4 Ŧ6

t

Fig. 4 The chaotic behavior of system (17).

It is worth noting that (17) is in the form of (1) with A=

0

1

−4 −2

,

√ whose eigenvalues are −1 ± i 3. It can be shown in a similar way to system (14) that conditions (C1)− (C4) are valid also for (17). According to our results, the collection C of bounded on R solutions of (17) for which (y1 (t), y2 (t)) belongs to B is a LiYorke chaotic set with infinitely many quasi-periodic motions. Utilizing the solution of (14) that is shown in Figure 3 as the perturbation in system (17), we depict in Figure 4 the solution of (17) with z1 (t0 ) = 0.37, z2 (t0 ) = −3.71, where t0 = 0.61. Figure 4 reveals that the system (17) is Li-Yorke chaotic. To confirm one more time that the 6−dimensional system (11) + (14) + (17) admits a chaotic attractor, we represent in Figure 5 the projection of the trajectory of (11) + (14) + (17) corresponding to the initial data x1 (t0 ) = 0.24, x2 (t0 ) = 0.16, y1 (t0 ) = 1.32, y2 (t0 ) = −2.25, z1 (t0 ) = 0.37, z2 (t0 ) = −3.71, where t0 = 0.61, on the x2 − y2 − z2 space. The simulation result confirms that system (11) + (14) + (17) possesses a chaotic attractor.


25

6 4

z2

2 0 Ŧ2 Ŧ4 Ŧ6 Ŧ1

0 Ŧ2

0

x

2

Ŧ4

1

y

2

Ŧ6

Fig. 5 The projection of the chaotic trajectory of system (11) + (14) + (17) on the x2 − y2 − z2 space.

6 Control of tori In this section, we will numerically demonstrate the stabilization of the unstable quasi-periodic motions embedded in the chaotic attractor of the unidirectionally coupled Duffing oscillators (11) + (14) + (17). We suppose that to stabilize the quasi-periodic solutions of (14) and (17) it is sufficient to control the chaos of system (11). For that purpose we will make use of the Ott-Grebogi-Yorke (OGY) control method [9] applied to the logistic map (10), which gives rise to the presence of chaos in (11). We proceed by briefly explaining the OGY control method for the map [70]. Suppose that the parameter λ in the logistic map (10) is allowed to vary in the range [3.9 − ε , 3.9 + ε ], where ε is a given small number. Consider an arbitrary solution {κi } , κ0 ∈ [0, 1], of the map and denote by κ ( j) , j = 1, 2, . . . , p, the target unstable p−periodic orbit to be stabilized. In the OGY control method [70], at each iteration step i after the control mechanism is switched on, we consider the map (10) with the parameter value λ = λ¯ i , where [2κ λ¯ i = 3.9(1 +

( j) − 1][κ

i−κ ( j) κ [1 − κ ( j)]

( j) ]

),

(18)

provided that the number on the right-hand side of the formula (18) belongs to the interval [3.9 − ε , 3.9 + ε ]. In ( j)

( j)

−1][κi −κ ] to the parameter λ = 3.9 of the logistic other words, we apply a perturbation in the amount of 3.9[2κκ( j) [1− κ ( j) ] map, if the trajectory {κi } is sufficiently close to the target periodic orbit. This perturbation makes the map behave regularly so that at each iteration step the orbit κi is forced to be located in a small neighborhood of a previously chosen periodic orbit κ ( j) . Unless the parameter perturbation is applied, the orbit κi moves away from ( j) −1][κ −κ ( j) ] i κ ( j) due to the instability. If | 3.9[2κκ( j) [1− | > ε , we set λ¯ i = 3.9, so that the system evolves at its original κ ( j) ] parameter value, and wait until the trajectory {κi } enters a sufficiently small neighborhood of the periodic orbit ( j)

( j)

−1][κi −κ ] κ ( j) , j = 1, 2, . . . , p, such that the inequality −ε ≤ 3.9[2κκ( j) [1− ≤ ε holds. If this is the case, the control of κ ( j) ] chaos is not achieved immediately after switching on the control mechanism. Instead, there is a transition time before the desired periodic orbit is stabilized. The transition time increases if the number ε decreases [54]. An unstable 2−periodic solution of (11) can be stabilized by controlling the 1−periodic orbit of the logistic map (10), i.e. the fixed point 2.9/3.9 of the map. We apply the OGY control method around the fixed point of the logistic map to stabilize the corresponding quasi-periodic solution of the 6−dimensional system (11) + (14) + (17). Figure 6 represents the x2 , y2 and z2 coordinates of (11) + (14) + (17) corresponding to the initial data x1 (t0 ) = 0.24, x2 (t0 ) = 0.16, y1 (t0 ) = 1.32, y2 (t0 ) = −2.25, z1 (t0 ) = 0.37, z2 (t0 ) = −3.71, where t0 = 0.61. The control mechanism is switched on at t = θ35 and switched off at t = θ95 . The value ε = 0.06 is used in the simulation. There is a transition time such that the control becomes dominant approximately at t = 53,

26


x2

1 0 Ŧ1 0

20

40

60

80

100

120

140

160

180

200

0

20

40

60

80

100

120

140

160

180

200

0

20

40

60

80

100

120

140

160

180

200

t

y2

0 Ŧ2

z2

Ŧ4

t

4 2 0 Ŧ2 Ŧ4 Ŧ6

t

Fig. 6 The application of the OGY control method around the fixed point 2.9/3.9 of the logistic map (10) for the stabilization of quasi-periodic motions of the coupled Duffing oscillators (11) + (14) + (17). The value ε = 0.06 is used in the simulation. Control is switched on at t = θ35 and switched off at t = θ95 . 0

z2

Ŧ1 Ŧ2 Ŧ3 Ŧ4 Ŧ1 Ŧ0.5

x

0

2

0.5 1

Ŧ4

Ŧ3.5

Ŧ3

Ŧ2.5

y

Ŧ2

Ŧ1.5

Ŧ1

2

Fig. 7 The stabilized torus of system (11) + (14) + (17).

and it prolongs approximately till t = 148, after which chaos develops again. It is seen in Figure 6 that the 2−periodic solution of system (11) is controlled for 53 ≤ t ≤ 148, and accordingly the corresponding quasiperiodic solutions of systems (14) and (17) are stabilized in the same interval of time. Moreover, we illustrate the stabilized torus of system (11) + (14) + (17) in Figure 7. Both of the Figures 6 and 7 support the result of Theorem 6 such that they manifest the presence of quasi-periodic motions embedded in the chaotic attractor of (11) + (14) + (17). 7 Conclusions The possibility of replacing infinitely many periodic solutions by more general types of regular motions in a chaotic attractor was considered in the studies [7, 8, 43]. The present paper provides a method to obtain chaotic attractors with infinitely many almost periodic motions instead of periodic ones. In this way, the complexity of chaos increases.


27

Chaos in the sense of Li-Yorke and the one obtained through a cascade of almost periodic motions are considered in the present study. One of the advantages of the proposed procedure is the controllability of the obtained chaos. A control technique for stabilizing the unstable almost periodic motions is presented, and control of tori is numerically demonstrated by means of the OGY control method [9] in unidirectionally coupled Duffing oscillators. The results reveal that the OGY control method is suitable to stabilize not only periodic motions but also almost periodic ones. Other chaos control methods such as the Pyragas method [71] can also be used for that purpose. It is worth noting that the presented method is appropriate to obtain route to chaos by intermittency [72] in which regular behavior of almost periodic type is interrupted by sporadic bursts of chaotic behavior. According to Watanabe et al. [53] chaos increases the capacity of memorizing in neural networks. One can suppose that chaos with almost periodic motions provides a memory with a larger capacity than that with periodic motions. Therefore, the obtained results may be useful for the theory of neural networks and investigations of brain activities.

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Bäcklund Transformation and Quasi-Integrable Deformation of Mixed Fermi-Pasta-Ulam and Frenkel-Kontorova Models Kumar Abhinav1†, A Ghose Choudhury2† , Partha Guha1† 1 2

SN Bose National Centre for Basic Sciences JD Block, Sector III, Salt Lake, Kolkata 700106, India Department of Physics, Surendranath College, 24/2 Mahatma Gandhi Road, Calcutta 700009, India Submission Info Communicated by D. Volchenkov Received 20 January 2017 Accepted 26 March 2017 Available online 1 April 2018 Keywords Fermi-Pasta-Ulam equation Frenkel-Kontorova Models Bäcklund transformation Quasi-integrable deformation

Abstract In this paper we study a non-linear partial differential equation (PDE), proposed by Kudryashov [arXiv:1611.06813v1[nlin.SI]], using continuum limit approximation of mixed Fermi-Pasta-Ulam and Frenkel-Kontorova Models. This generalized semi-discrete equation can be considered as a model for the description of non-linear dislocation waves in crystal lattice and the corresponding continuous system can be called mixed generalized potential KdV and sine-Gordon equation. We obtain the Bäcklund transformation of this equation in Riccati form in inverse method. We further study the quasi-integrable deformation of this model.


1 Introduction Recently Kudryashov [1], using the continuous limit approximation, has derived a nonlinear partial differential equation for the description of dislocations in a crystalline lattice which can be regarded as a generalization both the Frenkel-Kontorova [2,3] and the Fermi-Pasta-Ulam [4–6] models. This generalized model can be considered to describe nonlinear dislocation waves in the crystal lattice following, d 2 yi = (yi+1 − 2yi + yi−1 )[k + α (yi+1 − yi−1 ) + β (y2i+1 + y2i + y2i−1 − yi+1 yi − yi+1 yi−1 − yi yi−1 )] dt 2 2π yi ), − f0 sin( a

(1)

for all (i = 1, ..., N). Here yi denotes the displacement of the i-th mass from its original position, t is time, k, α , β , f0 and a are constant parameters of the system. This equation boils down to Frenkel-Kontorova form for the description of dislocations in the rigid body when α = 0 and β = 0. In the case of f = 0 and β = 0 the system of equations becomes the well-known Fermi-Pasta-Ulam model. One must note that the Fermi-PastaUlam model N → ∞ and h → 0 is transformed to the Korteweg-de Vries (KdV) equation. † Corresponding

author. Email address: [email protected](Kumar Abhinav), [email protected](A Ghose Choudhury), [email protected](Partha Guha) ISSN 2164-6376, eISSN 2164-6414/$-see front materials © 2018 L&H Scientific Publishing, LLC. All rights reserved. DOI:10.5890/DNC.2018.03.003

32

Kumar Abhinav et al. / Discontinuity, Nonlinearity, and Complexity 7(1) (2018) 31–41

Kudryashov showed that in the continuum limit the semi-discrete equation takes the following form: uxt + α ux uxx + 3β u2x uxx + γ uxxxx = δ sin u.

(2)

In the special case when α = 0 and γ = 2β it is shown that one can cast the equation in the AKNS scheme with the Lax pair given by A B −iλ q , M= , (3) L= r iλ C −A where q = −r = −ux /2 and the functions A, B and C are

δ A = − cos u(iλ )−1 + β u2x (iλ ) + 8β (iλ )3 , 4 δ β B = − sin u(iλ )−1 + (β uxxx + u3x ) − 2β uxx (iλ ) + 4β ux (iλ )2 , 4 2 δ β C = − sin u(iλ )−1 − (β uxxx + u3x ) − 2β uxx (iλ ) − 4β ux (iλ )2 . 4 2 Equation (2) follows from the usual zero-curvature condition, viz Ftx = Lt − Mx + [L, M] = 0.

(4)

As (2) is a continuous system one may derive Bäcklund transformations [7, 8] for the equation using the method proposed by Konno and Wadati [9, 10] which makes use of the Riccati equation. In the next section we study this procedure. Real physical systems contain finite number of degrees of freedom, thereby prohibiting the corresponding field-theoretical models from being integrable in principle, as the latter requires infinite conserved quantities. However, they are known to posses physically obtainable solitonic states, considerably similar in structure to integrable ones, like sine-Gordon (SG) [11]. Therefore, the study of physical continuous systems can be motivated as slightly deformed integrable models. Recently [12,13], the SG model was deformed into an approximate system, with a finite number of conserved charges. The corresponding connection (curvature) was almost flat, yielding an anomaly instead of the usual zero-curvature condition. This system was, therefore, dubbed as quasiintegrable (QI). In a recent paper [14] we obtain the quasi-integrable deformation of the KdV equation. In the present paper we outline the quasi-deformation of the new equation proposed by Kudryashov. 1.1

Lagrangian and Hamiltonian

The integrable sector of Eq. 2, represented as, uxt + 3β u2x uxx + 2β uxxxx = δ sin u,

(5)

∂L ∂L ∂L ∂L , )t + ( )x + ( )xxx = ∂ ut ∂ ux ∂ uxxx ∂u

(6)

follows from a Euler-Lagrange form, ( corresponding the Lagrangian,

β 1 (7) L = ux ut + u4x + β ux uxxx − δ cos u. 2 4 Therefore, the system described by Eq. 5 has a Lagrangian structure with respect to the variable u. However, the presence of higher order derivatives therein depicts time-evolution of ux instead. However, the canonical variable is identified to be u, leading to a Hamiltonian through the Legendre transformation, H :=

∂L β − L ≡ δ cos u − u4x − β ux uxxx . ∂ ut 4

(8)


33

2 Bäcklund transformation using the Riccati equation Classical stable solutions of integrable systems are of paramount interest owing to their Physical realizability. The standard way to arrive at such a solution is through Bäcklund transformation [8]. For this purpose, we write the scattering problem as, ∂ ψ1 ψ1 −iλ q = , (9) ψ2 r iλ ∂ x ψ2 ∂ ψ1 ψ1 A B = . (10) ψ2 C −A ∂ t ψ2 The consistency of (9) and (10) yields the equation (4) with the eigenvalues λ being time independent. By introducing the function ψ1 , (11) Γ := ψ2 we obtain from (9) and (10) the Riccati equations:

∂Γ = −2iλ Γ + q − rΓ2 , (12) ∂x ∂Γ = B + 2AΓ −CΓ2. (13) ∂t To derive a Bäcklund transformation one seeks a transformation Γ −→ Γ which satisfies an equation identical to (12) with q(x,t) replaced by (14) q (x,t) = q(x,t) + f (λ , Γ), for some suitable function f . Then upon eliminating Γ between (12) and (14) one arrives at the desired Bäcklund transformation. In our case as r(x,t) = −q(x,t) = ux /2 we shall take Γ =

1 ∂ and q (x,t) = q(x,t) − 2 tan−1 Γ. Γ ∂x

(15)

As

∂ (ΓΓ ) = −4iλ + (q + q )(Γ + Γ ), ∂x (where use has been made of (12) and its corresponding similar version with q and Γ ) we see that 0=

2Γ ∂ q + q = , q − q = 2 tan−1 Γ. 2iλ 1 + Γ2 ∂x

(16)

From the latter equation we notice that as r(x,t) = −q(x,t) = ux /2, 1 u − u ). tan−1 Γ = (u − u), or Γ = tan( 4 4 On the other hand the first part of (16) with Γ as given above yields, 2Γ u − u q + q ), = = sin( 2iλ 1 + Γ2 2 whence,

u − u ). 2 To find the time-part we use the form of Γ found above in (13) to obtain after simplification ux + ux = 4iλ sin(

u − u u − u ) − 2(C + B) cos( ). 2 2 Equations (17) and (18) constitute the desired Bäcklund transformation. ut − ut = 2(C − B) + 4A sin(

(17)

(18)

34


One-Soliton Solution: In order to construct a one-soliton solution of (2) from the Bäcklund transformation given by (17) and (18) we set λ = iμ , μ ∈ R and note that as u = 0 is a trivial solution of (2) we have u ux = 4μ sin , 2

ut = −4A sin

u 2

with

A=

It now follows that u = ±4 arctan (eθ ),

where

θ = 2μ x −

δ − 8β μ 3 . 4μ

(19)

δt + 16β μ 3t, 2μ

which matches the solution given in [1] when η = 2μ , δ → −δ and β → −β . 3 Derivation of the conservation laws Given the Lax pair one can quite easily derive an infinite number of conserved quantities for equation (2). Firstly it follows from the zero-curvature condition that Ax + rB − qC = 0,

(20)

qt − Bx − 2(iλ )B − 2qA = 0,

(21)

rt −Cx + 2(iλ )C + 2rA = 0.

(22)

Using (13) and the above set of equations one easily derives the following equation

∂ ∂ (rΓ) = (−A +CΓ), ∂t ∂x

(23)

which has the general form of an conservation law for the conserved densities and the flows. In fact from (12) we have (24) r[(rΓ)/r]x = −2(iλ )(rΓ) + qr − (rΓ)2 , where the suffix represents the usual partial differentiation with respect to x. We may expand (rΓ) in inverse powers of (iλ ) as rΓ =

∞

∑ fn (iλ )−n ,

(25)

n=1

which when inserted into (24) yields the following recurrence relation for the conserved densities, namely −2 fs+1 = r(

n−1 fs )x − qrδs,0 + ∑ fq fs−q , s = 0, 1, 2, . . . , r q=1

(26)

thereby confirming integrability of the system by yielding infinite number of conserved charges. The first few these conserved densities, at the lowest order, are given by,

f4 = − Consequently (23) becomes

1 f1 = qr, 2 1 f2 = − 2 rqx , 2 1 1 f3 = 3 rqxx − 3 (qr)2 , 2 2

(28)

1 ((qxxx − (q2 r)x ) − 2qrqx ). 24

(30)

(27)

(29)


∞

35

∂

∑ fn,t (iλ )−n = ∂ x [−A−1(iλ )−1 − A1(iλ ) − A3(iλ )3

n=1

∞

+(C−1 (iλ )−1 +C0 +C1 (iλ ) +C2 (iλ )2 ) ∑ ( n=1

fn )(iλ )−n )]. r

(31)

where the coefficients of the various powers of (iλ ) are as follows:

δ A1 = − cos u, A1 = β u2x , A3 = 8β , 4 δ β B−1 = C−1 = − sin u, B0 = −C0 = β uxxx + u3x , 4 2 B1 = C1 = −2β uxx , B2 = −C2 = 4β ux . It can be verified that for all positive powers of (iλ ) this relation are identically satisfied, while for various negative powers of (iλ ) we have the following relations: (iλ )−1 :

1 3β 4 ∂ ∂ δ β β (− u2x ) = ( cos u + ux uxxx − u2xx + u ), ∂t 8 ∂x 4 2 4 16 x

(32)

which leads to Eq. 5 itself. The next order continuity equation has the form, (iλ )−2 :

∂ ∂ [−(u2x )x ] = ux −2δ sin u + 5β uxx u2x + 4β uxxxx , ∂t ∂x

(33)

and so on.

4 QI deformation The most trivial way to QI-deform the present system is by deforming the periodic sin u function, like that in Ref. [12, 13] or any other way. Incorporating this into the time-component M of the Lax pair as usual, only the first terms of O (1/λ ) in B and C changes. As only C appears in the continuity Eq. 23, and further, as its O (1/λ ) part does not contribute while obtaining Eq. 32 from Eq. 31, the O (1/λ ) conserved quantity remains conserved even after the deformation. However, the higher order equations coming from Eq. 31 gets effected and hence they do not support conserved charges in general. Thus, quasi-integrability is achieved by definition. Moreover, the deformation need not to be small or finite, as far as Eq. 32 is concerned. As an attempt for non-trivial QI deformation of the present system, we opt to deform the highest powered ε term in ux in the temporal component of the Lax pair: u3x → u3+ x . This exclusively changes the C0 contribution in Eq. 31, leading to 1 ∂ ∂ δ β β β (− u2x ) = [ cos u + ux uxxx − u2xx + u4x (2uεx + 1)], ∂t 8 ∂x 4 2 4 16

(34)

that falls back to Eq. 32 for ε → 0. Here also, even finite values of ε will work for conservation of the O (1/λ ) charge. In fact, as the RHS of Eq. 23 is a total derivative, as long as a deformation does not induce boundary non-zeros, the system will remain integrable.

36


Thus, only sensible and simplest way to make the system QI for sure is to deform sin u in B and C in a way that it leads to non-zero boundary value, as the O (1/λ ) charge remains unaffected, and thus conserved still. This requirement is naturally satisfied by a shift deformation of the type,

δ (B, C)−1 (= − sin u) → (B, C)−1 + D−1 ; 4

D−1 :=

∑

r

m=1 fm−1

gm (u),

(35)

where gm (u) are functions of u and its derivatives, which is finite at the boundary. Then, for m = n + 1, n representing the summation index in Eq. 31, there will be a non-vanishing term on the RHS of Eq. 31. For m < n + 1, however, gm will be multiplied with positive powers of ux and its derivatives, leading to conservation again. For m > n + 1, negative powers of ux and its derivatives will come into play, leading to infinities (nonconserved charges again). One can very well truncate the vale of m to avoid such infinities. All in all, quasiintegrability will be obtained as desired. One can very well attribute this deformation to the level of the Lagrangian (Eq. 7) or the Hamiltonian (Eq. 8). As discussed in subsection 1.1, although the system dynamics contains higher derivatives, the canonical variable is still u, making δ sin u equivalent to the ‘potential’ of the system. Therefore, the above deformation of Eq. 35, being QI, in principle corresponds to that of the sine-Gordon system [12, 13]. However, the present deformation needs to be of shift nature (subsec. 1.1), unlike the power modification of the previous cases. 4.1

Loop-algebraic treatment

From Eq. 3 the Lax pair can be re-expressed as, L = −iλ σ3 + qσ+ + rσ−

and

M = Aσ3 + Bσ+ +Cσ− ,

(36)

where the Pauli matrices σ3,± satisfy SU(2) algebra: [σ3 , σ± ] = ±2σ±

and

[σ+ , σ− ] = σ3 ,

(37)

allowing for the construction of the SU(2) loop algebra [14],

m m+n , = 2F2,1 bn , F1,2

bn = λ n σ3 ,

[F1n , F2m ] = λ bm+n ; where, 1 1 F1n = √ λ n (λ σ+ − σ− ) and F2n = √ λ n (λ σ+ + σ− ) . 2 2

(38)

Such algebraic structure enables sl(2) gauge rotation of the Lax pair, elegantly demonstrating the quasi-integrability of the system [12]. We now consider the QI deformation of Eq. 35, modifying the curvature defined through the zero-curvature condition in Eq. 4; leading to, 1¯ 1¯ σ+ + E(u) σ+ + X σx , Ftx → F¯tx = − E(u) 2 2

σx = σ+ + σ− ,

(39)

¯ where E(u) is the Euler function yielding the Quasi-modified equation while equated to zero: uxt + 3β u2x uxx + 2β uxxxx = δ sin u − 4D−1 ,

(40)

and X is the anomaly term given as, X = iD−1,x .

(41)


37

Gauge transformation: Based on the standard gauge-fixing of the QI systems [12–14], we undertake a gauge transformation with respect to the operator, g = exp(

∞

∑

Jn ),

where,

Jn = an1 F1n + an2 F2n .

(42)

n=−∞

The coefficients an1,2 are to be chosen such that the new spatial Lax pair component L¯ = gLg−1 + gx g−1 contains only bn s: (43) L¯ ≡ ∑ βLn bn , n

making it diagonal in the SU(2) basis. From the BCH formula: eX Ye−X = Y + [X,Y ] +

1 1 [X, [X,Y ]] + [X, [X, [X,Y ]]] + · · · , 2! 3!

the gauge-transformed spatial component takes the form, 1 1 L¯ = Jn,x + L + [Jn, L] + [Jm , [Jn , L]] + [Jl , [Jm , [Jn , L]]] + · · · , 2! 3!

(44)

with summations understood over all semi-positive integers. Few of the lowest order commutators are, q r [Jn , L] = 2iλ (an1 F2n + an2 F1n ) + √ (an1 − an2 )bn + √ (an1 + an2 )bn+1 , 2 2 q n 1 m n m n m+n+1 n m+n m+n [Jm , [Jn , L]] = iλ (a1 a1 − a2 a2 )b − √ (a1 − a2 )(am + am ) 1 F2 2 F1 2! 2 r m+n+1 m+n+1 + am ), − √ (an1 + an2 )(am 1 F2 2 F1 2 2 1 n m n l l+m+n+1 [Jl , [Jm , [Jn , L]]] = −i λ (am + al2 F1l+m+n+1 ) 1 a1 − a2 a2 )(a1 F2 3! 3 q l m l+m+n+1 − √ (an1 − an2 )(al1 am 1 − a2 a2 )b 3 2 r l m l+m+n+2 − √ (an1 + an2 )(al1 am , 1 − a2 a2 )b 3 2 .. .

(45)

n s leads to the order-by-order consistency relations: The gauge-fixing condition of vanishing coefficients for F1,2

O(F10 ) : O(F20 ) : O(F11 ) : O(F21 ) :

q r q a01,x = √ + √ (a01 − a02 )a02 − 2iλ a02 − √ , 2 2 2λ q 0 r q 0 0 0 0 a2,x = − √ + √ (a1 − a2 )a1 − 2iλ a1 − √ , 2 2 2λ r q a11,x = √ (a01 + a02 )a02 + √ [(a01 − a02 )a12 + (a11 − a12 )a02 ] − 2iλ a12 + · · · , 2 2 r 0 q 1 0 0 a2,x = √ (a1 + a2 )a1 + √ [(a01 − a02 )a11 + (a11 − a12 )a01 ] − 2iλ a11 + · · · , 2 2 .. .

(46)

38


leaving behind the coefficients for bn s as, q βL0 = −iλ + √ a01 − a02 , 2 q 1 q r 1 βL = √ a1 − a12 + √ (a01 + a02 ) + [iλ − √ (a01 − a02 )] a01 a01 − a02 a02 , 2 2 3 2 q r q βL2 = √ (a21 − a22 ) + √ a11 + a12 + 2iλ a01 a11 − a02 a12 − √ [2 a01 − a02 a01 a11 − a02 a12 2 2 3 2 0 0 0 0 r 0 1 1 0 0 0 +(a1 − a2 ) a1 a1 − a2 a2 ] − √ a1 + a2 a1 a1 − a02 a02 + · · · 3 2 .. .

(47)

Therefore, the transformed spatial Lax component L¯ of Eq. 43 is completely determined [12–14]. The gaugetransformed temporal Lax component, M¯ = gMg−1 + gt g−1 = ∑ [βMn bn + ϕ1n F1n + ϕ2n F2n ] ,

(48)

n

can also be evaluated similarly. Some of the lowest order commutators are, B C [Jn , M] = −2A (an1 F2n + an2 F1n ) + √ (an1 − an2 ) bn + √ (an1 + an2 ) bn+1 , 2 2 B 1 n m n m+n+1 m+n m+n [Jm , [Jn , M]] = −A (am − √ (an1 − an2 ) am + am 1 a1 − a2 a2 ) b 1 F2 2 F1 2! 2 C n m+n+1 m+n+1 + am , − √ (a1 + an2 ) am 1 F2 2 F1 2 2 1 n m n l l+m+n+1 [Jl , [Jm , [Jn , M]]] = A (am + al2 F1l+m+n+1 ) 1 a1 − a2 a2 ) (a1 F2 3! 3 B l m l+m+n+1 − √ (an1 − an2 )(al1 am 1 − a2 a2 )b 3 2 C l m l+m+n+2 − √ (an1 + an2 )(al1 am , 1 − a2 a2 )b 3 2 .. .

(49)

that leads to the lowest order coefficients, B βM0 = A + √ a01 − a02 , 2 C B B βM1 = √ a11 − a12 + √ a01 + a02 − [A + √ (a01 − a02 )](a01 a01 − a02 a02 ), 2 2 3 2 B C B βM2 = √ a21 − a22 + √ a11 + a12 − 2A a01 a11 − a02 a12 − √ [2 a01 − a02 a01 a11 − a02 a12 2 2 3 2 1 C + a1 − a12 a01 a01 − a02 a02 ] − √ a01 + a02 a01 a01 − a02 a02 + · · · 3 2 .. .

(50)


39

B C B ϕ10 = a01,t − √ − √ a01 − a02 a02 − 2Aa02 + √ , 2 2 2λ C B a01 − a02 a12 + a11 − a12 a02 − 2Aa12 + · · · , ϕ11 = a11,t − √ a01 + a02 a02 − √ 2 2 .. .

(51)

C B B ϕ20 = a02,t + √ − √ a01 − a02 a01 − 2Aa01 − √ , 2 2 2λ C B a01 − a02 a11 + a11 − a12 a01 − 2Aa11 + · · · , ϕ21 = a12,t − √ a01 + a02 a01 + √ 2 2 .. .

(52)

The transformed curvature:

The above gauge transformation further yields the curvature as, F¯tx = gFtx g−1 ≡ X

∑ ( f0nbn + f1nF1n + f2nF2n ) ,

(53)

n

with the lowest order commutators, X [Jn , Ftx ] = √ [(an1 − an2 )bn + (an1 + an2 )bn+1 ], 2 X 1 m+n m+n [Jm , [Jn , Ftx ]] = − √ [(an1 − an2 )(am + am ) 1 F2 2 F1 2! 2 m+n+1 m+n+1 + am )], +(an1 + an2 )(am 1 F2 2 F1 X 1 l m n n l+m+n+1 [Jl , [Jm , [Jn , Ftx ]]] = − √ (al1 am + (an1 + an2 )bl+m+n+2 ], 1 − a2 a2 )[(a1 − a2 )b 3! 3 2 .. .

(54)

It is to be noted that the rotated curvature obtained above is on-shell, i.e., owing to the gauge-fixing above, Eq. 40 can be applied. Therefore, it depends only on the anomaly function X of the QI deformation, and duly vanishes for X = 0. A few lowest order co-efficients of the curvature are expressed below: 1 f00 = √ (a01 − a02 ), 2 1 1 1 f01 = √ (a11 − a12 ) + √ (a01 + a02 ) − √ (a01 − a02 )(a01 a01 − a02 a02 ), 2 2 3 2 1 1 1 f02 = √ (a21 − a22 ) + √ (a11 + a12 ) − √ [2(a01 − a02 )(a01 a11 − a02 a12 ) 2 2 3 2 +(a01 a01 − a02 a02 ){(a11 − a12 ) + (a01 + a02 )}] + · · · .. .

(55)

40


1 0 f1,2 = λ −1 ∓ 1 − √ a01 − a02 a02,1 , 2 1 1 a01 − a02 a12,1 + a11 − a12 a02,1 + a01 + a02 a02,1 , = −√ f1,2 2 1 1 2 f1,2 a1 − a12 a12,1 + a01 + a02 a12,1 + a11 + a12 a02,1 + · · · = −√ 2 .. .

(56)

¯ M] ¯ of the rotated curvature, by substituting the coefficients obFinally, from the definition F¯tx = L¯ t − M¯ x + [L, tained above, one finds the consistency conditions: n n βL,t − βM,x = X f0n

and

n ϕ−,x = −X f−n − 2βLn ∑ ϕ−m λ m ,

(57)

m

where,

ϕ−n := ϕ1n − ϕ− 2n

f−n := f1n − f2n ,

and

entirely evaluate all the coefficients with the aid of the gauge-fixing conditions in Eq.s 46 [14]. Quasi-conservation:

Following the treatment in Ref.s [12–14], the lowest order quasi-conserved charge is, ˆ ˆ 1 0 0 (58) Q := βL ≡ (−iλ + √ qa0− ); a0− := a01 − a02 , 2 x x

which is expected to be conserved for the system to be QI. This is consistent with the definition of the continuity expression in Eq. 32. From the lowest order (n = 0) contribution of first of the Eq.s 57, ˆ ˆ ˆ 0 1 1 i dQ0 0 √ √ √ = X a− = D−1,x a0− , (59) qa− t ≡ dt 2 x 2 x 2 x modulo vanishing total derivatives of functions of u, which is sensibly assumed to vanish asymptotically. For a given ux , a0− is uniquely determined from the gauge-fixing and consistency conditions. Then, it is always possible to choose the deformation functions as, gm (u) =

fm−1 0 p a− , r

p > 0,

(60)

for all m = 1, 2, 3, · · · , leading to, i 1 dQ0 =√ ∑ dt 2 m 1+ p

ˆ x

a0−

1+p x

≡ 0,

(61)

leading to a conserved charge all the time. This ensures at least one conserved charge for the deformed system, and thus, quasi-integrability. This has been obtained independent to the more comprehensive direct observation of the previous section without considering the loop algebraic structure.

5 Conclusions We have considered the combination of two well known dynamical systems, namely, Frenkel-Kontorova and Fermi-Pasta-Ulam models. This new dynamical system, proposed by Kudryashov, has been obtained by taking the continuum limit approximation for N → ∞ and h → 0. This continuous equation becomes the mixture of generalized potential KdV equation and the sine-Gordon equation. Using Wadati-Konno formalism we have


41

studied the Bäcklund transformation from Riccati form of inverse method. In the second half of the paper we have studied the quasi-integrable deformation of this new equation. Earlier we have shown that the quasiintegrable deformation of the KdV system [14] is indeed possible, provided the loop-algebraic generalization has been considered, and in fact we also know the quasi-integrable deformation of the sine-Gordon [12, 13] and super sine-Gordon equations [15]. Here, in succession, we have studied quasi-integrable deformation of the mixed generalized potential KdV and sine-Gordon equation.

Acknowledgement The authors are grateful to Professors Luiz. A. Ferreira, Wojtek J. Zakrzewski and Betti Hartmann for their encouragement, various useful discussions and critical reading of the draft. This paper is dedicated to the memory of our friend Anjan Kundu, his death cut short a productive career.

References [1] Kudryashov, N.A., Integrable model of nonlinear dislocations, arXiv:1611.06813v1[nlin.SI]. [2] Kontorova, T.A. and Frenkel, Y.I. (1938), On theory of plastic deformation, JETP, 8, 89, 1340, 1349 (in Russian). [3] Braun, O.M. and Kivshar, Y.S. (1998), Nonlinear dynamics of the Frenkel- Kontorova model, Physics Reports, 306, 1-108. [4] Fermi, E., Pasta, J.R., and Ulam, S. (Report LA-1940, 1955), Studies of nonlinear problems, Los Alamos: Los Alamos Scientific Laboratory. [5] Porter, M.A., Zabusky, N.J., Hu, B., and Campbell, D.K. (2009), Fermi, Pasta, Ulam and the Birth of Experimental Mathematics, American Scientist, 97(3), 214-221. doi:10.1511/2009.78.214. [6] Zabusky, N.J. and Kruskal, M.D. (1965), Interactions of solitons in a collisionless plasma and the recurrence of initial states, Physical Review Letters, 15(6), 240-243. [7] Faddeev, L.D. and Takhtajan, L.A. (1987), Hamiltonian Methods in the Theory of Solitons. Berlin: Springer-Verlag. [8] Das, A. (1989), Integrable models, World Scientific, Singapore. [9] Konno, K. and Wadati, M., Simple derivation of Bäcklund transformation from Riccati form of inverse method, Progress of Theoretical Physics, 53(6), 1652-1656. [10] Wadati, M., Sanuki, H., and Konno, K., Relationships among inverse method, Bäcklund transformation and an infinite number of conservation laws, Progress of Theoretical Physics, 53(2), 419-436. [11] Ferreira, S., Girardello, L., and Sciuto, S. (1978), An infinite set of conservation laws of the supersymmetric sinegordon theory, Phys. Lett. B, 76 303. [12] Ferreira, L.A. and Zakrzewski, W.J. (2011), The concept of quasi-integrability: a concrete example, JHEP, 05(130). [13] Ferreira, L.A., Luchini, G., and Zakrzewski, W.J. (2013), The concept of Quasi-integrability, Nonlinear and Modern Mathematical Physics AIP Conf. Proc., 1562(43). [14] Abhinav, K. and Guha, P. Quasi-Integrability of The KdV System, arXiv:1612.07499 [math-ph]. [15] Abhinav, K. and Guha, P. (2016), Quasi-Integrability in Supersymmetric Sine-Gordon Models, EPL 116 10004.



A New Comparison Theorem and Stability Analysis of Fractional Order Cohen-Grossberg Neural Networks Xiaolei Liu, Jian Yuan†, Gang Zhou, Wenfei Zhao School of Basic Sciences for Aviation, Naval Aviation University, Yantai 264001, P.R.China Submission Info Communicated by V. Afaimovich Received 5 October 2016 Accepted 26 March 2017 Available online 1 April 2018 Keywords Fractional calculus Comparison theorem Neural networks Stability analysis

Abstract This paper proposes a new comparison theorem and stability analysis of fractional order Cohen-Grossberg neural networks. Firstly, a new comparison theorem for fractional order systems is proved. Secondly, the stability of a class of fractional order Cohen-Grossberg neural networks with Caputo derivative is investigated on the basis of the above comparison theorem. Thirdly, sufficient conditions of stability of the neural networks are obtained utilizing the property of Mittag-Leffler functions, the generalized Gronwall-Bellman inequality and the method of the integral transform. Furthermore, a numerical simulation example is presented to illustrate the effectiveness of these results. ©2018 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Fractional calculus is an old and yet novel topic. It dates back to the end of 17th century and only really came to life over the last few decades, attracting interest of researchers in several areas including mathematics, physics, chemistry, material, engineering, finance and even social science [1–3]. In the past decades, stability conditions for linear fractional differential equations (FDEs) with or without delays have been obtained in [4–6]. The investigations of the stability of nonlinear fractional order system for Caputo’s derivative can be found in [7–9]. With the easy implementation of the fractional circuit and the rapid development of fractional differential equations, fractional calculus has broad applications in artificial neural networks. The existence and stability of the fractional order neural networks has been studied in [10–16]. In the neural areas, Boroomand and Menhaj proposed the fractional order Hopfield neural networks with fractance instead of common capacitor in the continuous network [17]. And the Mittag-Leffler stability of the fractional order nonlinear dynamic systems has been studied in [18–20]. Recently, some excellent results about fractional-order neural networks have been presented in [21, 22]. The finite-time stability of fractional delayed neural networks and the synchronization of fractional-order memristor-based neural networks have been studied in [18–22]. It is well known that the fractional order Hopfield neural networks have been studied in a majority of literatures. However, there are few literatures on fractional order Cohen-Grossberg neural networks up to now. So, in † Corresponding


44

Xiaolei Liu et al. / Discontinuity, Nonlinearity, and Complexity 7(1) (2018) 43–53

this paper we would study the stability of the following fractional-order Cohen-Grossberg neural networks: C α t0 Dt xi (t)

n

= −ai (xi (t))[bi (xi (t)) + ∑ ci j f j (x j (t))],

(1)

j=1

for i = 1, . . . , n, where Ct0 Dtα denotes Caputo fractional derivative with order α , 0 < α < 1, xi (t) denotes the state variable of the i−th neuron at time t, ai (·) denotes an amplification function, bi (·) is the behaved function, ci j (i = 1, . . . , n; j = 1, . . . , n) are connection weights of the neural networks, f j denotes the activation function of the jth neuron at the time t. We construct an auxiliary system, and attempt to convert the stability problem of system (1) into that of the auxiliary system. For this purpose, we need a corresponding comparison theorem for the fractional order differential systems with Caputo derivative. In fact, we can find some comparison theorem in [23–26]. Consider two fractional order differential systems: C α 0 Dt x(t) = f (t, x(t)), (2) x(0) = θ , C

and

α 0 Dt y(t)

= g(t, y(t)), y(0) = θ ,

(3)

where 0 < α < 1. The main result is that: if the condition f (t, ξ (t)) ≤ g(t, η (t)) is satisfied, where ξ (t) and η (t) are the solution of (2) and (3) respectively, then ξ (t) ≤ η (t). However, when we investigate the stability of a fractional order differential system, the solution is unknown. So the above condition is difficult to be verified. In order to solve this question, we will firstly give a new comparison theorem. The rest of this paper is organized as follows. Some necessary definitions and lemmas are presented in section 2. A new comparison theorem for fractional-order systems is proposed and proved in Section 3. The stability analysis of a class of fractional order Cohen-Grossberg neural networks is proposed in Section 4. Finally, numerical simulations are presented in Section 5 to illustrate the validity and feasibility of the results.

2 Preliminaries There are several definitions of the fractional order derivative, which are the extended concepts of integer order derivative. The commonly used definitions are Grünwald-Letnikov, Riemann-Liouville, and Caputo definitions. In this section, firstly we will recall the definitions of Riemann-Liouville and Caputo fractional derivative, the definition of Mittag-Leffler function and several important lemmas. Definition 1. The fractional integral of order α ∈ R+ of a function f (t) is defined as 1 I f (t) = Γ(α ) α

ˆ

where t ≥ t0 , Γ(·) is the gamma function, i.e., Γ(β ) =

t

(t − τ )α −1 f (τ )dτ ,

(4)

t0

´∞ 0

t β −1 e−t dt.

Definition 2. The Riemann-Liouville fractional derivative of order α ∈ R+ of a function f (t) is defined as RL α t0 Dt

where m ∈ N, m − 1 ≤ α < m,

d m 1 ( ) f (t) = Γ(m − α ) dt

d m dt

ˆ

t

f (τ )(t − τ )m−α −1 dτ ,

t0

is the mth derivative in the usual sense.

(5)


Definition 3. The Caputo fractional derivative of order α ∈ R+ of a function f (t) is defined as ˆ t m d f (τ ) 1 C α (t − τ )m−α −1 dτ , t0 Dt f (t) = Γ(m − α ) t0 dτ m where m ∈ N, m − 1 ≤ α < m,

dm f (τ ) dτ m

45

(6)

is the mth derivative of f (τ ) in the usual sense.

Lemma 1. [10] If α ∈ / N and f (t) is a function for which the Caputo fractional derivative Ca Dtα f (t) of order α α exists together with the Riemann-Liouville fractional derivatives RL a Dt f (t), then they are connected with each other by the following relations: C α a Dt

f (k) (a) (t − a)k−α , (n = [α ] + 1). Γ(k − α + 1) k=0

(7)

f (a) (t − a)−α . Γ(1 − α )

(8)

n−1

α f (t) = RL a Dt f (t) − ∑

In particular, when 0 < α < 1, we have C α a Dt

α f (t) = RL a Dt f (t) −

Definition 4. The Mittag-Leffler function Eα (z) and the two parameter Mittag-Leffler function Eα ,β (z) are defined as ∞ zk , α > 0, z ∈ C; (9) Eα (z) = ∑ k=0 Γ(kα + 1) ∞

Eα ,β (z) =

zk , α > 0, β > 0, z ∈ C. ∑ k=0 Γ(kα + β )

(10)

It is obvious that Eα (z) = Eα ,1 (z), andE1,1 (z) = ez . Lemma 2. [27] Let 0 < α < 1, then the following formula are valid α RL α a Dt Eα (λ (t − a) ) =

By (8) and Lemma 2, we have

(t − a)−α + λ Eα (λ (t − a)α ). Γ(1 − α )

α C α a Dt Eα (λ (t − a) )

= λ Eα (λ (t − a)α ).

(11)

(12)

In this paper, we make use of Dα instead of the symbol C0 Dtα . It is the Caputo derivative of order α ∈ R+ with the initial time 0. The Laplace integral transforms of the Caputo fractional derivative of order α ∈ R+ , and the Mittag-Leffler function are calculated respectively as follows [10]: L {Dα f (t)} = sα F(s) −

m−1

∑

f (k) (0)sα −k−1 ,

(13)

k=0

where F (s) = L { f (t)}, L {t β −1 Eα ,β (−λ t α )} =

1 sα −β , (R(s) > |λ | α ). α s +λ

(14)

Lemma 3. [29] If h(t) ∈ C1 ([0, +∞], R) denotes a continuously differentiable function, the following inequality holds almost everywhere: (15) Dα |h(t)| ≤ sgn(h(t))Dα h(t), 0 < α < 1.

46


Lemma 4. [10] Suppose that 0 < α < 2, β is an arbitrary positive real number, µ satisfies πα /2 < µ < min {π , πα }, then ( for K ∈ N) Eα ,β (z) =

K 1 1 1 (1−β )/α 1 + O( K+1 ), z exp(z1/α ) − ∑ k α z k=1 Γ(β − α k) z

(16)

where | arg(z)| ≤ µ , |z| → ∞. Lemma 5. [30] Suppose α > 0, a(t) is a nonnegative function locally integrable on 0 ≤ t < T (some T ≤ +∞) and g(t) is a nonnegative, non-decreasing continuous function defined on 0 ≤ t < T , g(t) and ´ t ≤ M(constant), α −1 suppose u(t) is nonnegative and locally integrable on 0 ≤ t < T with u(t) ≤ a(t) + g(t) 0 (t − s) u(s)ds on this interval. Then ˆ t ∞ (g(t)Γ(α ))n (t − s)nα −1 a(s)]ds. (17) u(t) ≤ a(t) + [∑ Γ(n α ) 0 n=1 Moreover, if a(t) is a non-decreasing function on [0, T ), then u(t) ≤ a(t)Eα ,1 (g(t)Γ(α )t α ) . Lemma 6. [31] There exists a constant K ≥ 1 such that for any 0 < α < 1,

kEα ,1 (At α )k ≤ K eAt , kEα ,α (At α )k ≤ K eAt ,

(18)

(19)

where A denotes a matrix, k·k denotes any vector or induced matrix norm.

3 A new comparison theorem for fractional-order systems In this section, we propose a new comparison theorem for fractional-order systems. Lemma 7. Consider fractional-order systems (2) and (3). If (i) g(t, x) is nondecreasing in x; (ii) f (t, x) ≤ g(t, x) for any (t, x) ∈ R+ × R; (iii) there exists a constant L > 0 such that g(t, x) − g(t, y) ≤ L(x − y), whenever x ≥ y; then ξ (t) ≤ η (t), wherever ξ (t), η (t) is the solution of the system (2) and (3) respectively. Let ηε (t) = η (t) + ε Eα ,1(2Lt α ), for small ε > 0. then we have limε →0 ηε (t) = η (t) and ηε (0) = η (0) + ε > η (0) = θ . In view of (12), under the conditions (i) and (iii), we get that Dα ηε (t) = Dα η (t) + Dα (ε Eα ,1 (2Lt α )) = g(t, η (t)) + 2ε LEα ,1 (2Lt α ) ≥ g(t, ηε (t)) + ε LEα ,1 (2Lt α )

> g(t, ηε (t)).

Hence,

ηε (t) − ηε (0) = I α Dα ηε (t) ˆ t 1 (t − τ )α −1 Dα ηε (τ )dτ = Γ(α ) 0 ˆ t 1 > (t − τ )α −1 g(τ , ηε (τ ))dτ , Γ(α ) 0


i.e.

ηε (t) > ηε (0) +

1 Γ(α )

ˆ

t

(t − τ )α −1 g(τ , ηε (τ ))dτ .

47

(20)

0

Because of ξ (t) is the solution of the system (2) and f (t, x(t)) ≤ g(t, x(t)), then we have ˆ t 1 ξ (t) = θ + (t − τ )α −1 f (τ , ξ (τ ))dτ Γ(α ) 0 ˆ t 1 (t − τ )α −1 g(τ , ξ (τ ))dτ . ≤θ+ Γ(α ) 0 We can claim ηε (t) > ξ (t). If this is not true, then it follows by ηε (0) > η (0) = θ = ξ (0), there exists a t1 > 0 such that ηε (t1 ) = ξ (t1 ) and ηε (t) > ξ (t), 0 < t < t1 . Because g(t, x) is nondecreasing in x, we have ˆ t1 1 (t1 − τ )α −1 g(τ , ηε (τ ))dτ ηε (t1 ) > ηε (0) + Γ(α ) 0 ˆ t1 1 > ξ (0) + (t1 − τ )α −1 g(τ , ξ (τ ))dτ Γ(α ) 0 ≥ ξ (t1 ), which is a contradiction with ηε (t1 ) = ξ (t1 ). So ηε (t) > ξ (t). Let ε → 0, we have η (t) ≥ ξ (t). Remark 1. The comparison theorem in [25] is that: if f (t, ξ (t)) ≤ g(t, η (t)), where ξ (t) and η (t) are the solutions of the system (2), (3) respectively, then ξ (t) ≤ η (t). However, this condition is not easy to be verified, because ξ (t) and η (t) are two different functions and we don’t know what they are. In the new theorem, the inequlity f (t, x(t)) ≤ g(t, x(t)) is compared concerning only one function x(t), and it can be illustrated more easily. So this comparison theorem can be applied more conveniently and effectively.

4 Stability of a class of fractional order Cohen-Grossberg neural networks In this section, we get the sufficient conditions for the stability of the fractional order Cohen-Grossberg neural networks (1). We make the following assumptions firstly. (H1) For each i = 1, 2, . . . , n, ai (·) is continuously bounded, and satisfies 0 < αim < ai (xi ) < αiM ; bi (·) satisfies i) 0 < βim < bi (x xi < +∞, for all xi ∈ R. (H2) λ = min {α1m β1m , α2m β2m , . . . , αnm βnm } > (Γ(α ))α . (H3) For each j = 1, 2, . . . , n, the activation functions f j satisfies the usual one-sided Lipschitz condition with Lipschitz constant L, i.e.

and satisfies

f j (x j ) − f j (y j ) ≤ L(x j − y j ), x j ≥ y j , L > 0,

(21)

f j (x j ) ≤ f j ( x j ).

(22)

T (H4) lim kF(x)k kxk = 0, where F(x) = ( f1 (x1 ), f2 (x2 ), . . . , fn (xn )) . x→0

Theorem 8. Under the assumptions (H1)-(H4), the system (1) is asymptotically stable. By Lemma 3, we have

48


Dα |xi (t)| ≤ sgn(xi (t))Dα (xi (t)) n

= sgn(xi (t))(−ai (xi (t))(bi (xi (t)) + ∑ ci j f j (x j (t)))). j=1

Case 1. when xi (t) > 0, from(H1) and (H3), we have n

Dα |xi (t)| ≤ −ai (xi (t))(bi (xi (t)) + ∑ ci j f j (x j (t))) j=1 n

≤ −ai (xi (t))(βim xi (t) + ∑ ci j f j (x j (t))) j=1

n

≤ −αim βim xi (t) + αiM ∑ |ci j || f j (x j (t))| =

j=1 n |ci j | f j (|x j (t)|). −αim βim |xi (t)| + αiM j=1

∑

Case 2. when xi (t) < 0, from (H1) and (H3), we have n

Dα |xi (t)| ≤ ai (xi (t))(bi (xi (t)) + ∑ ci j f j (x j (t))) j=1 n

≤ ai (xi (t))(βim xi (t) + ∑ ci j f j (x j (t))) j=1

n

≤ αim βim xi (t) + αiM ∑ |ci j || f j (x j (t))| j=1

n = −αim βim |xi (t)| + αiM ∑ ci j f j ( x j (t) ). j=1

So, we can get

n Dα |xi (t)| ≤ −αim βim |xi (t)| + αiM ∑ ci j f j ( x j (t) ).

(23)

j=1

According to (23), we construct the auxiliary system as follows:  n  Dα y (t) = −α m β m y (t) + α M ∑ ci j f j (y j (t)), i i i i i j=1  yi (0) = |xi (0)| , i = 1, 2, . . . , n. It is equivalent to a vector form:

Dα y(t) = −Ay(t) + BF(y(t)), y(0) = |x(0)| .

where y(t) = (y1 (t), y2 (t), . . . , yn (t))T ,

A = diag(α1m β1m , α2m β2m , . . . , αnm βnm ),

(24)

(25)


49



 α1M |c11 | α1M |c12 | · · · α1M |c1n |  α M |c21 | α M |c22 | · · · α M |c2n |  2 2 2 , B=  ··· ··· ··· ···  αnM |cn1 | αnM |cn2 | · · · αnM |cnn | By taking Laplace transform on Eq.(25), we get

Y (s) = (sα I + A)−1 sα −1 y(0) + L {BF(y(t))} ,

(26)

where Y (s) = L {y(t)}, I is an n × n identity matrix. By taking Laplace inverse transform on Eq. (26), we have the solution of (25), i.e. ˆ t α (t − τ )α −1 Eα ,α −A(t − τ )α BF (y (τ )) dτ . (27) y (t) = Eα ,1 (−At ) y(0) + 0

By Lemma 6, we have ky (t)k = kEα ,1 (−At α )k ky(0)k ˆ t

(t − τ )α −1 Eα ,α −A(t − τ )α kBF (y (τ ))k dτ + 0

≤ K kexp{−At}k ky(0)k ˆ t (t − τ )α −1 K kexp{−A (t − τ )}k kBk kF (y (τ ))k dτ , + 0

where K > 1. From (H4), we have limy→0 kyk < δ , we have kF(y)k ≤ Therefore,

kF(y)k kyk

= 0, so there exists a constant δ > 0, such that for any y satisfying

1 KkBk kxk.

ky (t)k ≤ K kexp{−At}k ky(0)k ˆ t (t − τ )α −1 kexp{−A (t − τ )}k kF (y (τ ))k dτ + 0 ˆ t (t − τ )α −1 e−λ (t−τ ) ky (τ )k dτ . ≤ Ke−λ t ky(0)k + 0

Multiplying by eλ t in both sides, we have eλ t ky (t)k ≤ K ky(0)k +

ˆ

t

ˆ

t

(t − τ )α −1 eλ τ ky (τ )k dτ .

(28)

(t − τ )α −1 u(τ )dτ .

(29)

0

Let u(t) = eλ t ky (t)k, from (28), we have u(t) ≤ K ky(0)k +

0

According to Lemma 5, u(t) ≤ K ky(0)k Eα ,1 (Γ(α )t α ), that is eλ t ky (t)k ≤ K ky(0)k Eα ,1 (Γ(α )t α ). Because Γ(α )t α → ∞ as t → +∞, by Lemma 5, we have 1 exp((Γ(α ))1/α t) α N 1 1 1 )), + O( −∑ N+1 k k α (Γ(α )) t α (N+1) k=1 Γ(1 − α k) Γ(α ) t

eλ t ky(t)k ≤ Kky(0)k(

(30)

50


0.1 x (t) 1

x2(t) x (t)

0.08

3

x(t)

0.06

0.04

0.02

0

0

1/3

2/3

1

t

Fig. 1 Waveform plot of the fractional order Cohen-Grossberg neural networks when q = 0.8.

i.e., 1 exp(((Γ(α ))1/α − λ )t) α N 1 1 −∑ e−λ t k kα Γ(1 − α k) Γ(α ) t k=1 1 +O( )e−λ t ). N+1 α (N+1) (Γ(α )) t

ky (t)k ≤ K ky(0)k (

(31)

By the assumption (H2), we have ky (t)k → 0, as t → +∞. Because |xi (t)|, yi (t) is the solution of the system (23) and (24) respectively, and |xi (0)| = yi (0), we have |xi (t)| ≤ yi (t) by the comparison theorem in Section 3. Therefore, x (t) → 0, as t → +∞, which implies that the system (1) is asymptotically stable.

5 Numerical simulations In this section, a numerical example is presented to illustrate the effectiveness of the theoretical result obtained in the above section. In system (1), let x(0) = [1, 1, 1]T , α = 0.8, A(x(t)) = diag (cos(3x1 ) + 2.3, sin(x2 ) + 1.4, sin(2x3 ) + 3.1) , B(x(t)) = (tanh(2x1 ), tanh(4x2 ), tanh x3 )T ,   2 1 3 C = (ci j ) =  −2 2 3  , 1 3 −1 T

F(x) = ( fi (xi )) = [(3/π ) arctan x31 , (3/π ) arctan x32 , (3/π ) arctan x33 ] . Then system (1) satisfies the condition of Theorem 1. Therefore x = 0 is a asymptotic stable equilibrium point (see Fig. 1). From Fig.1, we find that all of the solutions of system (1) with q = 0.8 will converge to the zero solution. Moreover, we display the trajectories of solutions of system(1) when q = 0.5, q = 0.7, q = 0.9 (see Figs.2-4).


51

0.1 α=0.7 α=0.5 α=0.9 0.08

1

x (t)

0.06

0.04

0.02

0

0

1/3

2/3

1

t

Fig. 2 Waveform plot of x1 (t) when q = 0.5, q = 0.7, q = 0.9. 0.1 α=0.7 α=0.5 α=0.9

0.08

2

x (t)

0.06

0.04

0.02

0

0

1/3

2/3

1

t

Fig. 3 Waveform plot of x2 (t)(t) when q = 0.5, q = 0.7, q = 0.9.

6 Conclusions This paper investigates the stability of a class of fractional order Cohen-Grossberg neural networks. We construct an auxiliary system, and convert the stability problems of the neural networks into that of auxiliary system on the basis of a new feasible comparison theorem of fractional system. By making use of the property of MittagLeffler function, the generalized Gronwall-Bellman inequality, and the method of the integral transform, some sufficient conditions are obtained to ensure the stability of the fractional order Cohen-Grossberg neural networks. Finally, we give a numerical example to illustrates the effectiveness of the developed theoretical results.

Acknowledgements All the authors acknowledge the valuable suggestions from the peer reviewers. This work was supported by the Natural Science Foundation of Shandong Province (Grant No. ZR2014AM006).

52


0.1 α=0.7 α=0.5 α=0.9 0.08

3

x (t)

0.06

0.04

0.02

0

0

1/3

2/3

1

t

Fig. 4 Waveform plot of x3 (t) when q = 0.5, q = 0.7, q = 0.9.

References [1] Yuan, L.G., Yang, Q.G., and Zeng, C.B. (2013), Chaos detection and Parameter identification in fractional-order chaotic systems with delay, Nonlinear Dynamics, 73, 439-448. [2] Tenreiro, M.J., Kiryakova, V., and Mainardi, F. (2011), Recent history of fractional calculus, Commun. Nonlinear Sci. Numer. Simul, 16, 1140-1153. [3] Sayevand, K., Golbabai, A., and Yildirim, A. (2011), Analysis of differential equations of fractional order, Applied Mathematical Modelling, 36, 4356-4364. [4] Bonnet, C. and Partington, J.R. (2000), Coprime factorizations and stability of fractional differential systems, Syst. Control Lett, 41, 167-174. [5] Deng, W.H., Li, C.P., and Lu, J.H. (2007), Stability analysis of linear fractional differential system with multiple time-delays, Nonlinear Dynamic, 48, 409-416. [6] Li, C.P. and Deng, W.H. (2007), Remarks on fractional derivatives, Appl. Math. Comput., 187, 777-784. [7] Sadati, S.J., Baleanu, D., Ranjbar, A., Ghaderi, R., and Abdeljawad, T. (2010), Mittag-Leffler stability theorem for fractional nonlinear systems with delay, Abstr. Appl. Anal., 108651. Doi:10.1155/2010/108651. [8] Liu, L. and Zhong, S. (2011), Finite-time stability analysis of fractional order with multi-state time delay, Word Acad. Sci. , Eng. Technol., 76, 874-877. [9] Lazarevic, M. (2007), Finite-time stability analysis of fractional order time delay systems: Bellman-Grronwall’s approach, Sci. Tech. Rev., 7, 8-15. [10] Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006), Theory and Applications of Fractional Differential Equations, Elsevier, B. V. [11] Yu, J., Hu, C., and Jiang, H.J. (2012), α -stability and α -synchronization for fractional-order neural networks, Neural Networks, 35, 82-87. [12] Delavari, H., Baleanu, D., and Sadati, J. (2012), Stability analysis of Caputo fractional-order nonlinear systems revisited, Nonlinear Dynamics, 67, 2433-2439. [13] Zhang, R.X., Tian, G., Yang, S.P., and Cao, H.F. (2015), Stability analysis of a class of fractional order nonlinear systems with order lying in (0, 2), ISA Transactions, http://dx.doi.org/10.1016/j.isatra.2014.12.006. [14] Jalilian, Y. and Jalilian, R. (2013), Existence of solution for delay fractional differential equations, Mediterr. J. Math., 10, 1731-1747. [15] Chen, L.P., Chai, Y., Wu, R.C., Ma, T.D., and Zhai, H.Z.(2013), Dynamic analysis of a class of fractional order neural networks with delay, Neurocomputing, 111, 190-194. [16] Li, Y., Chen, Y.Q., and Podlubny, I. (2009), Mittag-Leffler stability of fractional order nonlinear dynamic systems, Automatica, 45, 1965-1969. [17] Arefeh, B. and Mohammad, B.M. (2009), Fractional-Order Hopfield neural networks[J], M. Koppen et al. (Eds.): ICONIP 2008, Part I, LNCS 5506, 883-890. [18] Chen, J.J., Zeng, Z.G., and Jiang, P. (2014), Global Mittag-Leffler stability and synchronization of memristor-based fractional-order neural networks, Neural networks, 51, 1-8. [19] Sadati, S.J., Baleanu, D., Ranjber, A., Ghaderi, R., and Abdeljawad, T. (2010), Mittag-Leffler Stability Theorem for Fractional Nonlinear Systems with Delay, Abstract and Applied Analysis, 2010, Article ID108651, 7 pages. Doi:


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10.1155/2010/108651. [20] Yu, J.M., Hu, H., Zhou, S.B., and Lin, X.R. (2013), Generalized Mittag-Leffler stability of multi-variables fractional order nonlinear systems, Automatica, 49, 1798-1803. [21] Wu, R.C., Lu, Y.F., and Chen, L.P. (2014), Finite-time stability of fractional delayed neural networks, Neurocomputing, http://dx.Doi.org/10.1016/j.neucom.2014.07.060. [22] Bao, H.B. and Cao, J.D., Projective synchronization of fractional-order memristor-based neural networks, Neural Networks, 63, 1-9. [23] Dlavari, H., Baleanu, D., and Sadati, J. (2012), Stability analysis of Caputo frational-order nonlinear systems revisited, Nonlinear Dyn., 67, 2433-2439. [24] Wang, Z.L., Yang, D.S., and Ma, T.D. (2014), Stability analysis for nonlinear fractional-order systems based on comparison principle, Nonlinear Dyn., 75, 387-402. [25] Wang, H., Yu, Y.G., Wen, G.G., Zhang, S., and Yu, J.Z. (2015), Global stability analysis of fractional-order Hopfield neural networks with time delay, Neurocomputing, 154, 15-23. [26] Liang, S., Wu, R.C., and Chen. L.P. (2015), Comparison principles and stability of nonlinear fractional-order cellular neural networks with multiple time delays, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2015.05.063. [27] Gorenflo, R., Kilbas, A.A., Mainardi, F., and Rogosin, S.V., Mittag-Leffler Functions, Related Topics and Applications, Springer. [28] Stamova, I.M. (2015) , Mittag-Leffler stability of impulsive differential equations of fractional order, Quarterly of Applied Mathematics, http://dx.doi.org/10.1090/qam/1394. [29] Zhang, S., Yu, Y.G., and Wang, H. (2014), Mittag-Leffler stability of fractional-order Hopfield neural networks, Nonlinear Analysis: Hybrid Systems. http://dx.doi.org/10.1016/j.nahs.2014.10.001. [30] Ye, H., Gao, J., and Ding, Y. (2007), A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328, 1075-1081. [31] De la Sen, M. (2011), About robust stability of Caputo linear fractional dynamic systems with time delays through fixed point theory, Fixed Point Theory Appl., 867932. [32] Liang, Y.J., Chen, W., and Richard, L. (2016), Connecting complexity with spectral entropy using the laplace transformed solution to the fractional diffusion equation, Physica A Statistical Mechanics & Its Applications, 453, 327-335. [33] Chen, W. amd Pang, G. (2016), A new definiton of fractional Laplacian with application to modeling three-dimensional nonlocal heat conduction, Journal of Computational Physics, 309, 350-367. [34] Liang, Y. and Chen, W. (2015), Argularized Miner’s rule for fatigue reliability analysis with Mittag-Leffler statistics, International Journal of Damage Mechanics, 1056789515607610.



Existence of Solutions of Stochastic Fractional Integrodifferential Equations P. Umamaheswari†, K. Balachandran, N. Annapoorani Department of Mathematics, Bharathiar University, Coimbatore 641 046, India Submission Info Communicated by D. Volchenkov Received 21 February 2017 Accepted 3 April 2017 Available online 1 April 2018

Abstract In this paper, a general class of stochastic fractional integrodifferential equations is investigated. The Picard-Lindelöf successive approximation scheme is used to establish the existence of solutions. The uniqueness of the solution is also studied under suitable conditions.

Keywords Picard-Lindelöf successive approximation Integrodifferential equations Fractional differential equations Stochastic differential equations


1 Introduction In the last twenty years, there have been more and more researchers interested in studying the fractional calculus. Fractional Differential Equations (FDEs) [1–4] can efficiently describe dynamical behavior of real life phenomena more accurately than integer order equations because of its ability to describe systems with memory and hereditary properties. FDEs have numerous applications in many areas of science and engineering such as in viscoelasticity, electrochemistry, nonlinear oscillation of earthquake, the fluid-dynamic traffic model, flow in porous media, aerodynamics and in different branches of physical and biological sciences. In many problems, there are real phenomena depending on the effect of white noise random forces. Stochastic differential equations were first initiated and developed by Itô in 1942 [5]. Stochastic differential equations [6–8] now find applications in many disciplines including economics and finance, environmetrics, physics, population dynamics, biology and medicine. An important application of stochastic differential equations occurs in the modeling of problems associated with water catchment and the percolation of fluid through porous structures. A natural extension of deterministic ordinary differential equation model is the stochastic differential equation model, where relevant parameters are randomized or modeled as random processes of some suitable form or simply by adding a noise term to the driving equations of the system. Stochastic fractional differential equations are used to model dynamical systems affected by random noises. The theory and application of integrodifferential equations plays an important role in the mathematical modeling of many fields: physical, biological phenomena and engineering sciences in which it is necessary to take † Corresponding


56

P. Umamaheswari et al. / Discontinuity, Nonlinearity, and Complexity 7(1) (2018) 55–65

into account the effect of real world problems. Most of the practical systems are integrodifferential equations in nature and hence the study of integrodifferential equations becomes very important. The advantage of the integrodifferential equations representation for a variety of problem is witnessed by its increasing frequency in the literature and in many texts on methods of advanced applied mathematics [9–11]. We are surrounded by the phenomena and the properties which we call fluctuations, such as the density of traffic on a highway, the variations within a biological species, the up-and-down of stock markets, the twinkling of stars, and so on [12–15]. These fluctuations are related to the irregularity or stochasticity among the realizations, that is, among the individual specimens of the data. This has become the reason for the attention given to the theory and applications of stochastic differential equations [16–18]. Our motivation for considering FDEs with random elements comes from the fact that many phenomena in science that have been modeled by fractional differential equations have some uncertainty. Therefore for investigating more accurate solutions, we need the solutions of Stochastic Fractional Differential Equations(SFDEs). These equations have immense physical applications in many fields such as turbulence, heterogeneous flows and materials, viscoelasticity and electromagnetic theory [19]. Complex dynamic processes in science and engineering operating under internal structural and external environmental perturbations can be modelled by stochastic fractional differential equations by introducing the concept of dynamics processes operating under a set of linearly independent time-scales. The problem of existence and uniqueness of solutions of the initial value problem provide the basis for the model validation and further undertaking a study of the corresponding dynamic processes. By utilizing either approximation schemes or fixed point theory, the initial value problem is reduced to its equivalent integral equation problem. Here the Picard-Lindelöf successive approximation scheme is used to establish the existence and uniqueness of solutions of the general class of stochastic fractional integrodifferential equations.

2 Preliminaries In this section, we present a few well-known concepts and results in the field of fractional and stochastic differential equations. Definition 1 (Riemann-Liouville fractional integral). The Riemann-Liouville fractional integral operator of order α > 0 of a function f ∈ L1 (R+ ) is defined as ˆ t 1 α (t − s)α −1 f (s)ds, t > 0, (1) I0+ f (t) = Γ(α ) 0 where Γ(·) is the Euler gamma function. Definition 2 (Riemann-Liouville fractional derivative). The Riemann-Liouville fractional derivative of order α > 0, n − 1 < α < n, n ∈ N, is defined as ˆ d n t d n n−α 1 α ( ) (t − s)n−α −1 f (s)ds, (2) D0+ f (t) = ( ) I0+ f (t) = dt Γ(n − α ) dt 0 where the function f (t) has absolutely continuous derivatives upto order (n − 1). Definition 3 (Caputo fractional derivative). The Caputo fractional derivative of order α > 0, n − 1 < α < n, n ∈ N, is defined as ˆ t 1 C α D0+ f (t) = (t − s)n−α −1 f (n) (s)ds, (3) Γ(n − α ) 0 where the function f (t) has absolutely continuous derivatives upto order (n − 1).


57

Definition 4 (Chebyshev’s Inequality). If X is a random variable and 1 ≤ p < ∞, then P(|X| ≥ λ ) ≤

1 E(|X| p ) for all λ > 0. λp

Lemma 1 (Borel Cantelli Lemma). ∞

(i) If {Ak } ⊂ F and ∑ P(Ak ) < ∞, then k=1

P( lim sup Ak ) = 0. k→∞ ∞

(ii) If the sequence {Ak } ⊂ F is independent and ∑ P(Ak ) = ∞, then k=1

P( lim sup Ak ) = 1. k→∞

Definition 5 (Gronwall inequality). Let φ (t) be a nonnegative function defined for 0 ≤ t ≤ T and let C0 , A denote some positive constants. If ˆ t φ (t) ≤ C0 + A φ (s)ds for all 0 ≤ t ≤ T, 0

then

φ (t) ≤ C0 exp(At) for all 0 ≤ t ≤ T. Definition 6 (Multi-time scale integral [20]). For p ∈ N, p > 1, let {T1 , T2 , . . . , Tp } be a set of linearly independent time scales. Let f : (a, b)× Rp−1 → Rn be a continuous function defined by f (t) = f (T1 (t), T2 (t), . . . , Tp (t)). The multi-time scale integral of the composite function f over an interval [t0 ,t] ⊆ (a, b) is defined as a sum of p integrals with respect to the time-scales T1 , T2 , . . . , Tp . We denote it by I f , ˆ (I f )(t) =

p

t

f (s)ds =

t0

∑ (I j f )(t), j=1

where the sense of the integral

ˆ (I j f )(t) =

t

f (s)dTj (s),

t0

depends on the time scale Tj , for each j = 1, 2, . . . , p. Example 1. For p = 3, consider the linearly independent set consisting of time scales T1 (t) := t signifying the ideal and controlled environmental condition, T2 (t) := W (t), where W is the standard Wiener process and T3 (t) := t α , 0 < α < 1 indicates the time varying delay or lagged process. In this case, f (t) = f (T1 (t), T2 (t), T3 (t)) and (I f )(t) = (I1 f )(t) + (I2 f )(t) + (I3 f )(t), where the integrals

ˆ

t

(I1 f )(t) =

ˆ f (s)ds,

0

(I2 f )(t) =

t

f (s)dW (s),

0

ˆ

t

(I3 f )(t) = 0

(t − s)α −1 f (s) ds, Γ(α )

are Cauchy-Riemann/Lebesgue, Itô-Doob and Riemann-Liouville type respectively.

58


Under the set of time scales in Example 1, we have the following stochastic fractional differential equation dx(t) = b(t, x(t))dt + σ1 (t, x(t))dW (t) + σ2 (t, x(t))(dt)α ,

t ∈ J = [0, T ],

(4)

with the initial condition x(0) = x0 , where α ∈ (1/2, 1), b, σ2 ∈ C(J × Rn , Rn ), σ1 ∈ C(J × Rn , Rnm ) and W = {W (t),t ≥ 0} is an m-dimensional Brownian motion on a complete probability space Ω ≡ (Ω, F , P). From definition 2, we can rewrite (4) in its equivalent integral form as follows: ˆ t ˆ t ˆ t b (s, x(s)) ds + σ1 (s, x(s)) dW (s) + α (t − s)α −1 σ2 (s, x(s)) ds. x(t) =x0 + 0

0

0

Remark 1. We note that 1. If σ2 (., .) ≡ 0 in Example 1, then the initial value problem (4) is reduced to known Itô-Doop type stochastic initial value problem dx(t) = b(t, x(t))dt + σ1 (t, x(t))dW (t), x(0) = x0 , whose fundamental properties and applications have been well studied for more than half-century [21–23]. 2. For σ1 (., .) ≡ 0, we have the following generalized version of the classical deterministic fractional differential equations [1, 3]. dx(t) = b(t, x(t))dt + σ2 (t, x(t))(dt)α ,

x(0) = x0 .

3. If b(., .) ≡ 0 and σ1 (., .) ≡ 0, then (4) becomes the initial value problem with Caputo type fractional differential equations [1, 4]. C α D0 = σ2 (t, x(t)), x(0) = x0 . 4. If σ1 ≡ 0 and σ2 (., .) ≡ 0, then (4) is to the deterministic initial value problem dx(t) = b(t, x)dt,

x(0) = x0 .

3 Existence and uniqueness The problems of existence and uniqueness of solutions of initial value problem provide the basis for the model validation and further undertaking a study of the corresponding dynamic processes. Kloeden and Platen [24], Mao [25] and Øksendal [26] have established the existence and uniqueness results for stochastic differential equations. The results are obtained by using the successive approximation scheme. The same technique was used by Arnold [27] and Evans [28] to obtain the existence and uniqueness of solution for the stochastic differential equations. The existence and uniqueness of stochastic fractional differential equation was studied by Pedjeu and Ladde [20]. Kamrani [19] discussed the numerical solution of stochastic fractional differential equation. Here the classical Picard-Lindelöf method of successive approximation scheme [29,30] is used to obtain the existence and uniqueness of solution for the following stochastic fractional integrodifferential equation. Consider the stochastic fractional integrodifferential equation of the form ˆ t ˆ t ˆ t f1 (t, s, x(s))ds, f2 (t, s, x(s))ds, . . . , fn (t, s, x(s))ds)dt dx(t) = b(t, x(t), 0 0 0 ˆ t ˆ t ˆ t g1 (t, s, x(s))ds, g2 (t, s, x(s))ds, . . . , gn (t, s, x(s))ds)dW (t) + σ1 (t, x(t), ˆ0 t ˆ0 t ˆ0 t h1 (t, s, x(s))ds, h2 (t, s, x(s))ds, . . . , hn (t, s, x(s))ds)(dt)α , + σ2 (t, x(t), 0

0

0


59

with the initial condition x(0) = x0 , t ∈ J = [0, T ], where α ∈ (1/2, 1), b, σ2 ∈ C(J × Rn × Rn×n , Rn ), σ1 ∈ C(J × Rn × Rn×n, Rnm ), fi , gi , hi ∈ C(J × J × Rn , Rn ), i = 1, 2, . . . , n and W = {W (t),t ≥ 0} is an m-dimensional Brownian motion on a complete probability space Ω ≡ (Ω, F , P). We can rewrite the above equation in its equivalent integral form as follows [31, 32]: ˆ t ˆ s ˆ s ˆ s b(s, x(s), f1 (s, τ , x(τ ))d τ , f2 (s, τ , x(τ ))d τ , . . . , fn (s, τ , x(τ ))d τ )ds x(t) = x0 + 0

ˆ

t

+

ˆ

0 s

σ1 (s, x(s),

0

ˆ

+α

ˆ

(t − s)α −1 σ2 (s, x(s),

0

s

g1 (s, τ , x(τ ))d τ ,

0 t

0

s

s

g2 (s, τ , x(τ ))d τ , . . . ,

0

ˆ

0

ˆ ˆ

s

h1 (s, τ , x(τ ))d τ ,

0

gn (s, τ , x(τ ))d τ )dW (s)

0

ˆ

s

h2 (s, τ , x(τ ))d τ , . . . ,

0

hn (s, τ , x(τ ))d τ )ds.

0

Theorem 2 (Existence and Uniqueness). Assume that (t, x) ∈ J ×Rn , α ∈ (1/2, 1), b, σ2 ∈ C(J ×Rn ×Rn×n, Rn ), σ1 ∈ C(J × Rn×n × Rn , Rnm ), fi , gi , hi ∈ C(J × J × Rn , Rn ), i = 1, 2, . . . , n and W = {W (t),t ≥ 0} is an mdimensional Brownian motion on a complete probability space Ω ≡ (Ω, F , P). Suppose the following inequalities hold: (i) Linear growth condition : ˆ t ˆ t ˆ t f j (t, s, x(s))ds, G j = g j (t, s, x(s))ds, H j = h j (t, s, x(s))ds, j = 1, 2, . . . , n, For Fj = 0

0

0

|b(t, x, F1 , F2 , . . . , Fn )|2 + |σ1 (t, x, G1 , G2 , . . . , Gn )|2 + |σ2 (t, x, H1 , H2 , . . . , Hn )|2 n

n

n

j=1

j=1

j=1

≤ K 2 (1 + |x|2 + ∑ |Fj |2 + ∑ |G j |2 + ∑ |H j |2 ), |Fj |2 + |G j |2 + |H j |2 ≤ K 2j (1 + |x|2 ),

(5)

j = 1, 2, . . . , n,

(6)

for some constants K, K j > 0, j = 1, 2, . . . , n. (ii)The Lipschitz condition : ˆ t ˆ t ˆ t fi (t, s, x(s))ds, Gi = gi (t, s, x(s))ds, Hi = hi (t, s, x(s))ds, i = 1, 2, . . . , n, For Fi = ˆ

0

and F˜i = 0

t

ˆ

0 t

fi (t, s, x(s))ds, ˜ G˜ i = 0

ˆ

0 t

gi (t, s, x(s))ds, ˜ H˜ i =

hi (t, s, x(s))ds, ˜ i = 1, 2, . . . , n,

0

˜ F˜1 , . . . , Fñ )|2 + |σ1 (t, x, G1 , . . . , Gn ) − σ1 (t, x, ˜ G˜1 , . . . , Gñ )|2 |b(t, x, F1 , . . . , Fn ) − b(t, x, n

n

+ |σ2 (t, x, H1 , . . . , Hn ) − σ2 (t, x, ˜ H˜1 , . . . , Hñ )|2 ≤ L2 (|x − x| ˜ 2 + ∑ |Fi − F˜i|2 + ∑ |Gi − G˜ i |2 i=1

i=1

n

+ ∑ |Hi − H˜ i|2 ),

(7)

i=1

˜ 2 ), i = 1, 2, . . . , n, |Fi − F˜i|2 + |Gi − G˜ i|2 + |Hi − H˜ i |2 ≤ L2i (|x − x|

(8)

for some constants L, Li > 0, i = 1, 2, . . . , n. Let x0 be a random variable defined on (Ω, F , P) and independent of the σ -algebra Fst ⊂ F generated by {W (s),t ≥ s ≥ 0} and such that E|x0 |2 < ∞. Then the initial value problem

60


⎫ ⎪ ⎪ ⎪ dx(t) = b(t, x(t), f1 (t, s, x(s))ds, f2 (t, s, x(s))ds, . . . , fn (t, s, x(s))ds)dt ⎪ ⎪ ⎪ 0 0 0 ⎪ ˆ t ˆ t ˆ t ⎪ ⎪ ⎪ g1 (t, s, x(s))ds, g2 (t, s, x(s))ds, . . . , gn (t, s, x(s))ds)dW (t)⎬ + σ1 (t, x(t), ˆ0 t ˆ0 t ˆ0 t ⎪ ⎪ ⎪ ⎪ h1 (t, s, x(s))ds, h2 (t, s, x(s))ds, . . . , hn (t, s, x(s))ds)(dt)α , ⎪ + σ2 (t, x(t), ⎪ ⎪ ⎪ 0 0 0 ⎪ ⎪ ⎭ x(0) = x0 , ˆ

t

ˆ

ˆ

t

t

(9)

has a unique solution which is t-continuous with the property that x(t, ω ) is adapted to the filtration Ftx0 generated by x0 and {W (s)(·), s ≤ t} and sup E[|x(t)|2 ] < ∞.

(10)

0≤t≤T

Proof. Existence: First we follow the classical method of successive approximations to establish the existence of solution of the initial value problem (9). Let us define x(0) (t) = x0 and x(k) (t) = x(k) (t, ω ) inductively as follows: ˆ t ˆ s ˆ s (k+1) (k) (k) (t) = x0 + b(s, x (s), f1 (s, τ , x (τ ))d τ , . . . , fn (s, τ , x(k) (τ ))d τ )ds x 0 0 0 ˆ s ˆ s ˆ t σ1 (s, x(k) (s), g1 (s, τ , x(k) (τ ))d τ , . . . , gn (s, τ , x(k) (τ ))d τ )dW (s) + 0 0 0 ˆ s ˆ s ˆ t α −1 (k) (k) h1 (s, τ , x (τ ))d τ , . . . , hn (s, τ , x(k) (τ ))d τ )ds, (11) + α (t − s) σ2 (s, x (s), 0

0

0

for k = 0, 1, 2, . . . . If, for fixed k ≥ 0, the approximation x(k) (t) is Ft -measurable and continuous on J, then it follows from (5)-(8), that the integrals in (11) are meaningful and that the resulting process x(k+1) (t) is Ft measurable and continuous on J. As x(0) (t) is obviously Ft -measurable and continuous on J, it follows by induction that so too is each x(k) (t) for k = 1, 2, . . . . Since x0 is Ft -measurable with E(|x0 |2 ) < ∞, it is clear that sup E(|x(0) (t)|2 ) < ∞. 0≤t≤T

Applying the algebraic inequality (a + b + c + d)2 ≤ 4(a2 + b2 + c2 + d 2 ), the Cauchy-Schwarz inequality, the Itô isometry and the linear growth conditions (5) and (6) we obtain from (11) that E(|x(k+1) (t)|2 ) ≤4E[|x0 |2 ] ˆ s ˆ s ˆ t (k) (k) f1 (s, τ , x (τ ))d τ , . . . , fn (s, τ , x(k) (τ ))d τ )|2 ds] + 4tE[ |b(s, x (s), ˆ0 s ˆ0 s ˆ 0t g1 (s, τ , x(k) (τ ))d τ , . . . , gn (s, τ , x(k) (τ ))d τ )|2 ds] + 4E[ |σ1 (s, x(k) (s), 0 0 0 ˆ t ˆ s ˆ s 2α −1 t 2 (k) (k) E[ |σ2 (s, x (s), h1 (s, τ , x (τ ))d τ , . . . , hn (s, τ , x(k) (τ ))d τ )|2 ds]. + 4α 2α − 1 0 0 0 Therefore n

E(|x(k+1) (t)|2 ) ≤ 4E[|x0 |2 ] + 4K 2 (1 + ∑ Ki2 )(1 + t + α 2 i=1

t 2α −1 ) 2α − 1

ˆ

t 0

E[|x(k) (s)|2 ]ds,


61

for k = 0, 1, 2, . . . and i = 1, 2, . . . , n. By induction, we have sup E(|x(k) (t)|2 ) ≤ C0 < ∞, 0≤t≤T

for k = 1, 2, 3, . . .. Applying the Schwarz inequality and Itô isometry, we obtain E[|x(k+1) (t) − x(k) (t)|2 ] ˆ s ˆ s ˆ t (k) (k) E[|b(s, x (s), f1 (s, τ , x (τ ))d τ , . . . , fn (s, τ , x(k) (τ ))d τ ) ≤ 3t 0 0 0 ˆ s ˆ s f1 (s, τ , x(k−1) (τ ))d τ , . . . , fn (s, τ , x(k−1) (τ ))d τ )|2 ]ds − b(s, x(k−1) (s), 0 0 ˆ s ˆ s ˆ t (k) (k) g1 (s, τ , x (τ ))d τ , . . . , gn (s, τ , x(k) (τ ))d τ ) + 3 E[|σ1 (s, x (s), 0 0 0 ˆ s ˆ s (k−1) (k−1) − σ1 (s, x (s), g1 (s, τ , x (τ ))d τ , . . . , gn (s, τ , x(k−1) (τ ))d τ )|2 ]ds 0 0 ˆ ˆ s ˆ s α 2t 2α −1 t (k) (k) E[|σ2 (s, x (s), h1 (s, τ , x (τ ))d τ , . . . , hn (s, τ , x(k) (τ ))d τ ) +3 2α − 1 0 0 0 ˆ s ˆ s (k−1) (k−1) (s), h1 (s, τ , x (τ ))d τ , . . . , hn (s, τ , x(k−1) (τ ))d τ )|2 ]ds. − σ2 (s, x 0

0

By using the Lipschitz continuity assumptions (7), (8) and α > 1/2, we have E[|x(k+1) (t) − x(k) (t)|2 ] n

≤ 3L (1 + ∑ 2

L2i )[1 + t +

i=1

α 2t 2α −1 ] 2α − 1

ˆ

t

E[|x(k) (s) − x(k−1) (s)|2 ]ds.

0

Therefore E[|x

(k+1)

(k)

(t) − x (t)| ] ≤ 2

3C12 [1 + T

α 2 T 2α −1 ] + 2α − 1

ˆ

t

E[|x(k) (s) − x(k−1) (s)|2 ]ds,

(12)

0

where C12 = L2 (1 + ∑ni=1 L2i ). From (11), by applying again the Schwarz inequality, the Itô isometry together with the growth conditions (5) and (6) for k = 1, we have ˆ t ˆ s ˆ s (1) (0) 2 E[|b(s, x0 , f1 (s, τ , x0 )d τ , . . . , fn (s, τ , x0 )d τ )|2 ]ds E[|x (t) − x (t)| ] ≤ 3t 0 0 0 ˆ s ˆ s ˆ t E[|σ1 (s, x0 , g1 (s, τ , x0 )d τ , . . . , gn (s, τ , x0 )d τ )|2 ]ds +3 0 0 0 ˆ ˆ s ˆ s α 2 T 2α −1 t E[|σ2 (s, x0 , h1 (s, τ , x0 )d τ , . . . , hn (s, τ , x0 )d τ )|2 ]ds +3 2α − 1 0 0 0 ˆ t n α 2 T 2α −1 2 2 2 )3K (1 + ∑ Ki ) E(1 + |x0 |2 )ds ≤ 3C1 (1 + T + 2α − 1 0 i=1 ≤ 32C22C12 (1 + T +

α 2 T 2α −1 )(t)(1 + E|x0 |2 ), 2α − 1

(13)

with C22 = K 2 (1 + ∑ni=1 Ki2 ). Now, for k = 1, replacing E[|x(1) (t) − x(0) (t)|2 ] in the inequality (12) with the value on the right hand side of

62


inequality (13) and integrating, we obtain (2)

(1)

α 2 T 2α −1 + ) 2α − 1

ˆ

t

E[|x(1) (s) − x(0) (s)|2 ]ds ˆ α 2 T 2α −1 2 t ≤ C22 (1 + E|x0 |2 )[32C12 (1 + T + )] sds 2α − 1 0 α 2 T 2α −1 2 t 2 )] × . ≤ C22 (1 + E|x0 |2 )[32C12 (1 + T + 2α − 1 2!

E[|x (t) − x (t)| ] ≤ 2

3C12 (1 + T

0

(14)

For k = 2, proceeding as before, we have E[|x(3) (t) − x(2) (t)|2 ] ≤ C22 (1 + E|x0 |2 )[32C12 (1 + T +

α 2 T 2α −1 3 t 3 )] × . 2α − 1 3!

(15)

Thus, by the principle of mathematical induction, we have E[|x(k+1) (t) − x(k) (t)|2 ] ≤

BM k+1t (k+1) , k = 0, 1, 2, . . . , 0 ≤ t ≤ T, (k + 1)!

(16)

2 2α −1

where B = C22 (1 + E|x0 |2 ) and M = 32C12 (1 + T + α2αT −1 ) is a constant depending only on α , T,C12 and E|x0 |2 . Thus sup E[|x(k+1) (t) − x(k) (t)|2 ] ≤ 0≤t≤T

BM k+1 T (k+1) , k = 0, 1, 2, . . . . (k + 1)!

(17)

This implies the mean-square convergence of the successive approximations uniformly on J. That is, x(m) (t) − x(n) (t) 2L2 (P) ≤ ≤ =

m−1

∑ x(k+1) − x(k) 2L (P) 2

k=n m−1 ˆ T

∑

BM k+1t (k+1) dt (k + 1)!

k=n 0 m−1 BM k+1T (k+2)

∑

k=n

(k + 2)!

→ 0,

as m, n → ∞.

Then, by applying the Chebyshev’s inequality and summing up the resultant inequalities, we have ∞

1

∞

BM k+1 T (k+2) k4 , (k + 2)! k=1

sup (|x(k+1) (t) − x(k) (t)|2 ) > 2 ] ≤ ∑ ∑ P[0≤t≤T k

k=1

(18)

where the series on the right side converges by ratio test. Hence the series on the left side also converges, so by the Borel-Cantelli lemma, we conclude that sup0≤t≤T |x(k+1) (t) − x(k) (t)|2 converges to 0, almost surely, that is, the successive approximations x(k) (t) converge, almost surely, uniformly on J to a limit x(t) defined by n

lim (x(0) (t) + ∑ (x(k) (t) − x(k−1) (t))) = lim x(n) (t) = x(t).

n→∞

k=1

n→∞

(19)

Since x(t) is the limit of nonanticipating functions and the uniform limit of a sequence of continuous functions, it itself is nonanticipating and continuous. From (11), we have


ˆ

t

ˆ

ˆ

s

s

x(t) = x0 + b(s, x(s), f1 (s, τ , x(τ ))d τ , . . . , fn (s, τ , x(τ ))d τ )ds 0 0 0 ˆ s ˆ s ˆ t σ1 (s, x(s), g1 (s, τ , x(τ ))d τ , . . . , gn (s, τ , x(τ ))d τ )dW (s) + 0 0 0 ˆ s ˆ s ˆ t α −1 h1 (s, τ , x(τ ))d τ , . . . , hn (s, τ , x(τ ))d τ )ds, + α (t − s) σ2 (s, x(s), 0

63

0

(20)

0

for all t ∈ J. This completes the proof of the existence of solutions of the stochastic fractional integrodifferential equation (9). Uniqueness: The uniqueness follows from the Itô isometry, the Lipschitz conditions (7) and (8). Let x(t, ω ) and y(t, ω ) be solution processes through the initial data (0, x0 ) and (0, y0 ) respectively, that is, x(0, ω ) = x0 (ω ) and y(0, ω ) = y0 (ω ), ω ∈ Ω. Let ˆ t ˆ t f1 (s, τ , x(s))d τ , . . . , fn (s, τ , x(s))d τ ) a(s, ω ) = b(s, x(s), 0 0 ˆ t ˆ t f1 (s, τ , y(s))d τ , . . . , fn (s, τ , y(s))d τ ), − b(s, y(s), 0 0 ˆ t ˆ t γ1 (s, ω ) =σ1 (s, x(s), g1 (s, τ , x(s))d τ , . . . , gn (s, τ , x(s))d τ ) 0 0 ˆ t ˆ t g1 (s, τ , y(s))d τ , . . . , gn (s, τ , y(s))d τ ), − σ1 (s, y(s), 0 ˆ t ˆ t 0 γ2 (s, ω ) = σ2 (s, x(s), h1 (s, τ , x(s))d τ , . . . , hn (s, τ , x(s))d τ ) 0 0 ˆ t ˆ t h1 (s, τ , y(s))d τ , . . . , hn (s, τ , y(s))d τ ). − σ2 (s, y(s), 0

0

Then by virtue of the Schwarz inequality and the Itô isometry, we have ˆ t 2 2 E[|x(t) − y(t)| ] ≤ 4E[|x0 − y0 | ] + 4tE[ |a(s, ω )|2 ds] 0 ˆ t ˆ t α 2t 2α −1 2 E[ |γ2 (s, ω )|2 ds] +4E[ |γ1 (s, ω )| ds] + 4 2 α − 1 0 0 2t 2α −1 ˆ t α ) E[|x(s) − y(s)|2 ]ds. ≤ 4E[|x0 − y0 |2 ] + 4C12 (1 + t + 2α − 1 0 ´t We define v(t) = E[|x(t) − y(t)|2 ]. Then the function v satisfies v(t) ≤ F + A 0 v(s)ds, where F = 4E[|x0 − y0 |2 ] 2 2α −1 and A = 4C12 (1 + t + α2αt −1 ). By the application of the Gronwall inequality, we conclude that v(t) ≤ F exp(At). Now assume that x0 = y0 . Then F = 0 and so v(t) = 0 for all t ≥ 0. That is, E[|x(t) − y(t)|2 ] = 0. Hence

P |x(t) − y(t)| = 0

for all t ∈ Q ∩ J = 1,

where Q denotes the set of all rational numbers. Since Q is dense in R and since the solutions are continuous and coincide on a countably dense subset of [0, T ], they must coincide, almost surely, on the entire interval [0, T ]. for all t ∈ J = 1, P |x(t, ω ) − y(t, ω )| = 0 that is, the solution of (9) is unique. This completes the proof of existence and uniqueness of solution of the stochastic fractional integrodifferential equation (9).

64


4 Example Consider the following stochastic fractional integrodifferential equation of the form ⎫ ⎪ ⎪ ⎪ e ds)dt ⎪ ⎪ 0 ⎪ ⎪ ˆ t ⎪ ⎪ 1 ⎪ 2 −t − 3 x(s) + (log x(t) + (2t + 1)e + 3 e ds)dW (t)⎬ ˆ 0 ⎪ ⎪ ⎪ 1 t − 1 x(s) 1 α 4 ⎪ + e ds)(dt) , ⎪ + (−2x(t) + ⎪ ⎪ 1 + x(t) 4 0 ⎪ ⎪ ⎪ ⎭

e−t x(t) + dx(t) = ( (29 + e−t )(1 + x(t))

x(0) = x0 .

ˆ

t

1 − 10 x(s)

(21)

´t 1 ´t 1 −t x(t) Here b = ( (29+ee−t )(1+x(t)) + 0 e− 10 x(s) ds), σ1 = (log x(t) + (2t 2 + 1)e−t + 3 0 e− 3 x(s) ds) and σ2 = (−2x(t) + ´ 1 1 t − 14 x(s) + ds). It can be easily seen that b, σ1 and σ2 satisfy the condition (5)-(8) of Theorem 2. 4 0e 1+x(t) Hence, by the theorem 2 the stochastic fractional integrodifferential equation (21) has a unique solution.

Acknowledgement The authors are thankful to the referees for the improvements of the paper.

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Evolution Towards the Steady State in a Hopf Bifurcation: A Scaling Investigation Vin´ıcius B. da Silva†, Edson D. Leonel Departamento de F´ısica, UNESP - Univ Estadual Paulista, Av. 24A, 1515 - Bela Vista, 13506-900, Rio Claro, SP, BRA Submission Info Communicated by V. Afaimovich Received 30 January 2017 Accepted 13 April 2017 Available online 1 April 2018 Keywords Hopf bifurcation Scaling properties Critical exponents Normal forms

Abstract Some scaling properties describing the convergence for the steady state in a Hopf bifurcation are discussed. Two different procedures are considered in the investigation: (i) a phenomenological description obtained from time series coming from the numerical integration of the system, leading to a set of critical exponents and hence to scaling laws; (ii) a direct solution of the differential equations, which is possible only in the normal form. At the bifurcation, the convergence to the stationary state obeys a generalized and homogeneous function. For short time, the dynamics giving by the distance from the fixed point is mostly constant when a critical time is reached hence changing the dynamics to a convergence for the steady state given by a power law. Both the size of the constant plateau and the characteristic crossover time depend on the initial distance from the fixed point. Near the bifurcation, the convergence is described by an exponential decay with a relaxation time given by a power law. ©2018 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction In general, the dependence on one or more parameters characterizes most of the dynamical systems. In the simple harmonic oscillator, the natural frequency of oscillation is an example of such parameter. The parameter variation may change the qualitative structure of the flow of solutions related to a dissipative dynamical system therefore changing the topological solution of a given attractor. A bifurcation is the name given to the qualitative change in the dynamics [1] due to a parameter variation. Bifurcations are observed in a variety of systems including dynamical population [2, 3], electric circuits [4, 5], chemical reactions [6, 7], discrete mappings [8, 9], laser [10–12] and many others [13–15]. There are two different classes of bifurcations: (i) local and; (ii) global. In a local bifurcation the variation of a control parameter produces a change of stability of a fixed point, indeed an attractor, and hence the topological modifications in the system can be confirmed by an investigation near the fixed point, therefore a local analysis. For a global bifurcation, invariant structures collide with each other and this include a collision between an invariant manifold and chaotic attractor, yielding in a destruction of the chaotic attractor. As a consequence, a major change in the global topology of the system can not be foreseen by a local analysis of fixed point. † Corresponding


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Vin´ıcius B. da Silva, Edson D. Leonel / Discontinuity, Nonlinearity, and Complexity 7(1) (2018) 67–79

In this class of bifurcation lives the so called crisis events [16–19]. We concentrate in a special type of local bifurcation, namely Hopf bifurcation. The achievement of the stationary state of a dissipative system, in many cases, leads to attractors in the phase space. Such attractors might be of zero dimension, id est fixed points, or one-dimensional - limit cycles - or higher dimension attractors. The eigenvalues of the Jacobian matrix for the dynamical equations of the system, while evaluated at the attractors, give their stability. The attractor is reached while an initial condition is given along the basin of attraction of an attractor and evolved for a sufficient long time, therefore an asymptotic dynamics. By asymptotic, one understands as long enough time such as t → ∞. However a clear discussion regarding the dynamics evolving towards the attractor and its corresponding scaling properties were, so far, not made. Our attempt here is to fill up this gap. Therefore, the main goal of this work is to explore the evolution towards the steady state at and near at a Hopf bifurcation. We focus particularly in its normal form. To do so we shall apply a scaling formalism like the one used in statistical mechanics [20,21] to describe phase transitions [22–24]. We consider a dynamical system described by a set of ordinary differential equations written in the so-called normal form. It mimics the dynamics of a complex system however keeping only the lowest nonlinear terms as possible to reproduce the phenomenon. This implies that a complicated set of equations can be expanded in Taylor series and that the normal form reproduces all the dynamics of such set of equations near the criticality. Before the bifurcation, the dynamics converges to a fixed point which is asymptotically stable. At the bifurcation, the fixed point loses stability and after the bifurcation it repeals the dynamics which converges to a closed orbit in a plane, indeed a limit cycle. Because the attractor is a closed cycle in a plane, it turns out that the polar coordinates is the most convenient set of variables to describe the dynamics, hence the dynamics is obtained by the radius, ρ , and angle φ . At the bifurcation, we notice the following: given an initial condition close to the fixed point and inside of the basin of attraction of the fixed point, the dynamical variable ρ keeps almost constant in a long plateau until it suffers a changeover marked by a characteristic crossover time and decays to the fixed point with a behavior described by a power law. The size of the plateau depends on the initial distance from the fixed point as so the crossover time. This type of dynamics obeys a generalized and homogeneous function that leads to a set of three critical exponents yielding also in a scaling law. Near the bifurcation, the dynamics is no longer described by a homogeneous function, but rather by an exponential decay. The relaxation time is given by a power law whose argument corresponds to the distance, in the parameter, to where the bifurcation happened. This paper is organized as follows. In Section 2 we discuss the normal form of Hopf bifurcation. Section 3 is devoted to describe a phenomenological approach based on a set of three scaling hypotheses leading to critical exponents and hence to a scaling law. In Section 4 we dedicate to investigate an analytical description of the convergence to the steady state in the Hopf bifurcation confirming the results obtained by numerical simulation. Discussions and conclusions are made in Section 5.

2 The normal form of the Hopf Bifurcation A set of differential equations that describes a Hopf bifurcation [1] is written as x˙ = xμ − w0 y + (ax − by)(x2 + y2 ) + O(5),

(1)

y˙ = yμ + w0 x + (ay + bx)(x + y ) + O(5),

(2)

2

2

where x and y are dynamical variables, μ is a control parameter, a, b and w0 are constants. Here the term O(5) represents the higher-order term of the type xk1 and yk2 with k1 + k2 = 5. The following lemma is proved in Appendix this paper.


69

-1

10

Numerical data for μ < 0

10

-0

-1

10

-1

y

10

0

Numerical data for μ > 0

0 -1

-10

-1

-10

(b) -0

-1

-10

0

-1

10

-10

-10

-0

-10

0

-1

10

-1

x

-0

10

10

-0

-0

10

0

10

Fig. 1 Phase portraits for y vs. x for values of μ : (a) before and, (b) after the bifurcation, respectively.

Theorem 1. The system described by equations (1) and (2) is locally topologically equivalent near the origin to system described by equations (3) and (4). x˙ = xμ − w0 y + (ax − by)(x2 + y2 ),

(3)

y˙ = yμ + w0 x + (ay + bx)(x + y ).

(4)

2

2

Therefore, the higher-order terms in the set of equations do not interfere in the bifurcation behavior of the system. Since after the bifurcation the dynamics lives in a plane, it is convenient to use polar coordinates instead of rectangular variables. Fixing w0 = 1 and a = −1, equations (3) and (4) can be written in polar coordinates as dρ = μρ − ρ 3 , dt dφ = 1 + bρ 2 , dt

(5) (6)

where ρ and φ describe the radial and the angular coordinates, respectively. Besides that, μ controls the stability of the fixed point at the origin, w0 gives the frequency of infinitesimal oscillations, and b is a free parameter. In the next section we discuss the scaling properties of dynamics at a Hopf bifurcation.

3 A phenomenological description and scaling properties We discuss now a phenomenological approach to explore the evolution towards the steady state at and near at a Hopf bifurcation. As we could see in the previous section, the set of equations (5) and (6) are characterized by radial and angular variables. However, to explore the scaling properties that characterize the bifurcation, the equations are analyzed separately. We start first with the radial equation. The fixed points are obtained by solving f (ρ ) = μρ − ρ 3 = 0. The solutions have physical meaning only √ when ρ ≥ 0. We therefore end up with two fixed points ρ1∗ = 0, and ρ2∗ = μ . As we can see in Figure 1, when μ < 0 the origin ρ1∗ = 0 becomes a stable spiral whose sense of rotation depends on the sign of w0 . For μ = 0 the origin is still a stable spiral however the speed of convergence is different from μ < 0. Finally, for μ > 0 √ there is an unstable spiral at the origin and a stable circular limit cycle at ρ2∗ = μ , as discussed in [1]. The natural variable to describe the decay to the steady state is the distance from the fixed point [25]. So, for the fixed point ρ1∗ = 0 the distance taken from the stationary state is the own dynamical variable ρ (t). The decay to the steady state must also depend of the time t, the initial condition ρ0 , and the parameter μ . Since μ = 0 defines the bifurcation, the convergence to the fixed point is shown in Figure 2 for different initial conditions of ρ0 . We see from Figure 2 that depending on the initial condition ρ0 , the dynamics stays in a constant plateau for different intervals of time until eventually reaching a characteristic crossover time, tx , and the orbit changes

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Vin´ıcius B. da Silva, Edson D. Leonel / Discontinuity, Nonlinearity, and Complexity 7(1) (2018) 67–79 0

10

ρ0 =1

-1

10 10

βρ

-2

ρ0 =10

-1

ρ0 =10

-2

ρ0 =10

-3

ρ0 =10

-4

ρ (t)

10 10

-3

-4

10

-5

0

3

10

10

10

t

6

9

10

Fig. 2 Convergence to the steady state at ρ1∗ = 0 for different initial conditions as shown in the figure. The parameter used was μ = 0 and the differential equations (5) and (6) were numerical integrated using a 4th order Runge-Kutta algorithm. We chose also w0 = 1.0 and b = 1.0.

from a constant regime to a power law decay given by a critical exponent βρ . We notice that the length of the plateau shows a clear dependence on the initial condition ρ0 . Based on the behavior observed from Figure 2, we can propose the following scale hypotheses: 1. For a short interval of time t, say t tx , the convergence to the steady state is given by α

ρ (t) ∝ ρ0 ρ ,

t tx .

(7)

A quick analysis of Figure 2 allows us to conclude that αρ = 1. 2. For a sufficient large t, say t tx , the convergence to the steady state is given by

ρ (t) ∝ t βρ ,

t tx ,

(8)

where βρ gives the decay exponent. 3. The characteristic crossover time tx that describes the changeover from a constant regime to a power law decay is given by (9) tx ∝ ρ0 zρ , where zρ is called as the changeover exponent. The exponents zρ and βρ are obtained by their specific plots. After the constant plateau, a power law fitting gives βρ = −0.499(3) −0.5. To obtain the exponent zρ we need to analyze the behavior of tx vs. ρ0 , as shown in Figure 3. The slope obtained for tρ vs. ρ0 is zρ = −2.00360(6) −2. Based on the behavior shown in Figure 2 and considering the three scaling hypotheses, it is possible to describe the behavior of ρ as a homogeneous and generalized function of the variables t and ρ0 , when μ = 0, as

ρ (ρ0 ,t) = ρ (c ρ0 , d t),

(10)

where is a scaling factor, c and d are characteristic exponents. As is a scaling factor, we chose c ρ0 = 1, −1/c therefore leading to = ρ0 . By substituting this expression in equation (10) we end up with −1/c

ρ (ρ0 ,t) = ρ0

−d/c

ρ (1, ρ0

t).

(11)


10

71

8 Numerical data Power Law fitting

6

10

z

ρ

=

-2

.00

36

tx 104

0(

6)

2

10

0

10 -5 10

-4

10

-3

10

-2

ρ0

10

-1

10

0

10

Fig. 3 Plot of the crossover time tx against the initial condition ρ0 together with its power law fitting giving zρ = −2.00360(6). −d/c

We assume ρ (1, ρ0 t) as a constant for t tx . Comparing equation (11) with the first scaling hypothesis we conclude that αρ = −1/c. We now chose c ρ0 = 1 yielding = t −1/c . Substituting in equation (10) we obtain for t tx that

ρ (ρ0 ,t) ∝ t −1/d .

(12)

A direct comparison of this result with the second scaling hypothesis gives βρ = −1/d. Finally, by comparing α /βρ

the two expressions obtained for the scaling factor we arrive in tx = ρ0 ρ scaling hypothesis allows us to obtain the following scaling law zρ = αρ /βρ .

. A comparison with the third

(13)

The knowledge of any two exponents allows determining the third one by substituting equation (13). Besides α that, the exponents can also be used to rescale the variables ρ (t) and t in a convenient way such that ρ → ρ /ρ0 ρ z and t → t/ρ0ρ and overlap all curves of ρ (t) vs. t shown in Fig. 2 onto a single and hence universal curve, as shown in Figure 4. Once we have discussed the convergence to the steady state at the bifurcation point, we now discuss the dynamics for μ = 0 which characterizes the neighborhood of a Hopf bifurcation. The convergence to the steady state is marked by an exponential law of the type

ρ (t) − ρ ∗ (ρ0 − ρ ∗ )e−t/τ , where τ is the relaxation time described by

τ ∝ μδ ,

(14)

(15)

where δ is a relaxation exponent. The achievement of the relaxation time τ is discussed as follows. An initial condition is given along the basin of attraction of the attractor. The dynamics is integrated in time using a 4th order Runge-Kutta algorithm. As soon as the distance from the attractor reaches a threshold smaller than 10−8 , the integration is interrupted, the time until that point is registered and another simulation is started for a different value of μ . Figure 5 shows the behavior of τ vs. μ . A power law fitting gives δ = −0.969(9) −1.

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Vin´ıcius B. da Silva, Edson D. Leonel / Discontinuity, Nonlinearity, and Complexity 7(1) (2018) 67–79 0

10

-1

10 10

-2

αρ

ρ/ ρ0

10 10

-3

ρ0=10 ρ0=10

-4

10

ρ0=1

ρ0=10 ρ0=10

-5

-1 -2 -3 -4

0

-6

6

10

10

10

z ρ0 ρ

t/

Fig. 4 Overlap of all curves shown in Figure 2 onto a single and universal plot by considering a convenient rescale of the axis. 6

10

Numerical data Power law fitting

10

5

δ = -0.969(9)

τ 104 10

3

2

10 -5 10

-4

10

-3

10

μ

-2

10

10

-1

Fig. 5 Plot of the behavior of relaxation time τ against μ . A power law fitting furnishes δ = −0.969(9) −1.

Let us now discuss the scaling results obtained for the angular equation (see equation (6)). Figure 6 shows the behavior of φ vs. t for different initial conditions and considering the parameter μ = 0. We see that different initial conditions produce different curves. Depending on the initial value of φ0 , the orbits stay in a plateau of constant φ for different ranges of time. Moreover after reaching the crossover time, tx , the orbit changes from a constant regime to a power law growth characterized by the critical exponent βφ . The length of the plateau depends on the initial condition φ0 .


73

Fig. 6 Plot of the angular variable φ as a function of time t for different initial conditions, as labeled in the figure. The parameter used was μ = 0.

This behavior allows us to announce the hypotheses: 1. For t tx , the dynamical variable φ behaves as α

φ (t) ∝ φ0 φ ,

t tx ,

(16)

t tx .

(17)

where from Figure 6 we see easily that αφ = 1. 2. For large t, typically t tx , the dynamics is as follows

φ (t) ∝ t βφ ,

3. The characteristic crossover time tx that describes the changeover from a constant regime to a power law growth is given by (18) tx ∝ φ0 zφ , where zφ gives the changeover exponent. The exponents zφ and βφ are obtained by their specific plots. After the constant plateau, a power law fitting gives βφ = 0.996(1) 1. To obtain the exponent zφ we need to analyze the behavior of tx vs. φ0 , see Figure 7. The slope obtained for tx vs. φ0 is zφ = 1.00383(5) 1. These scaling hypotheses lead to the same scaling law as discussed before, therefore we obtain that zφ = αφ /βφ . When the dynamical variables are scaled as α z φ → φ /φ0 φ and t → t/φ0 φ , all curves of φ (t) vs. t are overlapped into a single and universal curve, see figure 8. The dynamics of φ vs. t for μ = 0 is remarkable similar to the results obtained above. In the next section, we discuss the convergence to the steady state for a Hopf bifurcation by considering an analytical approach that involves solving the differentials equations (5) and (6) for both μ = 0 and μ = 0. 4 An analytical approach to the steady state for the Hopf Bifurcation Let us now apply the scale formalism to explore the Hopf bifurcation considering the solution of the differential equations (5) and (6). We start with considering the evolution towards the fixed point at the bifurcation point

74


μ = 0. The differential equation is then written as dρ = −ρ 3 . dt A straightforward integration gives

ρ (t) =

ρ0 1 + 2t ρ02

(19)

.

(20)

Let now discuss the implications of equation (20) for specific ranges of t. Considering the case where 2t ρ02 1, which is equal to t tx , we realize that ρ (t) ∝ ρ0 . Therefore, according to the first scaling hypothesis for the radial coordinate we can say the critical exponent αρ = 1. However, in the case 2t ρ02 1 that corresponds to t tx we get ρ (t) ∝ t −1/2 . (21) A quick comparison with the second scaling hypothesis of the previous section tell us that βρ = −1/2. The last case is when 2t ρ02 = 1, which is the case of t = tx . Then we end up with tx ∝ ρ0−2 .

(22)

Finally, according to third scale hypothesis, we can say that zρ = −2. These results reinforce our phenomenological approach as discussed in the earlier sections. We then discuss the case of μ = 0, therefore considering the convergence to the steady state at a neighborhood of a Hopf bifurcation. We have to solve the following differential equation ddtρ = μρ − ρ 3 . A direct integration gives √

ρ (t) − μ

√

μ −2μ t e . 2

(23)

Comparing this result with equations (14) and (15) we get that the relaxation exponent δ = −1. Thus, the results obtained in this section by considering an analytical approach are in complete agreement with the numerical results shown in previous section. A next step is to investigate the angular equation. We consider first the case of μ = 0. The differential ρ2 equation, when incorporated the solution of ρ (t) is written as ddtφ = w0 + b 1+2t0 ρ 2 . After integration we obtain 0 the following b φ (t) = φ0 + w0t + ln (1 + 2t ρ02 ). (24) 2 Let now discuss the implications of equation (24) for specific ranges of t. Considering the case where w0t + b2 ln (1 + 2t ρ02 ) φ0 , which is equal to t tx , we realize that φ (t) ∝ φ0 leading to αφ = 1. However, in the case w0t φ0 + b2 ln (1 + 2t ρ02 ) that corresponds to t tx we get φ (t) ∝ t, giving βφ = 1. The last case is obtained when w0t = φ0 + b2 ln (1 + 2t ρ02 ) ∼ = φ0 , which is the case of t = tx . Then we end up with tx ∝ φ0 , leading to zφ = 1. This approximation is valid since the function ln(t) varies slowly as compared to the linear term t. 5 Discussions and conclusions In a recent result involving bifurcation in 1-D mappings, it was considered particularly a family of logistic-like mappings [26]. The convergence to the fixed point at a bifurcation was investigated using both the phenomenological approach, using a scaling function and obtained at the end a scaling law with three critical exponents as well as an analytical procedure, transforming the equation of differences in a differential equation; later on integrating it easily. The phenomenological investigation considered, as made here, a set of three scaling hypotheses and specific plots to get the critical exponents. It was proved there [26] the exponent α = 1 is a constant while


75

4

10

Numerical data Power Law fitting 10

tx

)

3(5

2

8 03

.0

=1 zφ

0

10

10

-2

-2

10

10

0

2

φ0

10

4

10

Fig. 7 Plot of the crossover time tx against the initial condition φ0 together with its power law fitting for μ = 0 with slope zφ = 1.00383(5).

10

8

φ0 =10 φ0 =10 φ0 =1 φ0 =10

φ(t)/ φ0

αφ

φ0 =10

4

φ0 =10

10

0

10

-4

10

-2 -1

2 3

0

4

10

t

z /φ0 φ

10

8

10

Fig. 8 Overlap of all curves shown in Figure 6 onto a single and therefore universal plot by considering the following α z transformations: φ → φ /φ0 φ and t → t/φ0φ .

β = −1/γ and z = −γ do depend on the nonlinearity of the mapping. The authors show that for a logistic-like γ map of the type xn+1 = Rxn (1 − xn ), where γ ≥ 1 the scaling law z = α /β is also observed. Soon after [26], the authors made a Taylor expansion of the second iterated of the mapping and described with success [27] the critical exponents for a period doubling bifurcation. For this bifurcation, the exponents do not depend on the nonlinearity of the mapping and are then universal: α = 1, β = −1/2 and z = −2. As discussed in Refs. [26, 27] the critical exponents are packed in Table 1. An extension of the procedure was made also for a set of bifurcations observed in ordinary differential equations (see Ref. [28]) considering a set of three important bifurcations of 1-D flow namely: saddle-node, transcritical and supercritical pitchfork. The results are summarized in Table 2. In the present paper, our results extend the formalism to be used in a Hopf bifurcation. Because after the bifurcation the attractor is a limit cycle in a plane, it turns out to be convenient to use polar coordinate to investigate the dynamics. Instead of a single set of critical exponent, the dynamics needs two sets, among the

76

Vin´ıcius B. da Silva, Edson D. Leonel / Discontinuity, Nonlinearity, and Complexity 7(1) (2018) 67–79 γ

Table 1 Table of critical exponents α , β , z and δ observed for a family of logistic-like map xn+1 = Rxn (1 − xn ), as discussed in Refs. [26, 27]. Bifurcation

α

β

z

δ

− 1γ − 1γ − 12

−γ

−1

−γ

−1

−2

−1

Pitchfork

1

Transcritical

1

Period doubling

1

Table 2 Table of critical exponents α , β , z and δ for the three bifurcations discussed in Ref [28]. Equation

Bifurcation

α

β

z

δ

μ − x2

Saddle-node

1

−1

−1

− 12

x˙ = μ x − x2

Transcritical

1

−1

−1

−1

1

− 12

−2

−1

x˙ = x˙ =

μ x − x3

Supercritical pitchfork

Table 3 Table of critical exponents αρ ,φ , βρ ,φ , zρ ,φ and δ for a Hopf bifurcation. Radial

Angular

α

1

1

β

−1/2

1

z

−2

1

δ

−1

–

exponent δ . One set describes the convergence in the radial coordinate while the other set describes the evolution of the angular variable. Our results can be compacted as shown in Table 3. The results discussed here allow to understand the scaling properties of dynamical variable at and near at a Hopf bifurcation. Our results can be an alternative form to investigate and classify the type of bifurcation in experimental systems, e.g., electrical circuits, when the set of equations describing the dynamics are not all known. Presently we are working in a Chua circuit to investigate such properties experimentally.

Acknowledgements V.B.S. thanks to FAPESP (2015/23142-0). E.D.L acknowledges support from FAPESP (2012/23688-5) and CNPq (303707/2015-1) Brazilian agencies.

Appendix: Proof of Lemma 1 Proof. In this appendix we discuss the proof of lemma 1. This proof is remarkably similar to the Kuznetsov’s procedure made in [29]. Writing equations (1) and (2) into the complex form, we have z˙ = z(μ + iw0 ) + |z| z2 (a + ib) + · · · ,

(25)

and while truncating the higher order terms we obtain z˙ = z(μ + iw0 ) + |z| z2 (a + ib).

(26)

Our objective is to prove that equation (25) is locally topologically equivalent near to the origin to equation (26). To do that we suppose:


77

Fig. 9 The Poincaré (return) map near Hopf Bifurcation (Source: Kuznetsov, 1998, p. 109).

Step 1 (existence and uniqueness of the limit cycle). Writing the equation (25) in polar coordinates (ρ , φ ) with z = ρ eiφ we end up with dρ = ρ (μ + aρ 2 ) + Φ(ρ , φ ), dt dφ = w0 + Ψ(ρ , φ ), dt where Φ and Ψ are smooth functions of (ρ , φ , μ ). The terms Φ and Ψ incorporate the higher order terms of the equation (25). Once both a and w0 are constants we can fix them as a = −1 and w0 = 1. This gives dρ = ρ (μ − ρ 2 ) + Φ(ρ , φ ), dt dφ = 1 + Ψ(ρ , φ ). dt

(27) (28)

For a given μ , an orbit of the system described by equations (27) and (28) starting from (ρ , φ ) = (ρ0 , 0) has a typical representation as shown in figure 9 where ρ = ρ (φ , ρ0 ), ρ (0, ρ0 ) = ρ0 with ρ satisfying the equation

ρ (μ − ρ 2 ) + R(ρ , φ ) dρ = ρ (μ − ρ 2 ) + R(ρ , φ ), = dφ 1 + Ψ(ρ , φ )

(29)

where R(ρ , φ ) is a smooth function that incorporates the higher order terms. Since ρ (φ , 0) ≡ 0 we have d2ρ d2ρ dρ (ρ , 0) = (ρ , 0) = 2 (ρ , 0) = · · · = 0. dφ dρ dφ dφ

(30)

A Taylor series expansion of ρ (φ , ρ0 ) is written as

where u1 (φ ) =

ρ (φ , ρ0 ) = u1 (φ )ρ0 + u2 (φ )ρ02 + u3 (φ )ρ03 + . . . ,

(31)

d ρ 1 d 2 ρ 1 d 3 ρ , u ( φ ) = , u ( φ ) = ,... 2 3 d ρ0 ρ0 =0 2! d ρ02 ρ0 =0 3! d ρ03 ρ0 =0

(32)

Substituting equations (32) in (31) and sorting terms according to power of ρ0 , we arrive in the following linear differential equations du1 (φ ) = μ u1 (φ ), dφ du2 (φ ) = μ u2 (φ ), dφ du3 (φ ) = μ u3 (φ ) − u1 (φ )3 , dφ

78


Fig. 10 Construction of the homeomorphism neat the Hopf bifurcation.

with the initial conditions u1 (0) = 1, u2 (0) = u3 (0) = 0. The solutions for the differential equations above are respectively u1 (φ ) = eμφ , u2 (φ ) = 0, eμφ (1 − e2μφ ) . u3 (φ ) = 2μ Notice that these expressions are independent of R(ρ , φ ). Thus the Poincaré return map ρ0 ρ1 = ρ (2π , ρ0 ) is written as ρ1 = ρ0 e2μπ − e2μπ (2π + O(μ ))ρ03 + H.O.T., (33) for all smooth R = H.O.T.. Now according to Ref. [29] this map can be analyzed for sufficiently small ρ0 and |μ | with the Implicit Function Theorem. By the application of the Implicit Function Theorem we can establish there is a neighborhood of the origin in which the map has only the trivial fixed point for small μ < 0 and an √ (0) additional fixed point, ρ0 = μ + . . ., for small μ > 0. Taking into account that a positive fixed point of the map corresponds to a limit cycle of the system, we can infer that equations (27) and (28) with any higher-order terms have a unique (stable) limit cycle bifurcating from the origin and existing for μ > 0 just as equation (26). Therefore, the higher-order terms do not affect the limit cycle bifurcation in some neighborhood of z = 0 for |μ | sufficiently small. Step 2 (Construction of a homeomorphism). Now we have to prove that the phase portrait of equations (1) and (2) are topologically equivalent to that one obtained from equations (3) and (4). In order to do so, we establish a small but positive μ . From the step 1 we verified the systems (25) and (26) both have a limit cycle in some neighborhood of the origin. Notice that the transition from equation (28) to (31) is equivalent to the introduction of a new time re-parametrization such that the return time to the half-axis φ = 0 (mod 2π ) is the same for all orbits starting on this axis with ρ0 > 0. Next, we apply a linear scaling of ρ -coordinate of the system √ (28) such that the point of intersection of the cycle and the horizontal half-axis be at ρ = μ . We now define a map x x by the following construction (see Ref. [29] for further details). Take a point x = (x, y) and find values (ρ0 , τ0 ), where τ0 is the minimal time required for an orbit of equation (26) to approach the point x starting from the horizontal half-axis with ρ = ρ0 and construct an orbit of the system (25) on the x, y) (see figure 10). Set x = 0 for time interval [0, τ0 ] starting at this point. Denote the resulting point by x = ( x = 0. In this way the map constructed is a homeomorphism that, for μ > 0, maps orbits of system (26) in some neighborhood of the origin into orbits of (25) preserving time direction.


79

References [1] Strogatz, S.H. (2015), Nonlinear dynamics and chaos: with applications to physics, biology, chemistry and engineering, Westview Press. [2] Bashkirtseva, I. and Ryashko, L. (2011), Sensitivity analysis of stochastic attractors and noise-induced transitions for population model with Allee effect, Chaos, 21, 047514. [3] Strogatz, S.H. (2000), From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators, Physica D: Nonlinear Phenomena, 143, 1. [4] Georgiou, I.T. and Romeo, F. (2015), Multi-physics dynamics of a mechanical oscillator coupled to an electro-magnetic circuit, Int. J. Nonlinear. Mech., 70, 153. [5] Gardine, L., Fournier-Prunaret, D., and Charge, P. (2011), Border collision bifurcations in a Two-dimensional PWS map from a simple circuit, Chaos, 21, 023106. [6] Inarrea, M., Palacian, J.F., and Pascual, A.I. (2011), Bifurcations of dividing surfaces in chemical reactions, J. Chem. Phys., 135, 014110. [7] Bakes, D., Schreiberova, L., and Schreiber, I. (2008), Mixed-mode oscillations in a homogeneous pH-oscillatory chemical reaction system, Chaos 18, 015102. [8] Guckenheimer, J. (2008), Return maps of folded nodes and folded saddle-nodes, Chaos, 18, 015108. [9] Philominathan, P., Santhiah, M., Mohamed, IR., Murali, K., and Rajasekar, S. (2011), Chaotic dynamics of a simple parametrically driven dissipative circuit, Int. J. Bifurcations and Chaos, 21, 1927. [10] Virte, M., Panajotov, K., and Thienpont, H. (2013), Deterministic polarization chaos from a laser diode, Nat. Photonics, 7, 60. [11] Doedel, EJ. and Pando-L, CL. (2011), Isolas of periodic passive Q-switching self-pulsations in the three-level:two-level model for a laser with a saturable absorber, Phys. Rev. E., 84, 056207. [12] Cavalcante, H.L.D.S. and Leite-Rios, J.R. (2008), Experimental bifurcations and homoclinic chaos in a laser with a saturable absorber, Chaos, 18, 023107. [13] Guo, Y. and Luo, A.C.J. (2012), Parametric analysis of bifurcation and chaos in a periodically driven horizontal impact, Int. J. Bifurc. Chaos, 22, 1250268. [14] Luo, A.C.J. and O’Connor, D. (2009), Periodic motions and chaos with impacting chatter with stick in a gear transmission system, Int. J. Bifurc. Chaos, 19, 2093. [15] Yang, J.H., Sanjuan, M.A.F., Liu, H.G., and Cheng G. (2015), Bifurcation Transition and Nonlinear Response in a Fractional-Order System, J. Comput. Nonlinear. Dyn., 10, 061017. [16] Grebogi, C., Ott, E., and Yorke, J.A. (1983), Chaotic attractors in crisis, Phys. Rev. Lett., 48, 1507. [17] Grebogi, C., Ott, E., and Yorke, J.A. (1983), Crises, sudden changes in chaotic attractors, and trasient chaos, Physica D, 7, 181. [18] Leonel, E.D. and McClintock, P.V.E. (2005), A crisis in the dissipative Fermi accelerator model, J. Phys. A: Math. Gen, 38, L425. [19] Oliveira, D.F.M., Leonel, E.D., and Robnik, M. (2011), Boundary crisis and transient in a dissipative relativistic standard map, Phys. Lett. A, 375, 3365. [20] Helrich, C.S. (2009), Modern thermodynamics with statistical mechanics, Heidelberg: Springer-Verlag. [21] Reif, F. (1965), Fundamentals of statistical and thermal physics, New York: McGraw-Hill. [22] Pottier, N. (2010), Nonequilibrium statistical physics, linear irreversible processes, Oxford: Oxford University Press. [23] Cardy, J. (1996), Scaling and renormalization in statistical physics, Cambridge: Cambridge University Press. [24] Kadanoff, L.P. (1999), Statistical physics: statics, dynamics and renormalization, Singapore: World Scientific. [25] Leonel, E.D., da Silva, J.K.I., and Kamphorst, S.O. (2002), Relaxation and transients in a time-dependent logistic map, Int. J. Bifurc. Chaos, 12, 1667. [26] Teixeira, R.M.N., Rando, D.S., Geraldo, F.C., Costa Filho, R.N., Oliveira, J.A., and Leonel, E.D. (2015), Convergence towards asymptotic state 1-D mappings: A scaling investigation, Phys. Lett. A, 379, 1246 [27] Teixeira, R.M.N., Rando, D.S., Geraldo, F.C., Costa Filho, R.N., Oliveira, J.A., and Leonel, E.D. (2015), Addendum to: “Convergence towards asymptotic state 1-D mappings: A scaling investigation”, Phys. Lett. A, 379, 1246, 1796. [28] Leonel, E.D. (2016), Defining universality classes for three different local bifurcations, Commun. Nonlinear Sci. Numer. Simulat., 39, 520. [29] Kuznetsov, Y.A. (1998), Elements of applied bifurcation theory, New York: Springer Science & Business Media, 112.



Integrability of a Time Dependent Coupled Harmonic Oscillator in Higher Dimensions Ram Mehar Singh1 , S B Bhardwaj2†, Kushal Sharma3 , Richa Rani2 , Fakir Chand2 , Anand Malik4 1

Department of Physics, Chaudhary Devi Lal University Sirsa-125055, India Department of Physics, Kurukshetra University Kurukshetra-136119, India 3 Department of Mathematics, National Institute of Technology, Hamirpur-177005, India 4 Department of Physics, Chaudhary Bansi Lal University, Bhiwani-127021, India 2

Submission Info

Abstract

Keywords

Within the frame work of extended complex phase space characterized by x = x1 + ip4 , y = x2 + ip5 , z = x3 + ip6 , px = p1 + ix4 , py = p2 + ix5 and pz = p3 + ix6 , we investigate the exact dynamical invariant for a coupled harmonic system in three dimensions. For this purpose Lie-algebraic method is employed and the invariant obtained in this work may play an important role in reducing the order of differential equations, solution of Cauchy system and to check the accuracy of a numerical simulation.

Exact Invariant PT - Symmetric Hamiltonian Lie-algebraic method


Communicated by V. Afaimovich Received 14 February 2017 Accepted 18 May 2017 Available online 1 April 2018

1 Introduction Various physical properties of a dynamical system can be explained by the real Hamiltonian systems [1, 2] but there are some other aspects of a dynamical system which can not be explained in this way. It is the complex Hamiltonian which explain such aspects of a system successfully [1–4]. The complex Hamiltonian helps to understand the several phenomena in the area of physical sciences [5, 6] like phenomena pertaining to resonance scattering in atomic, molecular and nuclear physics, population growth in biology [7] and study of delocalization transition for the unbinding of vortices in type-II superconductors [8] and to some chemical reactions also. In addition to this complex Hamiltonian has been used to study some other theoretical concepts i.e. complex trajectories with regard to the calculation of a semiclassical coherent state propagator in the path integral method have attracted a particular interest in laser physics [9]. There are various ways of complexifying a given Hamiltonian, but here we use the scheme given by Xavier and de Aguir [9], used to develop an algorithm for the computation of semiclassical coherent state propagator to transform potentials in extended complex phase space approach (ECPSA). The real and imaginary parts of x † Corresponding


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Ram Mehar Singh et al. / Discontinuity, Nonlinearity, and Complexity 7(1) (2018) 81–94

and p are introduced as x = Re x + i Im x , p = Re p + iIm p, if, we define x1 = Re x, p2 = Im x, p1 = Re p, x2 = Im p, then x and p can be defined as x = x1 + ip2 , p = p1 + ix2 .

(1)

The presence of variables (x1 , p2 , x2 , p1 ) in the above transformations may be regarded as some sort of coordinate momentum interactions of a dynamical system. Note that, in this complexification scheme, the degrees of freedom of the underlying system just become double. Another class of complex Hamiltonians called the PT - symmetric Hamiltonians in which the Hamiltonian is non-hermitian but eigenvalue spectrum, for certain parametric domain, is real and this reality of the spectrum is a consequence of the combined action of parity and time reversal invariance of H [10–12]. The parity operator Pˆ and time reversal operator Tˆ are defined by their action on position and momentum operators as Pˆ : xˆ −→ −xˆ ; pˆ −→ − p, ˆ ˆ T : xˆ −→ xˆ ; pˆ −→ − pˆ ; iˆ −→ −iˆ.

(2) (3)

However, the combined parity and time, (Pˆ Tˆ ), operator has the following effects Pˆ Tˆ : xˆ −→ −xˆ ; pˆ −→ pˆ ; iˆ −→ −iˆ.

(4)

The quantum aspect of a complex Hamiltonian has been studied extensively at various levels [11–15] but the classical aspect of these systems is not still explored to the same extent. In classical aspect, the integrability of the dynamical system has a significant contribution. The integrability of a dynamical system plays an important role to study a dynamical system and lack of it leads to a phase space with non-zero measure of chaotic trajectories. For defining the integrability of a dynamical system, there is some criteria for real and involuntary invariants of a dynamical system [1, 16, 17], but there is no such prescription for the invariants associated with a complex Hamiltonian system in spite of the fact that these Hamiltonians have been in use for quite long time [18–20]. It is worth to mention that not only the complex Hamiltonians are expected to admit complex invariants but real Hamiltonians are also found to exhibit complex invariants. The complex invariants play a pivotal role in the domain of variety of fields [21] mainly molecular Physics and accelerator Physics etc in so far as they are capable of fascinating reduction of the problem of the solutions of the equations of motion, fermion masses and quark mixing in particle physics, CP-conservation, two-Higgs-doublet model scalar potential [22–24], in classification of Yang-Mill fields [25, 26]. Thus it is evident from the fact that integrability of complex dynamical system require further investigation and invariants of a dynamical system offers further insight into the detailed nature of the system under consideration. As exact invariants are possible only for few systems and most of the systems in the literature have been studied using approximation or perturbation methods and accordingly one deals with approximate invariant or first integral of motion but in the present study, we restrict ourselves to the study of exact invariants of a dynamical system. In this part, several attempts have been made to construct the invariants in lower dimensions [27–29], but no attempt has been made in higher dimensions. With the same motivation, we construct the complex invariant for a time dependent coupled harmonic oscillator in three dimensions. The paper is described as follow: in section-2, we are devoted with the mathematical formalism of the Lie-algebraic method to construct the exact invariants for a dynamical system. Under the same mathematical prescriptions, results and discussion for the construction of invariants for a three-dimensional coupled harmonic oscillator are addressed in section-3 and finally, concluding remarks are given in section-4.


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2 Lie-algebraic approach For a three-dimensional complex system described by H(x, y, z, px , py , pz ), the transformations for the position and momenta variables is defined as x = x1 + ip4 ,

y = x2 + ip5 , z = x3 + ip6 ,

px = p1 + ix4 , py = p2 + ix5 , pz = p3 + ix6 .

(5)

The presence of variables (x3 , p3 , x4 , p4 ) in the above transformations may be regarded as some sort of coordinatemomentum interactions of a dynamical system. Under the transformation(5), we derive d 1 ∂ 1 ∂ 1 ∂ d d ∂ ∂ ∂ = ( = ( = ( −i ), −i ), −i ), dx 2 ∂ x1 ∂ p4 dy 2 ∂ x2 ∂ p5 dz 2 ∂ x3 ∂ p6 1 ∂ d 1 ∂ d 1 ∂ ∂ ∂ ∂ d = ( −i ), = ( −i ), = ( −i ). d px 2 ∂ p1 ∂ x4 d py 2 ∂ p2 ∂ x5 d pz 2 ∂ p3 ∂ x6

(6)

The Hamiltonian H(x, y, z, px , py , pz ) for a three-dimensional system in complex phase space is expressed by H(x, y, z, px , py , pz ) = ∑ hm (t)Γm (x1 , p4 , x2 , p5 , x3 , p6 , p1 , x4 , p2 , x5 , p3 , x6 ),

(7)

m

where Γm ’s depend upon time implicitly and hm (t) are the complex coefficients function of t. Here the dynamical algebra is the Lie algebra of the Γm ’s and closed with respect to the Poisson bracket, namely k Γk , [Γm , Γn ] = ∑ Cmn

(8)

k

k are complex constants of the algebra. As the complex invariant I is a member of the dynamical where Cmn algebra and it can be written as

I = ∑ λn (t)Γn (x1 , p4 , x2 , p5 , x3 , p6 , p1 , x4 , p2 , x5 , p3 , x6 ),

(9)

n

where λn (t) are complex coefficients functions of t. The time dependence of the dynamical invariant I(x, y, z, px , py , pz ) must satisfy the relation dI ∂ I = + [I, H] = 0, dt ∂t

(10)

where [., .] is the poisson bracket and [I, H] = [I, H](x,px ) + [I, H](y,py ) + [I, H](z,pz) . Then employing the transformation (5) on eq.(10) give rise to [A, B] = [A, B](x1 ,p1 ) − i[A, B](x1 ,x4 ) − i[A, B](p4 ,p1 ) − [A, B](p4 ,x4 ) [A, B](x2 ,p2 ) − i[A, B](x2 ,x5 ) − i[A, B](p5 ,p2 ) − [A, B](p5 ,x5 ) [A, B](x3 ,p3 ) − i[A, B](x3 ,x6 ) − i[A, B](p6 ,p3 ) − [A, B](p6 ,x6 ) .

(11)

On substituting eqs.(7) and (9) in eq.(11), we find k λn hm Γk = 0, ∑ λ˙k Γk + ∑ ∑ Cmn k

(12)

k n,m

which for a fixed k reduces to a set of coupled ordinary differential equations, namely k λ˙k + ∑ ∑ Cmn λn hm = 0.

(13)

m n

Now, one can determine the λi s appearing in eqs.(9). Once λk s are available from (13), then their substitution in (9) yield the Invariant I.

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3 Illustrative example On exploiting the Lie-algebraic approach as described in section-2, the complex dynamical invariant for a coupled harmonic oscillator is constructed as The PT - Symmetric Hamiltonian for a time-dependent coupled harmonic oscillator is written as k2 (t) 1 H(x, y, z) = (p2x + p2y + p2z ) + 1 (x2 + y2 + z2 ) + k2 (t)(xy + yz + zx). 2 2

(14)

Then, under the transformation (5), Hamiltonian (14) turns out to be 1 H = (p21 − x24 + p22 − x25 + p23 − x26 ) + i(p1 x4 + p2 x5 + p3 x6 ) + ik12 (x1 p4 + x2 p5 + x3 p6 ) 2 k2 + 1 (x21 + x22 + x23 − p24 − p25 − p26 ) + k2 [x1 x2 − p4 p5 + x2 x3 − p5 p6 + x1 x3 − p4 p6 ] 2 +ik2 [x1 p5 + x2 p4 + x2 p6 + x3 p5 + x3 p4 + x1 p6 ] =

30

∑ hm (t)Γm (x1 , p4 , x2 , p5 , x3 , p6 ),

(15)

m=1

where, the coefficients Γm ’s and hm ’s are given by 1 2 1 1 1 1 p1 , Γ2 = x24 , Γ3 = p1 x4 , Γ4 = x1 p4 , Γ5 = x21 , Γ6 = p24 , Γ7 = p22 ; 2 2 2 2 2 1 2 1 2 1 2 1 2 1 Γ8 = x5 , Γ9 = p2 x5 , Γ10 = x2 p5 , Γ11 = x2 , Γ12 = p5 , Γ13 = p3 , Γ14 = x26 ; 2 2 2 2 2 1 2 1 2 Γ15 = p3 x6 , Γ16 = x3 p6 , Γ17 = x3 , Γ18 = p6 , Γ19 = x1 x2 , Γ20 = p4 p5 , Γ21 = x1 p5 ; 2 2 Γ22 = x2 p4 , Γ23 = x2 x3 , Γ24 = p5 p6 , Γ25 = x2 p6 , Γ26 = x3 p5 , Γ27 = x1 x3 , Γ28 = p4 p6 ; Γ1 =

Γ29 = x3 p4 , Γ30 = x1 p6 ,

(16)

h1 = 1, h2 = −1, h3 = i, h4 = h9 = i, h10 =

ik12 ,

h11 =

k12 ,

ik12 ,

h12 =

h5 =

k12 ,

−k12 , h13

h6 =

−k12 ,

h7 = 1, h8 = −1;

= 1, h14 = −1, h15 = i, h16 = ik12 ;

h17 = k12 , h18 = −k12 , h19 = k2 , h20 = −k2 , h21 = ik2 , h22 = ik2 , h23 = k2 ; h24 = −k2 , h25 = ik2 , h26 = ik2 , h27 = k2 , h28 = −k2 , h29 = ik2 , h30 = ik2 .

(17)

It is to be noted that the phase space algebra in this case is not closed unless one adds 48 more phase space functions Γi s namely Γ31 = p1 p2 ; Γ32 = p1 p3 ; Γ33 = p1 p4 ; Γ35 = p1 p5 ; Γ36 = p1 p6 ; Γ37 = p1 x1 ; Γ38 = p1 x2 ; Γ39 = p1 x3 ; Γ41 = p1 x5 ; Γ42 = p1 x6 ; Γ43 = p2 p3 ; Γ44 = p2 p4 ; Γ45 = p2 p5 ; Γ46 = p2 p6 ; Γ47 = p2 x1 ; Γ48 = p2 x2 ; Γ49 = p2 x3 ; Γ50 = p2 x4 ; Γ51 = p2 x6 ; Γ52 = p3 p4 ; Γ53 = p3 p5 ; Γ54 = p3 p6 ; Γ55 = p3 x1 ; Γ56 = p3 x2 ; Γ57 = p3 x3 ; Γ58 = p3 x4 ; Γ59 = p3 x5 ; Γ60 = x1 x4 ; Γ61 = x1 x5 ; Γ62 = x1 x6 ; Γ63 = x2 x4 ; Γ64 = x2 x5 ; Γ65 = x2 x6 ; Γ66 = x3 x4 ; Γ67 = x3 x5 ; Γ68 = x3 x6 ; Γ69 = p4 x4 ; Γ70 = p4 x5 ; Γ71 = p4 x6 ; Γ72 = p5 x4 ; Γ73 = p5 x5 ; Γ74 = p5 x6 ; Γ75 = p6 x4 ; Γ76 = p6 x5 ; Γ77 = p6 x6 ; Γ78 = x4 x5 ; Γ79 = x4 x6 ; Γ80 = x5 x6 .

(18)

The corresponding hi s are to be set zero i.e. h31 = · · · = h31 = 0. Under the definition of (11), the non-vanishing poisson bracket for Γi s are written as


85

[Γ1 , Γ4 ] = iΓ37 − Γ33 ; [Γ1 , Γ5 ] = −Γ37 ; [Γ1 , Γ6 ] = iΓ33 ; [Γ1 , Γ19 ] = −Γ38 ; [Γ1 , Γ20 ] = iΓ35 ; [Γ1 , Γ22 ] = iΓ38 ; [Γ1 , Γ27 ] = −Γ39 ; [Γ1 , Γ28 ] = iΓ36 ; [Γ1 , Γ29 ] = iΓ39 ; [Γ1 , Γ30 ] = −Γ36 ; [Γ1 , Γ33 ] = 2iΓ1 ; [Γ1 , Γ37 ] = −2Γ1 ; [Γ1 , Γ44 ] = iΓ31 ; [Γ1 , Γ47 ] = −Γ31 ; [Γ1 , Γ52 ] = iΓ32 ; [Γ1 , Γ55 ] = −Γ32 ; [Γ1 , Γ60 ] = −Γ3 ; [Γ1 , Γ61 ] = −Γ41 ; [Γ1 , Γ62 ] = −Γ42 ; [Γ1 , Γ69 ] = iΓ3 ; [Γ1 , Γ70 ] = iΓ41 ; [Γ1 , Γ71 ] = iΓ42 ; ‘[Γ2 , Γ4 ] = iΓ69 + Γ60; [Γ2 , Γ5 ] = iΓ60 ; [Γ2 , Γ6 ] = Γ69 ; [Γ2 , Γ19 ] = iΓ63 ; [Γ2 , Γ20 ] = Γ72 ; [Γ2 , Γ21 ] = iΓ72 ; [Γ2 , Γ22 ] = Γ63 ; [Γ2 , Γ27 ] = iΓ66 ; [Γ2 , Γ29 ] = Γ66 ; [Γ2 , Γ30 ] = iΓ75 ; [Γ2 , Γ33 ] = Γ3 ; [Γ2 , Γ37 ] = iΓ3 ; [Γ2 , Γ44 ] = Γ50 ; [Γ2 , Γ47 ] = iΓ50 ; [Γ2 , Γ52 ] = Γ58 ; [Γ2 , Γ55 ] = iΓ58 ; [Γ2 , Γ60 ] = 2iΓ2 ; [Γ2 , Γ61 ] = iΓ78 ; [Γ2 , Γ62 ] = iΓ79 [Γ2 , Γ69 ] = 2Γ2 ; [Γ2 , Γ70 ] = Γ78 ; [Γ2 , Γ71 ] = Γ79 ; [Γ3 , Γ4 ] = iΓ33 + Γ37 + iΓ60 − Γ69 ; [Γ3 , Γ5 ] = iΓ37 − Γ60 ; [Γ3 , Γ6 ] = Γ33 + iΓ69 ; [Γ3 , Γ19 ] = iΓ33 − Γ63; [Γ3 , Γ20 ] = Γ35 + iΓ72 ; [Γ3 , Γ21 ] = iΓ35 − Γ72 ; [Γ3 , Γ22 ] = Γ38 + iΓ63 ; [Γ3 , Γ27 ] = iΓ39 − Γ66; [Γ3 , Γ28 ] = Γ36 + iΓ75 ; [Γ3 , Γ29 ] = Γ39 + iΓ66 ; [Γ3 , Γ30 ] = iΓ36 − Γ75; [Γ3 , Γ33 ] = 2Γ1 + iΓ3 ; [Γ3 , Γ37 ] = 2iΓ1 − Γ3 ; [Γ3 , Γ44 ] = Γ31 + iΓ50 ; [Γ3 , Γ47 ] = iΓ31 − Γ50; [Γ3 , Γ52 ] = Γ32 + iΓ58 ; [Γ3 , Γ55 ] = iΓ32 − Γ58 ; [Γ3 , Γ60 ] = −2Γ2 + Γ3 ; [Γ3 , Γ61 ] = iΓ41 − Γ78 ; [Γ3 , Γ62 ] = iΓ42 − Γ79 ; [Γ3 , Γ69 ] = 2iΓ2 + Γ3 ; [Γ3 , Γ70 ] = Γ41 + iΓ78 ; [Γ3 , Γ71 ] = Γ41 + iΓ79 ; [Γ4 , Γ31 ] = −iΓ47 + Γ44 ; [Γ4 , Γ32 ] = −iΓ55 + Γ52 ; [Γ4 , Γ33 ] = 2Γ6 − iΓ4 ; [Γ4 , Γ35 ] = −iΓ21 + Γ20 ; [Γ4 , Γ36 ] = −Γ30 + Γ28 ; [Γ4 , Γ37 ] = −2iΓ5 + Γ4 ; [Γ4 , Γ38 ] = Γ22 − iΓ19 ; [Γ4 , Γ39 ] = Γ29 − iΓ27 ; [Γ4 , Γ41 ] = −iΓ61 + Γ70 ; [Γ4 , Γ42 ] = Γ71 − iΓ62 ; [Γ4 , Γ50 ] = −Γ47 − iΓ44 ; [Γ4 , Γ58 ] = −Γ55 − iΓ52; [Γ4 , Γ60 ] = −2Γ5 − iΓ4 ; [Γ4 , Γ63 ] = −iΓ22 − Γ19 ; [Γ4 , Γ66 ] = −iΓ29 − Γ27; [Γ4 , Γ69 ] = −2iΓ6 − Γ4 ; [Γ4 , Γ72 ] = −Γ21 − iΓ20 ; [Γ4 , Γ75 ] = −Γ30 − iΓ28; [Γ4 , Γ78 ] = −iΓ70 − Γ61; [Γ4 , Γ79 ] = −iΓ71 − Γ62 ; [Γ5 , Γ31 ] = Γ47 ; [Γ5 , Γ32 ] = Γ55 ; [Γ5 , Γ33 ] = Γ4 ; [Γ5 , Γ35 ] = Γ21 ; [Γ5 , Γ36 ] = Γ30 ; [Γ5 , Γ37 ] = 2Γ5 ; [Γ5 , Γ38 ] = Γ19 ; [Γ5 , Γ39 ] = Γ27 ; [Γ5 , Γ41 ] = Γ61 ; [Γ5 , Γ42 ] = Γ62 ; [Γ5 , Γ50 ] = −iΓ47 ; [Γ5 , Γ58 ] = −iΓ55 ; [Γ5 , Γ60 ] = −2iΓ5 ; [Γ5 , Γ63 ] = −iΓ19 ; [Γ5 , Γ66 ] = −iΓ27 ; [Γ5 , Γ69 ] = −iΓ4 ; [Γ5 , Γ72 ] = −iΓ21 ; [Γ5 , Γ75 ] = −iΓ30 ; [Γ5 , Γ78 ] = −iΓ61 ; [Γ5 , Γ79 ] = −iΓ62 ; . . . ; [Γ75 , Γ77 ] = −Γ75 ; [Γ75 , Γ79 ] = −2Γ2 ; [Γ75 , Γ80 ] = −Γ78 ; [Γ76 , Γ77 ] = −Γ76 ; [Γ76 , Γ79 ] = −Γ78 ; [Γ76 , Γ80 ] = −2Γ8 ; [Γ77 , Γ79 ] = −Γ79 ; [Γ77 , Γ80 ] = −Γ80 .

(19)

It is to be noted that for complex Hamiltonian, not only the computation of poisson bracket turns out to be complicated but also the number of non-vanishing brackets increases enormously in comparison of real Hamiltonian and makes the construction of invariants difficult for the three-dimensional system. For example, for the real H, the number of non-vanishing poisson was found to be 3, whereas, for its complex version it turns out to be 78. Using these poisson brackets in eq.(13) yield the following PDE’s

λ˙1 = 4(iλ33 − λ37 ), λ˙2 = −4(iλ60 + λ69), λ˙3 = −2(λ33 + iλ37 + λ60 − iλ69 ), λ˙4 = 2k12 (λ33 + iλ37 + λ60 − iλ69 ) + 2k2 (λ44 + iλ47 + λ52 + iλ55 + λ61 + λ62 − iλ70 − iλ71 ),

(20) (21) (22) (23) (24)

86


λ˙5 = 4k12 (λ37 − iλ60 ) + 4k2 (λ47 + λ55 − iλ61 − iλ62 ), λ˙6 = 4k12 (iλ33 + λ69) + 4k2 (iλ44 + iλ52 + λ70 + λ71 ), λ˙7 = 4(iλ45 − λ48 ), λ˙8 = −4(iλ64 + λ73 ), λ˙9 = −2(λ45 + λ64 + iλ48 − iλ73 ), λ˙10 = 2k12 (λ45 + iλ48 + λ64 − iλ73 ) + 2k2 (λ35 + iλ38 + λ53 + iλ56 + λ65 + λ63 − iλ72 − iλ74 ), ˙ λ11 = 4k12 (λ48 − iλ64 ) + 4k2 (λ38 + λ56 − iλ63 − iλ65 ), λ˙12 = 4k12 (iλ45 + λ73 ) + 4k2 (iλ35 + iλ53 + λ72 + λ74), λ˙13 = 4(iλ54 − λ57), λ˙14 = −4(iλ68 + λ77 ), λ˙15 = −2(λ54 + λ68 + iλ57 − iλ77 ), λ˙16 = 2k12 (λ54 + iλ57 + λ68 − iλ77 ) + 2k2 (λ46 + iλ49 + λ36 + iλ39 + λ66 + λ67 − iλ75 − iλ76 ), ˙ λ17 = 4k12 (λ57 − iλ68 ) + 4k2 (λ39 + λ49 − iλ66 − iλ67 ), λ˙18 = 4k12 (iλ54 + λ77 ) + 4k2 (iλ36 + iλ46 + λ75 + λ76), λ˙19 = 2k12 (λ38 + λ47 − iλ61 − iλ63 ) + 2k2 (λ37 + iλ48 + λ55 + iλ56 − iλ60 − λ62 − iλ64 − iλ65 ), λ˙20 = 2k12 (iλ35 + iλ44 + λ70 + λ72 ) + 2k2 (iλ33 + iλ45 + iλ52 + iλ53 + λ69 + λ71 + λ73 + λ74 ), ˙ λ21 = 2k12 (λ35 + iλ47 + λ61 − iλ72 ) + 2k2 (iλ37 + λ45 + λ53 + iλ55 + λ60 + λ62 − iλ73 − iλ74 ), ˙ λ22 = 2k12 (iλ38 + λ44 + λ63 − λ70) + 2k2 (λ33 + iλ48 + λ52 + iλ56 + λ64 + λ65 − iλ69 − iλ71 ), ˙ λ23 = 2k12 (λ49 + λ56 − λ65 − iλ67 ) + 2k2 (λ38 + λ39 + λ48 + λ57 − λ63 − iλ64 − iλ66 − iλ68 ), λ˙24 = 2k12 (iλ46 + iλ53 + λ74 + λ76 ) + 2k2 (iλ35 + iλ36 + iλ45 + iλ54 + λ72 + λ73 + λ75 + λ77 ), ˙ λ25 = 2k12 (λ46 + iλ56 + λ65 − iλ76 ) + 2k2 (λ36 + iλ38 + iλ48 + λ54 + λ63 + λ64 − iλ75 − iλ77 ), ˙ λ26 = 2k12 (iλ46 + iλ53 + λ74 + λ76 ) + 2k2 (iλ35 + iλ36 + iλ45 + iλ54 + λ72 + λ73 + λ75 + λ77 ), λ˙27 = 2k12 (λ39 + λ55 − iλ62 − iλ66 ) + 2k2 (λ37 + λ47 + λ49 + λ57 − iλ60 − iλ61 − iλ67 − iλ68 ), λ˙28 = 2k12 (iλ36 + iλ52 + λ71 + λ75 ) + 2k2 (iλ33 + iλ44 + iλ46 + iλ54 + λ69 + λ70 + λ76 + λ77 ), ˙ λ29 = 2k12 (iλ39 + λ52 + λ66 + iλ71 ) + 2k2 (λ33 + λ44 + iλ49 + iλ57 + λ67 + λ68 − iλ69 − iλ70 ), ˙ λ30 = 2k12 (λ36 + iλ55 + λ62 − iλ75 ) + 2k2 (iλ37 + λ46 + iλ47 + λ54 + λ60 + λ61 − iλ76 − iλ77 ),

(25) (26) (27) (28) (29) (30) (31) (32) (33) (34) (35) (36) (37) (38) (39) (40) (41) (42) (43) (44) (45) (46) (47) (48) (49) (50)


87

λ˙31 = −2(−λ38 + iλ35 + iλ44 − λ47 ), λ˙32 = −2(iλ36 − λ39 + iλ52 − λ55 ), λ˙33 = 2(−λ4 + iλ6 ) + 2k12 (iλ1 + λ3 ) + 2k2 (iλ31 + iλ32 + λ41 + λ42), λ˙35 = 2(−λ21 + iλ20 ) + 2k12 (iλ31 + λ41) + 2k2 (iλ32 + λ42 + iλ1 + λ3 ), λ˙36 = 2(iλ28 − λ30 ) + 2k12 (iλ32 + λ42 ) + 2k2 (iλ31 + λ41 + iλ1 + λ3 ), λ˙37 = 2(iλ4 − λ5 ) + 2k12 (λ1 − iλ3 ) + 2k2 (λ31 + λ32 − iλ41 − iλ42 ), λ˙38 = 2(−λ19 + iλ22 ) + 2k12 (λ31 − iλ41 ) + 2k2 (λ32 − iλ42 + λ1 − iλ3 ), λ˙39 = 2(iλ4 − λ5 ) + 2k12 (λ32 − iλ42 ) + 2k2 (λ31 − iλ41 + λ1 − iλ3 λ42 ), λ˙41 = −2(λ35 + iλ38 + λ61 − iλ70 ), λ˙42 = −2(λ36 + iλ39 + λ62 − iλ71 ), λ˙43 = 2(iλ46 − λ49 + iλ53 − λ56 ), λ˙44 = 2(iλ20 − λ22 ) + 2k12 (iλ31 + λ50 ) + 2k2 (iλ7 + λ9 + iλ43 + λ51 ), λ˙45 = 2(−λ10 + iλ42 ) + 2k12 (iλ7 + λ9 ) + 2k2 (iλ31 + λ50 + iλ43 + λ51 ), λ˙46 = 2(iλ24 − λ25 ) + 2k12 (iλ43 + λ51 ) + 2k2 (iλ7 + λ9 + iλ31 + λ50 ), λ˙47 = −2(λ19 − iλ21 ) + 2k12 (λ31 − iλ50 ) + 2k2 (λ7 − iλ9 + λ43 − iλ51 ), λ˙48 = 2(iλ10 − λ11 ) + 2k12 (λ7 − iλ9 ) + 2k2 (λ31 − iλ50 + λ43 − iλ51 ), λ˙49 = −2(λ23 − iλ26 ) + 2k12 (λ43 − iλ51 ) + 2k2 (λ7 − iλ9 + λ31 − iλ50 ), λ˙50 = −2(λ44 + iλ47 + λ63 − iλ72 ), λ˙51 = −2(λ46 + iλ49 + λ65 − iλ74 ), λ˙52 = 2(iλ28 − λ29 ) + 2k12 (iλ32 + λ58 ) + 2k2 (iλ13 + λ15 + iλ43 + λ52 ), λ˙53 = 2(−λ26 + iλ24 ) + 2k12 (iλ43 + λ59) + 2k2 (iλ13 + λ15 + iλ32 + λ58 ), λ˙54 = 2(iλ18 − λ16 ) + 2k12 (iλ13 + λ15 ) + 2k2 (iλ32 + iλ43 + λ58 + λ59 ), λ˙55 = −2(λ27 − iλ30 ) + 2k12 (λ32 − iλ58 ) + 2k2 (λ13 − iλ15 + λ43 − iλ59 ), λ˙56 = −2(λ33 − iλ25 ) + 2k12 (λ43 − iλ59 ) + 2k2 (λ13 − iλ15 + λ32 − iλ58 ), λ˙57 = 2(iλ16 − λ17 ) + 2k12 (λ13 − iλ15 ) + 2k2 (λ32 + λ43 − iλ58 − iλ59 ), λ˙58 = −2(λ52 + iλ55 + λ66 − iλ75 ), λ˙59 = −2(λ53 + iλ56 + λ67 − iλ76 ), λ˙60 = −2(λ4 + iλ5 ) + 2k12 (−iλ2 + λ3 ) + 2k2 (λ50 + λ58 − iλ78 − iλ79 ), λ˙61 = −2(iλ19 + λ21 ) + 2k12 (λ41 − iλ78 ) + 2k2 (−iλ8 + λ9 + λ59 − iλ80 ), λ˙62 = −2(iλ27 + λ30 ) + 2k12 (λ42 − iλ79 ) + 2k2 (λ15 + λ51 − iλ80 − iλ14 ), λ˙63 = −2(λ22 + iλ19 ) + 2k12 (λ50 − iλ78 ) + 2k2 (−iλ2 + λ3 + λ58 − iλ79 ), λ˙64 = −2(λ10 + iλ11 ) + 2k12 (λ9 − iλ8 ) + 2k2 (λ41 − iλ78 + λ59 − iλ80 ), λ˙65 = −2(λ25 + iλ23 ) + 2k12 (λ51 − iλ80 ) + 2k2 (−iλ14 + λ15 + λ42 − iλ79 ), λ˙66 = −2(iλ27 + λ29 ) + 2k12 (−iλ79 + λ58 ) + 2k2 (−iλ2 + λ3 + λ50 − iλ78 ), λ˙67 = −2(iλ23 + λ26 ) + 2k12 (λ59 − iλ80 ) + 2k2 (−iλ8 + λ9 + λ41 − iλ78 ), λ˙68 = −2(iλ17 + λ16 ) + 2k12 (λ15 − iλ14 ) + 2k2 (λ42 + λ51 − iλ80 − iλ79 ), λ˙69 = −2(λ6 + iλ4 ) + 2k12 (λ2 + iλ3 ) + 2k2 (iλ50 + iλ58 + λ78 + λ79), λ˙70 = −2(λ20 + iλ22 ) + 2k12 (λ78 + iλ41 ) + 2k2 (λ8 + iλ9 + iλ59 + λ80 ), λ˙71 = −2(λ28 + iλ29 ) + 2k12 (iλ42 + λ79) + 2k2 (λ14 + iλ15 + λ80 + iλ51 ), λ˙72 = −2(λ20 + iλ21 ) + 2k12 (iλ50 + λ78) + 2k2 (λ2 + iλ3 + iλ58 + λ79 ),

(51) (52) (53) (54) (55) (56) (57) (58) (59) (60) (61) (62) (63) (64) (65) (66) (67) (68) (69) (70) (71) (72) (73) (74) (75) (76) (77) (78) (79) (80) (81) (82) (83) (84) (85) (86) (87) (88) (89) (90)

88


λ˙73 = −2(iλ10 + λ12 ) + 2k12 (iλ9 + λ8 ) + 2k2 (iλ41 + iλ59 + λ78 + λ80 ), λ˙74 = −2(λ24 + iλ26 ) + 2k12 (iλ51 + λ80 ) + 2k2 (λ14 + iλ15 + iλ40 + λ79 ), λ˙75 = −2(λ28 + iλ30 ) + 2k12 (iλ58 + λ79 ) + 2k2 (λ2 + iλ3 + iλ50 + λ78 ), λ˙76 = −2(λ24 + iλ25 ) + 2k12 (iλ59 + λ80 ) + 2k2 (λ8 + iλ9 + λ41 + λ78 ), λ˙77 = −2(iλ16 + λ18 ) + 2k12 (λ14 + iλ15 ) + 2k2 (iλ42 + iλ57 + λ80 + λ79 ), λ˙78 = −2(iλ61 + iλ63 + λ70 + λ72 ), λ˙79 = −2(λ75 + iλ66 + iλ62 + λ71 ), λ˙80 = 2(iλ65 + λ74 + iλ67 + λ76 ),

(91) (92) (93) (94) (95) (96) (97) (98)

As such the solutions of these 78 coupled ordinary differential equations for complex λ s is very difficult task. Therefore, we make an ansatz for some of λ s, Then from eqs.(20)-(22) one gets 2λ˙3 = iλ˙1 − iλ˙2 .

(99)

Assume λ3 = constant (say c3 ), then eq.(99) yields

λ˙1 − λ˙2 = 0 or λ˙1 = λ˙2 It allows us to make an ansatz for λ1 and λ2 as

λ1 = η1 (t) + c1 , λ2 = η1 (t) + c2 ,

(100)

where c1 and c2 are arbitrary complex constants and η1 (t) is some arbitrary complex function of t. Then in the similar fashion, from eqs.(23)-(38), one finds

λ4 = c4 , λ5 = η2 (t) + c5 , λ6 = η2 (t) + c6 , λ9 = c9 , λ7 = η3 (t) + c7 , λ8 = η3 (t) + c8 , λ10 = c10 , λ11 = η4 (t) + c11 , λ12 = η4 (t) + c12 , λ15 = c15 , λ13 = η5 (t) + c13 , λ14 = η5 (t) + c14 , λ16 = c16 , λ17 = η6 (t) + c17 , λ18 = η6 (t) + c18 ,

(101) (102) (103) (104) (105)

where, c4 to c15 are arbitrary complex constants and η2 (t) to η6 (t) are some arbitrary complex functions of t. From eq.(53) and (56), we have iλ˙33 + λ˙37 = −2(λ5 + λ6 ),

(106)

iλ˙33 + λ˙37 = −2(2η2 + c5 + c6 ),

(107)

on employing eq.(101) in (106), we have

on the other hand, single differentiation of (20) yields

λ¨1 η¨1 = . iλ˙33 − λ˙37 = 4 4

(108)

From eqs.(107) and (108), one immediately finds

λ33 =

−i −1 [η˙1 (t) − 8σ1 ] + c33 , λ37 = [η˙1 (t) + 8σ1 ] + c37 , 8 8

(109)


89

´ where, σ1 (t) = (2η2 (t) + c5 + c6 )dt. Similarly from the set of eqs.(21, 78, 87), (27, 63, 66), (28, 82, 91), (33, 72 , 75), (34, 86, 95), we have i −1 [η˙1 (t) + 8σ1 ] + c69 , λ60 = [η˙1 (t) − 8σ1 ] + c60 , λ69 = (110) 8 8 −i −1 λ45 = [η˙3 (t) − 8σ2 ] + c45 , λ48 = (111) [η˙3 (t) + 8σ2 ] + c48 , 8 8 i −1 [η˙3 (t) + 8σ2 ] + c73 , λ64 = [η˙3 (t) − 8σ2 ] + c64 , λ73 = (112) 8 8 −i −1 [η˙5 (t) + 8σ3 ] + c57 , λ54 = [η˙5 (t) − 8σ3 ] + c54 , λ57 = (113) 8 8 i −1 [η˙5 (t) + 8σ3 ] + c77 , λ68 = [η˙5 (t) − 8σ3 ] + c68 , λ77 = (114) 8 8 ´ ´ where, σ2 (t) = (2η4 (t) + c11 + c12 )dt and σ3 (t) = (2η6 (t) + c17 + c18 )dt. From eqs.(39)-(42), we have

λ˙19 = λ˙20 − iλ˙21 − iλ˙22 ,

(115)

let λ˙20 = iλ˙22 = η˙6 (t) then eqs.(115) yields

λ˙19 = −iλ˙21 = η˙7 (t),

(116)

and after solving eqs.(116), one finds

λ20 = η6 + c20 , λ22 = −iη6 + c22 λ19 = η7 + c19 , λ21 = iη7 + c21 .

(117)

Similarly from the set of eqs.(43-46), (47-50), (51, 59, 68, 96), (52, 60, 76, 79) and (61, 69, 77, 98) we have

λ23 = η9 + c23 , λ24 = η8 + c24 , λ25 = iη9 + c25 , λ26 = −iη8 + c26 , λ27 = η10 + c27 , λ28 = η11 + c28 , λ29 = iη10 + c29 , λ30 = −iη11 + c30 , λ31 = η12 + c31 , λ50 = iη12 + c50 , λ78 = η13 + c78 , λ41 = −iη13 + c41 , λ79 = η14 + c79 , λ42 = −iη14 + c42 , λ32 = η15 + c32 , λ58 = η15 + c58 , λ80 = η16 + c80 , λ51 = −iη16 + c51 , λ43 = η17 + c43 , λ59 = η17 + c59 ,

(118) (119) (120) (121) (122)

where, c23 —c32 , c41 , c42 , c50 , c51 , c78 —c80 are arbitrary complex constants and η8 (t), —, η17 (t) are some arbitrary complex functions of t. From eqs.(54) and (57), we have

λ˙35 − iλ˙38 = 2[iλ20 − λ21 + iλ19 + λ22 ] = 2[ic20 − c21 + ic19 + c22 ],

(123)

let, c20 = ic22 and c19 = −ic21 then eq.(123) gives λ˙35 = iλ˙38 = η˙18 (t) or

λ˙35 = iλ˙38 = η˙18 (t),

(124)

λ35 = η18 + c35 , λ38 = −iη18 + c38 .

(125)

after solving eq.(124), we have

90


Similarly, from the set of eqs.(55) & (58), (62) & (65), (64) &(67), (71) & (74), (70) & (73), (79) & (88), (80) & (89), (81) & (90), (83) & (92), (84) & (93) and (64), (67), we have

λ36 = η19 + c36 , λ44 = η20 + c44 , λ46 = η21 + c46 , λ53 = η22 + c53 , λ52 = η23 + c52 , λ61 = η24 + c61 , λ62 = η25 + c62 , λ63 = η26 + c63 , λ65 = η27 + c65 , λ66 = η28 + c66 , λ67 = η29 + c67 ,

λ39 = −iη19 + c39 , λ47 = −iη20 + c47 , λ49 = −iη21 + c49 , λ56 = −iη22 + c56 , λ55 = −iη23 + c55 , λ70 = iη24 + c70 , λ71 = iη25 + c71 , λ72 = iη26 + c72 , λ74 = iη27 + c74 , λ75 = iη28 + c75 , λ76 = iη29 + c76 .

(126) (127) (128) (129) (130) (131) (132) (133) (134) (135) (136)

Note that in all the above solutions the ci j s in general are complex constants. Thus, we have solved eqs.(20)(98) in terms of twenty nine arbitrary functions ηi (t) and seventy eight complex constants i.e. ci j s . Use of these results for λ1 to λ80 in eqs.(20)-(98) will further provide constraints on the choice of these arbitrary quantities. For example eqs.(20)-(21) gives ic33 = c37 ,

ic60 = −c69 ,

ic68 = −c77 ,

ic22 = c20 ,

c28 = ic30 ,

ic64 = −c73 ,

− ic21 = −c19 ,

c27 = −ic29 ,

ic54 = c57 ,

c19 = −c20 ,

c21 = c22 ,

c27 = −c28 ,

c29 = c30 ,

c24 = ic26 ,

c25 = c26 ,

c31 = c78 ,

c41 = −c50

c43 = c80 ,

c51 = −c59 ,

c53 = ic56 ,

c65 = −c74 , c52 = ic55 ,

c62 = −ic71 .

c23 = −ic25 , c32 = c79 ,

ic45 = c48 ,

c23 = −c24 , c42 = −c58 ,

Also other constraining relations are c5 − c6 = 2ic4 , c7 − c8 = 2ic9 ,

η6 = η7 ,

η8 = − η9 ,

c1 − c2 = 2ic3 , c17 − c18 = 2ic16 ,

η10 = −η11 , λn s

and Finally, substituting the values of various tonian in its most general form can be written as

Γn s

c11 − c12 = 2ic10 , c13 − c14 = 2ic15 ,

η12 = η13 ,

η14 = η15 ,

η16 = η17 .

in eq.(9), the invariant I for the PT - Symmetric Hamil-

1 1 1 1 I = η1 (p21 + x24 ) + (c1 p21 + c2 x24 ) + c3 p1 x4 + c4 x1 p4 + η2 (x21 + p24 ) + (c5 x21 + c6 p24 ) 2 2 2 2 1 1 1 1 2 2 2 2 2 2 + η3 (p2 + x5 ) + (c7 p2 + c8 x5 ) + c9 p2 x5 + c10 x2 p5 + η4 (x2 + p5 ) + (c11 x22 2 2 2 2 1 1 1 +c12 p25 ) + η5 (p23 + x26 ) + (c13 p23 + c14 x26 ) + c15 p3 x6 + c16 x3 p6 + η6 (x23 + p26 ) 2 2 2 i i 1 2 2 + (c17 x3 + c18 p6 ) + (η˙1 − 8σ1 )(x1 x4 − p1 p4 ) + (c33 p1 p4 + c60 x1 x4 ) + (η˙3 − 8σ2 ) 2 8 8 i (x2 x5 − p2 p5 ) + (c45 p2 p5 + c64 x2 x5 ) + (η˙5 − 8σ3 )(x3 x6 − p3 p6 ) + (c54 p3 p6 8 1 1 +c68 x3 x6 ) − (η˙1 + 8σ1 )(p1 x1 + p4 x4 ) + (c37 p1 x1 + c69 p4 x4 ) − (η˙3 + 8σ2 ) 8 8


91

1 ×(p2 x2 + p5 x5 ) + (c48 p2 x2 + c73 p5 x5 ) − (η˙5 + 8σ3 )(p3 x3 + p6 x6 ) + (c57 p3 x3 8 +c77 p6 x6 ) + η6 (p4 p5 − ix2 p4 ) + (c20 p4 p5 + c22 x2 p4 ) + η7 (x1 x2 + ix1 p5 ) + (c19 x1 x2 +c21 x1 p5 ) + η8 (p5 p6 − ix3 p5 ) + (c24 p5 p6 + c26 x3 p5 ) + η9 (x2 x3 + ix2 p6 ) + (c23 x2 x3 +c25 x2 p6 ) + η10 (x1 x3 + ix3 p4 ) + (c27 x1 x3 + c29 x3 p4 ) + η11(p4 p6 − ix1 p6 ) + (c28 p4 p6 +c30 x1 p6 ) + η12 (p1 p2 + ix4 p2 ) + (c31 p1 p2 + c50 x4 p2 ) + η13 (x4 x5 − ix5 p1 ) + (c41 p1 x5 +c78 x4 x5 ) + η14 (x4 x6 − ix6 p1 ) + (c42 p1 x6 + c79 x4 x6 ) + η15(p1 p3 + ix4 p3 ) + (c32 p1 p3 +c58 x4 p3 ) + η16 (x5 x6 − ix6 p2 ) + (c51 p2 x6 + c80 x5 x6 ) + η17(p2 p3 + ix5 p3 ) + (c43 p2 p3 +c59 x5 p3 ) + η18 (p1 p5 − ix2 p1 ) + (c35 p1 p5 + c38 x2 p1 ) + η19 (p1 p6 − ix3 p1 ) + (c36 p1 p6 +c39 x3 p1 ) + η20 (p2 p4 − ix1 p2 ) + (c44 p2 p4 + c47 x1 p2 ) + η21 (p2 p6 − ix3 p2 ) + (c46 p2 p6 +c49 x3 p2 ) + η22 (p3 p5 − ix2 p3 ) + (c53 p3 p5 + c56 x2 p3 ) + η23 (p3 p4 − ix1 p3 ) + (c52 p3 p4 +c55 x1 p3 ) + η24 (x1 x5 + ix5 p4 ) + (c61 x1 x5 + c70 p4 x5 ) + η25(x1 x6 + ip4 x6 ) + (c62 x1 x6 +c71 x6 p4 ) + η26 (x2 x4 + ix4 p5 ) + (c63 x2 x4 + c72 p5 x4 ) + η27(x2 x6 + ip5 x6 ) + (c65 x2 x6 +c74 p5 x6 ) + η28 (x3 x4 + ip6 x4 ) + (c66 x3 x4 + c75 p6 x4 ) + η29(x3 x5 + ix5 p6 ) + (c67 x3 x5 +c76 p6 x5 ).

(137)

After rearranging the various terms in eq.(137), we have I=

1 1 −1 η˙1 [(x1 p1 + x4 p4 ) + i(p1 p4 − x1 x4 )] + [2η1 + c1 + c2 ](p21 + x24 ) + σ˙1 (x21 + p24 ) 8 4 4 1 i −σ1 [x1 p1 + x4 p4 − i(p1 p4 − x1 x4 )] + (c1 − c2 )(p21 − x24 − 2ip1 x4 ) + c4 (x21 − p24 4 4 i i −2ix1 p4 ) + (c33 − c60 )[x1 p1 + x4 p4 − i(p1 p4 − x1 x4 )] + (c33 + c60 )[x1 p1 − x4 p4 2 2 1 1 −i(p1 p4 + x1 x4 )] − η˙3 [(x2 p2 + x5 p5 ) + i(p2 p5 − x2 x5 )] + [2η3 + c7 + c8 ](p22 + x25 ) 8 4 1 1 2 2 + σ˙2 (x2 + p5 ) − σ2 [x2 p2 + x5 p5 − i(p2 p5 − x2 x5 )] + (c7 − c8 )(p22 − x25 − 2ip2 x5 ) 4 4 i i i + c10 (x22 − p25 − 2ix2 p5 ) + (c45 − c64 )[x2 p2 + x5 p5 − i(p2 p5 − x2 x5 )] + (c45 + c64 ) 4 2 2 1 1 [x2 p2 − x5 p5 − i(p2 p5 + x2 x5 )] − η˙5 [(x3 p3 + x6 p6 ) + i(p3 p6 − x3 x6 )] + [2η5 + c17 8 4 1 1 +c18 ](p23 + x26 ) + σ˙3 (x23 + p26 ) − σ3 [x3 p3 + x6 p6 − i(p3 p6 − x3 x6 )] + (c17 − c18 ) 4 4 i i 2 2 2 2 ×(p3 − x6 − 2ip3 x6 ) + c16 (x3 − p6 − 2ix3 p6 ) + (c54 − c68 )[x3 p3 + x6 p6 − i(p3 p6 − x3 x6 )] 4 2 i + (c54 + c68 )[x3 p3 − x6 p6 − i(p3 p6 + x3 x6 )] + (η7 + c19 )[x1 x2 − p4 p5 + i(x2 p4 + x1 p5 )] 2 +(η9 + c23 )[x2 x3 − p5 p6 + i(x2 p6 + x3 p5 )] + (η10 + c27 )[x1 x3 − p4 p6 + i(x3 p4 + x1 p6 )] +(η12 + c31 )[p1 p2 + x4 x5 + i(x4 p2 − x5 p1 )] + (η14 + c32 )[p1 p3 + x4 x6 + i(x4 p3 − x6 p1 )] +(η16 + c43 )[p2 p3 + x5 x6 + i(x5 p3 − x6 p2 )] − i(η18 + c35 )[p1 x2 − x4 p5 + i(x2 x4 + p1 p5 )] −i(η19 + c36 )[p1 x3 − x4 p6 + i(x3 x4 + p1 p6 )] − i(η20 + c44 )[p2 x1 − p4 x5 + i(x1 x5 + p2 p4 )] −i(η21 + c46 )[p2 x3 − x5 p6 + i(x3 x5 + p2 p6 )] − i(η22 + c53 )[p3 x2 − p5 x6 + i(x2 x6 + p3 p5 )] −i(η23 + c52 )[p3 x1 − x6 p4 + i(x1 x6 + p3 p4 )].

(138)

Note that this form of I conforms to (12) with the poisson bracket (13). It is not difficult to express (138) in

92


terms of x, x∗ , y, y∗ , z, z∗ , px , p∗x , py , p∗y , pz and p∗z as I=

1 1 1 i −1 η˙1 p∗x x + [2η1 + c1 + c2 ]px p∗x + σ˙1 xx∗ − σ1 x∗ px + (c1 − c2 )p∗x 2 + c4 x∗ 2 8 4 4 4 4 i i 1 1 1 ∗ ∗ ∗ ∗ ∗ + (c33 − c60 )x px + (c33 + c60 )x px − η˙3 py y + [2η3 + c7 + c8 ]py py + σ˙2 yy∗ 2 2 8 4 4 1 i i i 1 −σ2 y∗ py + (c7 − c8 )p∗y 2 + c10 y∗ 2 + (c45 − c64 )y∗ py + (c45 + c64 )y∗ p∗y − η˙5 p∗z z 4 4 2 2 8 1 1 i i 1 ∗ ∗ ∗ ∗2 ∗2 + [2η5 + c13 + c14 ]pz pz + σ˙3 zz σ3 z pz + (c13 − c14 )pz + c16 z + (c54 − c68 )z∗ pz 4 4 4 4 2 i + (c54 + c68 )z∗ p∗z + (η7 + c19 )xy + (η9 + c23 )yz + (η10 + c27 )zx + (η12 + c31 )px p∗y 2 +(η14 + c32 )pz p∗x + (η16 + c43 )py p∗z − i(η18 + c35 )px y − i(η19 + c36 )px z − i(η20 + c44 )py x −i(η21 + c46 )py z − i(η22 + c53 )pz y − i(η23 + c52 )pz x,

(139)

for a particular case when c1 = c2 (implying c3 = 0), c7 = c8 (implying c9 = 0), c13 = c14 (implying c15 = 0), c33 = c60 , c45 = c64 and c54 = c68 then (139) takes a simple form viz I=

1 1 i −1 η˙1 p∗x x + [η1 + c1 ]px p∗x + σ˙1 xx∗ − σ1 x∗ px + c4 x∗ 2 + ic33 x∗ p∗x 8 2 4 4 1 1 1 i ∗ ∗ ∗ ∗ − η˙3 py y + [η3 + c7 ]py py + σ˙2 yy − σ2 y py + c10 y∗ 2 + ic45 y∗ p∗y 8 2 4 4 1 1 1 i − η˙5 p∗z z + [η5 + c13 ]pz p∗z + σ˙3 zz∗ − σ3 z∗ pz + c16 z∗ 2 + ic54 z∗ p∗z 8 2 4 4 +(η7 + c19 )xy + (η9 + c23 )yz + (η10 + c27 )zx + (η12 + c31 )px p∗y +(η14 + c32 )pz p∗x + (η16 + c43 )py p∗z − i(η18 + c35 )px y − i(η19 + c36 )px z −i(η20 + c44 )py x − i(η21 + c46 )py z − i(η22 + c53 )pz y − i(η23 + c52 )pz x.

(140)

which conform the integrability condition.

4 Conclusions In the present work, we have made an attempt to investigate the classical aspect of a time dependent dynamical system using ECPSA approach. The main emphasis in this study is to construct the exact dynamical invariant for the coupled harmonic oscillator under the elegance of Lie-algebraic method. Generally, it is considered that there exist atleast certain criteria regarding the existence of the number of real invariants for a given system in terms of its dimensionality and subsequently on this basis one can think about the integrability of the given real system, but the same criteria does not seem to work for a complex system [17, 28]. It is to be noted that due to complexification, the degrees of freedom get doubled and construction of the complex invariants in higher dimensions becomes very difficult [1, 30, 31]. However, by making some restrictions on the constants of integration and arbitrary functions of time, the complex invariant of a given dynamical system is constructed.

Acknowledgments The authors are thankful to the learned referees for several useful comments which helped in considerably improving and fine-tuning some of the ideas in the original version of the paper.


93

References [1] Kaushal, R.S. (1998), Classical and Quantum Mechanics of Noncentral Potentials, Narosa Publishing House, New Delhi. [2] Singh, R.M., Bhardwaj, S.B. and Mishra, S.C. (2013), Closed-form solutions of the Schrödinger equation for a coupled harmonic potential in three dimensions, Computers & Math. with Appl., 66, 537-541. Singh, R.M., Chand, F. and Mishra, S.C. (2009), The solution of the Schrödinger equation for coupled quadratic and quartic potentials in two dimensions, Pramana J Phys., 72, 647-654. [3] Khare, A. and Mandal, B.P. (2000), A PT-invariant potential with complex QES eigenvalues, Phys. Lett. A 272, 53-56. [4] Singh, S. and Kaushal, R.S. (2003), Complex dynamical invariants for one-dimensional classical systems, Phys. Scr. 67, 181-185. [5] Bender, C.M. (2007), Making sense of non-Hermitian Hamiltonians, Reports on Progress in Physics, 70, 947-1018. [6] Kaushal, R.S. and Korsch, H.J. (2000), Some remarks on complex Hamiltonian systems, Phys. Lett. A, 276, 47-51. [7] Nelson, D.R. and Snerb, N.M. (1998), Non-Hermitian localization and population biology, Phys. Rev. E, 58, 13831403. [8] Hatano, N. and Nelson, D.R. (1997), Vortex depinning and non-Hermitian quantum mechanics, Phys. Rev., 56, 86518673. [9] Xavier, Jr. A.L. and de Aguiar, M.A.M. (1996), Complex trajectories in the quartic oscillator and its semiclassical coherent state, Ann. Phys.(N.Y.), 252, 458-478. [10] Bender, C.M. , Boettcher, S. and Meisinger, P.N. (1999), PT-symmetric quantum mechanics, J. Math. Phys., 40, 22012229. [11] Bhardwaj, S.B. , Singh, R.M. and Mishra, S.C.(2014), Eigenspectra of a complex coupled harmonic potential in three dimensions, Comp. & Math. Appl., 68, 2068-2079. [12] Bhardwaj, S.B. , Singh, R.M. and Mishra, S.C.(2016), Quantum mechanics of PT and non-PT -symmetric potentials in three dimensions, Pramana-J Phys., 87, 1-10. [13] Parthasarthi and Kaushal, R.S.(2003), Quantum mechanics of complex sextic potentials in one dimension, Phys Scr., 68, 115-127. [14] Bhardwaj, S.B. , Singh, R.M., Mishra , S.C. and Sharma, K. (2015), On Solving the Schrödinger Equation for ThreeDimensional Noncentral Potential, J. Adv. Phys., 4, 215-218. Bhardwaj, S.B. and Singh, R.M. (2016), Exact solutions of 3-dimensional Schrödinger equation with a coupled quartic potential, J. Adv. Phys., 5, 44-46. [15] Bender, C.M. and Turbiner, A. (1993), Analytic continuation of eigenvalue problems, Phys. Lett. A, 173, 442-446. [16] Hietarinta, J. (1987), Direct methods for the search of the second invariant, Phys. Rep. 147, 87-154. [17] Whittaker, E.T. (1960), A treatise on the analytical dynamics of particle and rigid bodies, Cambridge University press, London, p246. [18] Singh, R.M.(2015), Integrability of a coupled harmonic oscillator in extended complex phase space, Discontinuity, Nonlinearity and Complexity, 4, 35-48. Rao N.N., Buti B. and Khadkikar S.B.(1986), Hamiltonian systems with indefinite kinetic energy, Pramana J. Phys., 27, 497-505. [19] Colgerave, R.K., Croxson, P., and Mannan, M.A.(1988), Complex invariants for the time-dependent harmonic oscillator, Phys. Lett. A, 131, 407-410. [20] Moiseyev, N.(1998), Quantum theory of resonances: calculating energies, widths and cross-sections by complex scaling, Phys.Rep., 302, 212-293. Virdi J.S., Chand F. , Kumar C.N., and Mishra S.C. (2012), Complex dynamical invariants for two-dimensional nonhermitian Hamiltonian systems, Can. J. Phys., 90, 151-157. [21] Whiteman, K.J. (1977), Invariants and stability in classical mechanics, Rep. Prog. Phys., 40, 1033-1069. [22] Kusenko, A. and Shrock, R. (1994), General determination of phases in quark mass matrices, Phys. Rev. D, 50, R30R33. [23] Gunion, J.F. and Haber, H.E.(2005), Conditions for CP-violation in the general two-Higgs-doublet model, Phys. Rev. D, 72, 095002-21 . [24] Plebanski, J.F. and Demianski, M.(1976), Rotating, charged, and uniformly accelerating mass in general relativity, Ann. Phys.(N.Y.), 98, 98-127. [25] Kumar, C.N. and Khare, A.(1989), Chaos in gauge theories possessing vortices and monopole solutions, J. Phys. A: Math.& Gen., 22, L849-L853. [26] Savidy G.K. (1983), The Yang-Mills classical mechanics as a Kolmogorov K-system, Phys. Lett. B, L30, 303-307. [27] Virdi, J.S., Chand, F., Kumar, C.N. and Mishra, S.C. (2012), Complex dynamical invariants for two-dimensional nonhermitian Hamiltonian systems, Can. J. Phys., 90, 151-157. Mishra, S.C. and Chand, F. (2006), Construction of exact dynaical invariants of two-dimensional classical system, Pramana J. Phys., 66, 601-607.

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[28] Chand, F., Kumar, N. and Mishra, S.C. (2015), Exact fourth order invariants for one-dimensional time-dependent Hamiltonian systems, Indian J. Phys., 89, 709-712. [29] Kaushal, R.S. and Singh, S. (2001), Construction of complex invariants for classical dynamical systems, Ann. Phys. (N.Y.), 288, 253-276. [30] Bhardwaj, S.B., Singh, R.M. and Sharma, K. (2017), Complex dynamical invariants for a PT - symmetric Hamiltonian system in higher dimensions, Chinese J. Phys., (55), 533-542. [31] Tabour, M. (1989), Chaos and Integrability in Nonlinear Dynamics, Wiley publications, New York .



Approximation of Random Fixed Point Theorems Salahuddin† Department of Mathematics, Jazan University, Jazan, Kingdom of Saudi Arabia Submission Info Communicated by D. Volchenkov Received 10 May 2017 Accepted 10 June 2017 Available online 1 April 2018

Abstract The aim of this paper is to establish and discuss the approximation of Caristi’s random fixed point theorems. Our theorem is used to determine a large numbers of nonlinear stochastic problems.

Keywords Measurable spaces Approximation theory Caristi’s random fixed point Fixed point Measurable selection Random variable Complete separable metric spaces Random point to point mappings Random point to set mappings


1 Introduction Random fixed point theorems are stochastic generalizations of fixed point theorems, and are required for the theory of random equations, just as in the theory of deterministic equations and fixed point theorems. In Polish spaces, i.e., separable complete metric spaces, random fixed point theorems for contraction mappings were proved by Spaeek [1] and Hans [2,3], Tsokos [4], Tsokos and Padgett [5,6], Cho et al. [7], Ahmad and Salahuddin [8], Itoh [9], Bharucha-Reid [10, 11] gave a random fixed point theorem of Schauder type on an atomic probability measure space. The applications of fixed point theory in different branches of mathematics, statistics, engineering and economics relating to problems associated with approximation theory, theory of differential equations, theory of integral equations, etc., has been recognized in the existing literature, see [12–17]. Motivated and inspired by mentioned research works [1–31], in this paper we established and discussed the approximation of Caristi’s random fixed point theorems. Our theorem is useful to determine a large numbers of nonlinear stochastic problems. † Corresponding


96

Salahuddin / Discontinuity, Nonlinearity, and Complexity 7(1) (2018) 95–105

2 Preliminaries Throughout this paper, (Ω, Σ) denotes a measurable space. Let X be a metric space with the metric ρ . Let 2X be the family of all subsets of X, CD(X) all nonempty closed subsets of X, CB(X) all nonempty bounded closed subsets of X, C(X) all nonempty compact subsets of X, respectively. A mapping F : Ω → 2X is called / ∈ Σ. This type of measuraΣ-measurable if for any open subset B of X, F−1 (B) = {ω ∈ Ω : F(ω ) ∈ B = 0} bility is usually called weakly measurable (see, Himmelberg [22]), but in this paper we always use this type of measurability, thus we omit the term weakly for simplicity. Notice that, when F(ω ) ∈ C(X) for all ω ∈ Ω, then F is measurable if and only if F−l (K) ∈ Σ for every closed subset K of X. A measurable mapping x : Ω → X is called a measurable selector of a measurable mapping F : Ω → CD(X) if x(ω ) ∈ F(ω ) for each ω ∈ Ω. A mapping f : Ω × X → X is called a random operator if for any x(ω ) ∈ X, f (·, x(ω )) is measurable. A mapping F : Ω × X → CD(X) is a random operator if for every x(ω ) ∈ X, F(·, x(ω )) is measurable. A measurable mapping x : Ω → X is called a random fixed point of a random operator f : Ω × X → X (or F : Ω × X → CD(X)) if for every ω ∈ Ω, f (ω , x(ω )) = x(ω ) (or x(ω ) ∈ F(ω , x(ω ))). A random mapping f : Ω × X → X is called compact if f is continuous and f (X) is precompact. A random mapping F : Ω × X → CD(X) is called compact if Fω (X) = x∈X Fω (x) is precompact. F is called random asymptotically regular if for each ω ∈ Ω, x(ω ) ∈ X, there exists a random sequence {xn (ω )} such that x0 (ω ) = x(ω ), xn+1 (ω ) ∈ F(ω , xn (ω )) and d(xn (ω ), xn+1 (ω )) → 0 as n → ∞. F(ω ) is called upper (lower) semicontinuous if for any closed (open) subset K of X, F−1 (K) is closed (open). F(ω ) is called continuous if F(ω ) is both upper and lower semicontinuous. Throughout this paper we always assume that x = x(ω ), a random point and Tω = T (ω ). First we recalling the following Caristi’s theorems. Theorem 1. [13, 19] Let (X, ρ ) be a complete metric space and T : X → X be a continuous mapping which satisfies for some φ : X → [0, ∞), ρ (x, T x) ≤ φ (x) − φ (T x), x ∈ X. Then the sequence of iterates {T n x}∞ n=1 converges to a fixed point of T for each x ∈ X. Theorem 2. [13, 19] Let (X, ρ ) be a complete metric space and φ : X → R1 be a lower continuous function which is bounded from below and T : X → X satisfies

ρ (x, T x) ≤ φ (x) − φ (T x), x ∈ X. Then T has a fixed point. / graph(T ) = {(x, y) ∈ X × X : Theorem 3. [12, 16] Let (X, ρ ) be a complete metric space T : X → 2X \ {0}, y ∈ T (x)} be the closed and φ : X → R1 ∪ {∞} be a bounded from below, and for each x ∈ X, inf{φ (y) + ρ (x, y) : y ∈ T (x)} ≤ φ (x). ∞ Let {εn }∞ n=0 ⊂ (0, ∞), Σn=0 εn < ∞, and let x0 ∈ X satisfy φ (x0 ) < ∞. Assume that for each integer n ≥ 0,

xn+1 ∈ T (xn ) and

φ (xn+1 ) + ρ (xn , xn+1 ) ≤ inf{φ (y) + ρ (xn , y) : y ∈ T (xn )} + εn . Then {xn }∞ n=0 converges to a fixed point of T . For each nonempty set Y and each function h : Y → R1 ∪ {∞}, set inf(h) = inf{h(y) : y ∈ Y }. Denote by card(E), the cardinality of a set E.


97

3 Approximation of random fixed points Theorem 4. Let Ω, Σ) be a measurable space and (X, ρ ) be a complete separable metric space. Assume that T : X × Ω → 2X \ {0} / is a random point to set mapping and φ : X × Ω → R1 ∪ {∞} is a randomly bounded below and for each ω ∈ Ω, x ∈ X (1) inf{φω (y) + ρ (x, y) : y ∈ Tω (x)} ≤ φω (x). Let ε : Ω → (0, ∞) be a measure and a measure δ ∈ (0, min{ε , 1}), let for each ω ∈ Ω, a random point x ∈ X satisfy φω (x0 ) < ∞, and let a natural number n0 satisfy n0 >

φω (x0 ) − inf(φω ) + 1 + 1. ε −δ

(2)

Assume that for all integers i ∈ {0, . . . , n0 − 1}, xi+1 ∈ Tω (xi )

(3)

φ (xi+1 ) + ρ (xi , xi+1 ) ≤ inf{φω (y) + ρ (xi , y) : y ∈ Tω (xi )} + δ .

(4)

and Then there exists an integer j ∈ {1, . . . , n0 } such that

ρ (x j , x j+1 ) > ε . Proof. From (1) and (4), since φω (xi ) < ∞, i = 0, . . . , n0 . Assume on the contrary, the theorem does not hold. Then for each j ∈ {1, . . . , n0 }, we have ρ (x j , x j+1 ) > ε . (5) Therefore from (1) and (4), we have

φω (x1 ) ≤ φω (x0 ) + δ ≤ φω (x0 ) + 1.

(6)

It follow from (1) and (4) that for each integers j ∈ {1, . . . , n0 − 1}, we have

φω (x j+1 ) + ρ (x j , x j+1 ) ≤ inf{φω (y) + ρ (x j , y) : y ∈ Tω (x j )} + δ ≤ φω (x j ) + δ .

(7)

From (5) and (7), for each integers j ∈ {1, . . . , n0 − 1}, we have

φω (x j+1 ) ≤ φω (x j ) + δ − ε . It follows from (6) and (8) that

φω (x0 ) + 1 − inf(φω ) ≥ φω (x1 ) − φω (xn0 ) =

n0 −1

∑ (φω (xi ) − φω (xi+1 )) ≥ (n0 − 1)(ε − δ )

i=1

implies that n0 ≤

φω (x0 ) − inf(φω ) + 1 + 1, ε −δ

this is a contradiction of (2). Hence theorem is proved.

(8)

98


4 Random fixed points of point to point mappings Assume that (Ω, Σ) is a measurable space and (X, ρ ) is a complete separable metric space. Let ϕ : [0, ∞) → [0, ∞) be a measurable function satisfy inf{ϕ (t) : t ∈ [s, ∞)} > 0 (9) for each s > 0 and lim inf + t→0

ϕ (t) > 0. t

(10)

Theorem 5. Let (Ω, Σ) be a measurable space and (X, ρ ) be a complete separable metric space. Let T : Ω × X → X be a continuous random mapping satisfies

ϕ (ρ (x, Tω (x))) ≤ φω (x) − φω (Tω (x)), for ω ∈ Ω, x ∈ X,

(11)

where φ : X × Ω → [0, ∞) be a measurable mapping. Then a random sequence {Tωn x}∞ n=1 converges to a random fixed point of Tω for each random point x ∈ X and ω ∈ Ω. Proof. Without any loss of generality, we may assume that ϕ (0) = 0. Let x ∈ X be a random point for each ω ∈ Ω. Set Tω0 (x) = x. From (11), for each integer i ≥ 0, 0 ≤ ϕ (ρ (Tωi (x), Tωi+1 (x))) ≤ φω (Tωi (x)) − φω (Tωi+1 (x)).

(12)

From (12), for each natural number n, we have

φω (x) ≥ φω (x) − φω (Tωn+1 (x)) n

n

i=0

i=1

= ∑ (φω (Tωi (x)) − φω (Tωi+1 (x))) ≥ ∑ ϕ (ρ (Tωi (x), Tωi+1 (x))).

(13)

Let ε : Ω → (0, ∞) be a measure and set

ζ = inf{ϕ (t) : t ∈ [ε , ∞)}.

(14)

From (9) and (14),we get

ζ > 0. From (13) and (14), for each natural number n, we have

φω (x) ≥ ∑{ϕ (ρ (Tωi (x), Tωi+1 (x))) : i ∈ {0, . . . , n} and ρ (Tωi (x), Tωi+1 (x)) ≥ ε } ≥ ζ card({i ∈ {0, . . . , n} : ρ (Tωi (x), Tωi+1 (x)) ≥ ε })

and card({i ∈ {0, . . . , n} : ρ (Tωi (x), Tωi+1 (x)) ≥ ε }) ≤

φω (x) ζ

for each natural number n. This implies that card({i ∈ {0, . . . , n} : ρ (Tωi (x), Tωi+1 (x)) ≥ ε }) ≤

φω (x) . ζ

Since the above inequality hold for any measure ε : ω → (0, ∞), we have lim ρ (Tωi (x), Tωi+1 (x)) = 0.

i→∞

(15)


99

From (10) there exist numbers t0 > 0 and λ0 > 0 such that

ϕ (t) ≥ λ0 t, ∀t ∈ [0,t0 ].

(16)

From (15) there exists an integer p0 ≥ 1 such that for each natural number i ≥ p0 , we have

ρ (Tωi (x), Tωi+1 (x)) ≤ t0 .

(17)

It follows from (16) and (17) that for each natural number i ≥ p0 , we have

ϕ (ρ (Tωi (x), Tωi+1 (x))) ≥ λ0 ρ (Tωi (x), Tωi+1 (x)).

(18)

For each natural number n, (12) and (18), we have p0 +n

φω (Tωp0 (x)) ≥ φω (Tωp0 (x)) − φω (Tωp0 +n+1(x)) =

∑ (φω (Tωi (x)) − φω (Tωi+1 (x)))

i=p0

≥

p0 +n

∑

ϕ (ρ (Tωi (x), Tωi+1 (x))) ≥

i=p0

This implies that

p0 +n

∑

λ0 ρ (Tωi (x), Tωi+1 (x)).

i=p0 ∞

1

∑ ρ (Tωi (x), Tωi+1(x)) ≤ λ0 φω (Tωp (x)), 0

i=p0

and the random sequence such that

{Tωi (x)}∞ i=0

is a randomly Cauchy sequence. Therefore there exists a random point x∗ x∗ = lim Tωi (x). i→∞

Since the random mapping Tω is a randomly continuous and x∗ = Tω (x∗ ). Hence result is proved. Theorem 6. Let (Ω, Σ) be a measurable space and (X, ρ ) be a complete separable metric space. Let a randomly lower semi continuous mapping φ : X × Ω → R1 ∪ {+∞} be a bounded below and not identically equal to ∞. If a random mapping T : Ω × X → X satisfies

ϕ (ρ (x, Tω (x))) + φω (Tω (x)) ≤ φω (x), for ω ∈ Ω, x ∈ X.

(19)

Then, it has a random fixed point. Proof. Without any loss of generality, we may assume that ϕ (0) = 0. From (10) there exist numbers t0 > 0 and λ0 > 0 such that ϕ (t) ≥ λ0 t, ∀t ∈ [0,t0 ]. (20) Set

t0 α0 = inf{ϕ (t) : t ∈ [ , ∞)}. 2

(21)

From (9), we have

α0 > 0. By using the Ekeland variational principle [15], for each ω ∈ Ω there exists a random point x¯ ∈ X such that

φω (x) ¯ ≤ inf(φω ) +

α0 , 2

(22)

and

φω (x) + λ0 ρ (x, x) ¯ > φω (x), ¯ ∀x ∈ X \ {x}. ¯

(23)

100


From (19) we have

ϕ (ρ (x, ¯ Tω (x))) ¯ + φω (Tω (x)) ¯ ≤ φω (x). ¯

(24)

¯ Tω (x)) ¯ ≥ t0 then from (21), we have If ρ (x,

ϕ (ρ (x, ¯ Tω (x))) ¯ ≥ α0 , which together with (24) implies that

α0 + φω (Tω (x)) ¯ ≤ φω (x). ¯ But this inequality contradicts (22). From this contradiction, we have

ρ (x, ¯ Tω (x)) ¯ < t0 .

(25)

ϕ (ρ (x, ¯ Tω (x))) ¯ ≥ λ0 ρ (x, ¯ Tω (x)). ¯

(26)

From (20) and (25), we have Again from (24) and (26), we have

φω (x) ¯ ≥ ϕ (ρ (x, ¯ Tω (x))) ¯ + φω (Tω (x)) ¯ ≥ λ0 ρ (x, ¯ Tω (x)) ¯ + φω (Tω (x)). ¯ ¯ then the above inequality contradicts (23). Thus x¯ = Tω (x) ¯ and the theorem is proved. If x¯ = Tω (x), 5 Random fixed points of point to set mappings Assume that (Ω, Σ) is a measurable space and (X, ρ ) is a complete separable metric space. Let ϕ : [0, ∞) → [0, ∞) be a measurable function satisfy (27) inf{ϕ (t) : t ∈ [s, ∞)} > 0, for each s > 0. Theorem 7. Let (Ω, Σ) be a measurable space and (X, ρ ) be a complete separable metric space. Suppose that lim inf + t→0

ϕ (t) > 0. t

(28)

/ be a continuous random mapping and the graph(Tω ) = {(x, y) ∈ X × X : y ∈ Tω (x)} is Let T : Ω × X → 2X \ {0} randomly closed. Let a random mapping φ : X × Ω → R1 ∪ {∞} be a bounded below and for each ω ∈ Ω, x ∈ X, inf{φω (y) + ϕ (ρ (x, y)) : y ∈ Tω (x)} ≤ φω (x). Let {εn }∞ n=0 ⊂ (0, ∞),

(29)

∞

∑ εn < ∞,

(30)

n=0

and for each ω ∈ Ω, let a random point x0 ∈ X satisfy φω (x0 ) < ∞. Assume that for each integer n ≥ 0, we have xn+1 ∈ Tω (xn )

(31)

φω (xn+1 ) + ϕ (ρ (xn , xn+1 )) ≤ inf{φω (y) + ϕ (ρ (xn , y)) : y ∈ Tω (xn )} + εn .

(32)

and Then the random sequence {xn }∞ n=0 converges to a random fixed point of Tω .


101

Proof. Without any loss of generality, we may assume that ϕ (0) = 0. It is clear that φω (xn ) < ∞, for all integer n ≥ 0. From (29) and (32), we have

ϕ (ρ (xn , xn+1 )) ≤ −φω (xn+1 ) + εn + inf{φω (y) + ϕ (ρ (xn , y)) : y ∈ Tω (xn )} ≤ −φω (xn+1 ) + φω (xn ) + εn , for each integer n ≥ 0.

(33)

From (33), for each integer m ≥ 0, we have m

∞

∞

i=0

i=0

i=0

∑ ϕ (ρ (xi , xi+1 )) ≤ φω (x0 ) − φω (xm ) + ∑ εi ≤ φω (x0 ) − inf(φω ) + ∑ εi.

(34)

Combining (30) and (34), we have ∞

∞

i=0

i=0

∑ ϕ (ρ (xi, xi+1 )) ≤ φω (x0 ) − inf(φω ) + ∑ εi < ∞.

(35)

Let ε : Ω → (0, ∞) be a measure, from (27) there exists a measure αε : Ω → (0, ∞) such that

ϕ (t) ≥ αε , ∀t ≥ ε .

(36)

It follows from (35) and (36) that ∞

αε card({i ∈ {0, 1, · · · } : ρ (xi , xi+1 ) ≥ ε }) ≤ φω (x0 ) − inf(φω ) + ∑ εi < ∞. i=0

This implies that for all sufficiently large natural number i, we have

ρ (xi , xi+1 ) < ε . Since ε is an arbitrary positive measure, we conclude that lim ρ (xi , xi+1 ) = 0.

(37)

i→∞

From (28) there exist numbers t0 > 0 and λ0 > 0 such that

ϕ (t) ≥ λ0t, for t ∈ [0,t0 ].

(38)

From (37), there exists an integer p0 ≥ 1 such that

ρ (xi , xi+1 ) ≤ t0 , for each integer i ≥ p0 . It follows from (35), (38) and (39) that ∞

∞ > φω (x0 ) − inf(φω ) + ∑ εi ≥ i=0

∞

∑

ϕ (ρ (xi , xi+1 )) ≥

i=p0

∞

∑ λ0ρ (xi , xi+1 ).

i=p0

This implies that the random sequence {xi }∞ i=0 is a random Cauchy sequence. Therefore there exists x∗ = lim Tω (xi ). i→∞

Since graph(Tω ) is closed, hence x∗ ∈ Tω (x∗ ), and theorem is proved.

(39)

102


Theorem 8. Let (Ω, Σ) be a measurable space and (X, ρ ) be a complete separable metric space. Suppose that / be a continuous random mapping and φ : X × Ω → R1 ∪ {∞} be a bounded below and for T : Ω × X → 2X \ {0} each ω ∈ Ω, x ∈ X (40) inf{φω (y) + ϕ (ρ (x, y)) : y ∈ Tω (x)} ≤ φω (x). Let M, ε : Ω → (0, ∞) be the measure, then there exist a measure δ > 0 and a natural number n0 such that the following assertion hold: Suppose that for each ω ∈ Ω, x0 ∈ X satisfies

φω (x0 ) ≤ M,

(41)

xi+1 ∈ Tω (xi ),

(42)

φω (xi+1 ) + ϕ (ρ (xi, xi+1 )) ≤ inf{φω (y) + ϕ (ρ (xi , y)) : y ∈ Tω (xi )} + δ .

(43)

and for each integer i ∈ {0, . . . , n0 − 1}, we have

and Then there exists an integer j ∈ {0, . . . , n0 − 1} such that

ρ (x j , x j+1 ) ≤ ε . Proof. From (27), there exists a number αε ∈ (0, 1) such that

ϕ (t) ≥ αε , ∀t ≥ ε . Set

δ= and for a natural number n0 we have n0 ≥

(44)

αε , 2

2(M − inf(φω )) + 1. αε

(45)

For ω ∈ Ω, let a random point x0 ∈ X satisfy (41) and assume that for each integer i ∈ {0, . . . , n0 − 1} both (42) and (43) hold. In order to complete the proof of the theorem, it is sufficient to show that there exists an integer j ∈ {0, . . . , n0 − 1} such that ρ (x j , x j+1 ) ≤ ε . Contrary assume that for all integers i ∈ {0, . . . , n0 − 1},

ρ (xi , xi+1 ) > ε .

(46)

From (40), (41) and (43), we have

φω (xi ) < ∞, i = 0, . . . , n0 . By (40) and (43), for all integers i = 0, . . . , n0 − 1, we have

ϕ (ρ (xi , xi+1 )) ≤ −φω (xi+1 ) + inf{φω (y) + ϕ (ρ (xi , y)) : y ∈ Tω (xi )} + δ ≤ −φω (xi+1 ) + φω (xi ) + δ .

(47)

It follows from (44), (46), (47) and the choice of measure δ that for all integers i = 0, . . . , n0 − 1, we have

αε ≤ −φω (xi+1 ) + φω (xi ) + δ and

αε ≤ −φω (xi+1 ) + φω (xi ). 2

(48)


103

By (41) and (48), we have M − inf(φω ) ≥ φω (x0 ) − φω (xn0 ) =

n0 −1

∑ (φω (xi ) − φω (xi+1 )) ≥

i=0

αε n0 , 2

and

2(M − inf(φω )) . αε This inequality contradicts (45), hence proof is completed. n0 ≤

Theorem 9. Let (Ω, Σ) be a measurable space and (X, ρ ) be a complete separable metric space. Suppose that lim inf + t→0

ϕ (t) > 0. t

(49)

/ be a continuous random mapping and Tω (x) be a compact subset of X for ω ∈ Ω, x ∈ X. Let T : Ω × X → 2X \{0} Let a random lower semi continuous mapping φ : X × Ω → R1 ∪ {∞} be a bounded below and not identically equal to ∞. If for each ω ∈ Ω, x ∈ X inf{φω (y) + ϕ (ρ (x, y)) : y ∈ Tω (x)} ≤ φω (x).

(50)

Then there exists a random point x¯ ∈ X such that ¯ x¯ ∈ Tω (x). Proof. We may assume that without loss of generality that

ϕ (0) = 0. From (49) there exists a numbers t0 > 0 and λ0 > 0 such that

ϕ (t) ≥ λ0 t, ∀t ∈ [0,t0 ].

(51)

From (27) there exists a number α0 > 0 such that

ϕ (t) ≥ α0 , ∀t >

t0 . 2

From Ekeland variational principle, see [15] there exists a random point x¯ ∈ X, ω ∈ Ω such that

φω (x) ¯ ≤ inf(φω ) +

α0 , 2

(52)

and

φω (x) + λ0 ρ (x, x) ¯ ≥ φω (x), ¯ ∀x ∈ X \ {x}. ¯

(53)

From (50), for each natural number n there exists a random point ¯ xn ∈ Tω (x),

(54)

such that

1 φω (xn ) + ϕ (ρ (xn , x)) ¯ ≤ φω (x) ¯ + . n It immediately follows from (55) that for each natural number n, we have 1 ϕ (ρ (xn , x)) ¯ ≤ φω (x) ¯ + − φω (xn ). n

(55)

(56)

104


From (52) and (56), we have

ϕ (ρ (xn , x)) ¯ ≤

1 α0 + , for each natural number n. n 2

(57)

By the choice of α0 and (57), for each natural number n > α20 , we have

ϕ (ρ (xn , x)) ¯ ≤ α0 , and

ρ (xn , x) ¯ ≤

t0 . 2

(58)

It follows from inequalities (51), (56) and (58) that for each natural number n > α20 , we have 1 λ0 ρ (xn , x) ¯ ≤ ϕ (ρ (xn , x)) ¯ ≤ φω (x) ¯ + − φω (xn ). n

(59)

¯ is a randomly compact, extracting a subsequence and reindexing if necessary, we may assume Since Tω (x) without loss of generality that there exists ¯ x∗ = lim xn ∈ Tω (x). i→∞

(60)

From (59), (60) and the lower semi continuity of φω , we have

φω (x∗ ) + λ0 ρ (x∗ , x) ¯ ≤ lim inf(φω (xn ) + λ0 ρ (xn , x)) ¯ ≤ φω (x). ¯ n→∞

Combining (53) and (60), this implies that

¯ x∗ = x¯ ∈ Tω (x).

This completes the proof.

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[15] Mordukhovich, B.S. (2006), Variational Analysis and Generalized Differentiation, 1, Basi Theory, Springer, Berlin. [16] Reich, S. and Zaslavski, A.J. (2014), Genericity in Nonlinear Analysis, Developments in Mathematics, 34, Springer, New York. [17] Reich, S. and Zaslavski, A.J. (2015), Variants of Caristi’s fixed point theorem, PanAmer. Math. J., 25(1), 42-52. [18] Ahmad, M.K. (2008), and Salahuddin, Random variational like inequalities, Adv. Nonlinear Var. Inequal., 11(2), 15-24. [19] Caristi, J. (1976), Fixed point theorems for maps satisfying inwardness conditions, Trans. Amer. Math. Soc., 215, 241-251. [20] Kim, J.K. and Salahuddin (2015), The existence of deterministic random generalized vector equilibrium problems, Nonlinear Funct. Anal. Appl., 20(3), 453-464. [21] Goudarzi, H.R. (2014), Random fixed point theorems in Frechet spaces with their applications, J. Math. Ext., 8(2), 71-81. [22] Himmelberg, C.J. (1975), Measurable relations, Fund. Math., 87, 53-72. [23] Hussain, S., Khan, M.F., and Salahuddin (2004), Strongly nonlinear mixed random variational inequalities, International J. Math. Sci., 3(2), 361-368. [24] Khan, M.F. and Salahuddin (2006), Completely generalized nonlinear random variational inclusions, Southeast Asian Bull. Math., 30(5), 261-276. [25] Papagergiou, N.S. (1986), Random fixed point theorems for measurable multifunction in Banach spaces, Proc. Amer. Math. Soc., 97(1), 507-514. [26] Salahuddin (2016), General random variational inequalities and applications, Transact. Math. Prog. Appl., 4(1), 25-33. [27] Siddiqi, A.H., Ahmad, M.K., and Salahuddin (2008), Applications of randomly pseudomonotone operators with randomly upper semicontinuity in generalized random quasi-variational inequalities, J. Appl. Funct. Anal., 3(1), 33-50. [28] Tan, K.K. and Yuan, X.Z. (1994), Random fixed point theorems and approximation in cones, J. Math. Anal Appl., 185, 378-390. [29] Verma, R.U., Khan, M.F., and Salahuddin, (2006), Generalized random variational like inequalities with randomly pseudomonotone multivalued mappings, PanAmer. Math. J., 16(3), 33-46. [30] Verma, R.U. and Salahuddin, (2013), A common fixed point theorem for fuzzy mappings, Transact. Math. Prog. Appl., 1(1), 59-68. [31] Verma, R.U. and Salahuddin (2016), Existence of random variational inequalities, PanAmer. Math. J., 26(2), 95-103.



Three-point Multi-term Fractional Integral Boundary Value Problems of Fractional Functional Differential Equations with Delay K. Sathiyanathan†, V. Krishnaveni, M. Sivabalan Department of Mathematics, SRMV College of Arts and Science, Coimbatore - 641020, India Submission Info Communicated by J A.T. Machado Received 23 March 2017 Accepted 16 June 2017 Available online 1 April 2018

Abstract In this paper, we study fractional functional differential equations with three-point multi-term boundary conditions. Our method of analysis is based on the reduction of the given system to an equivalent system of integral equations. Existence and uniqueness results are obtained by using Schauder fixed point theorem and contraction principle. An illustrative example is also presented.

Keywords Fractional differential equations Three point boundary conditions Contraction principle Schauder fixed point theorem


1 Introduction In recent years, the interest in the study of fractional differential equations has been growing rapidly. Fractional differential equations have arisen in mathematical models of systems and process in various fields such as aerodynamics, acoustics, mechanics, electromagnetism, signal processing, control theory, robotics, population dynamics, finance, etc, we can refer to [1–7]. A variety of results on initial and boundary value problems of fractional differential equations and inclutions can easily be found in the literature on the topic. For some recent results, we can refer to [8–21]. But the results dealing with delay are relatively scarce [1, 19, 22–26]. It is well known that in practical problems, the behavior of systems not only depends on the status just at the present, but also on the status in the past. Thus, in many cases, we must consider fractional functional differential equations with delay in order to solve practical problems. Consequently, our aim in this paper is to study the existence of solutions for boundary value problems of fractional functional differential equations. For 0 < r < 1, we denote by Cr the Banach space of all continuous functions ϕ : [−r, 0] → ℜ endowed with the sup-norm kϕ k[−r,0] := sup {|ϕ (s)| : s ∈ [−r, 0]} . † Corresponding


108

K. Sathiyanathan et al. / Discontinuity, Nonlinearity, and Complexity 7(1) (2018) 107–118

If u : [−r, 0] → ℜ, then for any t ∈ [0, 1], we denote by ut the element of Cr defined by ut (θ ) = u(t + θ ),

for θ ∈ [−r, 0].

In this paper we study the following three-point multi-term fractional integral boundary value problem for the fractional functional differential equation: C

Dα u(t) = f (t, ut ,C Dβ u(t)), 0 < t < 1,

(1)

where 1 < α < 2, 0 < β < 1 and C Dα , C Dβ denote Caputo fractional derivatives, f (t, ut ,C Dβ u(t)) is a continuous function associated with the boundary conditions m

u(0) = 0, u(1) = ∑ γi (I pi u) (η ) , 0 < η < 1,

(2)

i=1

where I pi is the Riemann-Liouville fractional integral of order pi > 0, i = 1, 2, ..., m and γi ≥ 0, i = 1, 2, ..., m, are γi η pi Γ(q) real constants such that ∑m i=1 Γ(pi +q) < 1. And u0 = ϕ , where η ∈ (0, 1) and ϕ is an element of the space Cr+ (0) := {ψ ∈ Cr |ψ (s)| ≥ 0, s ∈ [−r, 0], ψ (0) = 0,C Dβ ψ (s) = 0}. 2 Preliminaries In this section, we introduce some notations and definitions of fractional calculus [22,23] and present preliminary results needed in our proofs later. Definition 1. [1] The fractional integral of order α > 0 of a function f : (t0 , +∞) → ℜ is given by I α f (t) =

1 Γ(α )

ˆ

0

t

f (s) ds,t > t0 , (t − s)1−α

where Γ(.) is the gamma function, provided that the right-hand side is point-wise defined on (t0 , +∞). Definition 2. The Caputo fractional derivative of order α (n − 1 < α < n) of a function f : (t0 , +∞) → ℜ is given by ˆ t f n (s) 1 C α ds,t > t0 , D f (t) = Γ(n − α ) t0 (t − s)α +1−n where Γ(.) is the gamma function, provided that the right-hand side is point-wise defined on (0, ∞). Obviously, the Caputo derivative for every constant function is equal to zero. From the definition of the Caputo derivative, we can acquire the following statement. Lemma 1. Let f (t) ∈ L1 [t0 , ∞). Then C

Lemma 2. Let α > 0. Then

Dα (I α f (t)) = f (t),t > t0 and 0 < α < 1.

I α C Dα f (t) = f (t) − c1 − c2t − · · · − cnt n−1 ,

for some ci ∈ ℜ, i = 1, 2, ..., n, where n = [α ] + 1 and denotes the integer part of α .


109

The following property (Dirichlet’s formula) of the fractional calculus is well known [1]: I γ I β u(t) = I γ +β u(t), t ∈ [0, 1], u ∈ L(0, 1), γ + β ≥ 1, which has the form ˆ

1

γ −1

(t − s) 0

ˆ s ˆ Γ(γ )Γ(β ) t β −1 ( (s − τ ) u(τ )d τ )ds = (t − s)γ +β −1 u(s)ds. Γ( γ + β ) 0 0

For convenience we put

γi η pi Γ(α ) . i=1 Γ(pi + α ) m

Ω = 1− ∑

(3)

Next, we introduce the Green function of fractional functional differential equations boundary value problems. γi η Γ(α ) Lemma 3. Let ∑m i=1 Γ(pi +α ) < 1, γi ≥ 0, pi > 0, i = 1, 2, ..., m, and h ∈ ([0, 1], ℜ). The unique solution of the boundary value problem pi

C

Dα u(t) + h(t) = 0, t ∈ (0, 1), α ∈ (1, 2),

(4)

m

u(0) = 0, u(1) = ∑ γi (I pi u)(η ), 0 < η < 1,

(5)

i=1

is the integral equation u(t) =

ˆ

1

G(t, s)h(s)ds,

(6)

0

where G(t, s) is the Green’s function given by

γit gi (η , s), i=1 ΩΓ(pi + q) m

G(t, s) = g(t, s) + ∑

(7)

where  t(1 − s)α −1 − (t − s)α −1   ; 0 ≤ s ≤ t ≤ 1,  Γ(α ) g(t, s) = t(1 − s)α −1    ; 0 ≤ t ≤ s ≤ 1, Γ(α )

(8)

and ( gi (η , s) =

η pi +α −1 (1 − s)α −1 − (η − s) pi +α −1 ; 0 ≤ s ≤ η ≤ 1, η pi +α −1 (1 − s)α −1 ;

0 ≤ η ≤ s ≤ 1,

(9)

Proof. Using Lemmas 1 and 2, problem (4) − (5) can be expressed as an equivalent integral equation u(t) = c1t + c2 −

ˆ

0

t

(t − s)α −1 h(s)ds. Γ(α )

(10)

110


for c1 , c2 ∈ ℜ. The first condition of (5) implies that c2 = 0. Taking the Riemann-Liouville fractional integral of order pi > 0 for (10) and using Dirichlet’s formula, we get that ˆ s (s − r)α −1 (t − s) pi −1 (c1 s − h(r)dr)ds (I u) (t) = Γ(pi ) Γ(α ) 0 0 ˆ t ˆ t ˆ (t − s) pi −1 s (t − s) pi −1 s (s − r)α −1 = c1 ds − h(r)drds Γ(pi ) Γ(pi ) Γ(α ) 0 0 0 ˆ t 1 t pi Γ(α ) − (t − s) pi +α −1 ds. = c1 Γ(pi + α ) Γ(pi + α ) 0 pi

ˆ

t

The second condition of (5) yields c1 −

1

ˆ

0

m (1 − s)α −1 αi η pi Γ(α ) m γi h(s)ds = c1 ∑ −∑ Γ(α ) i=1 Γ(pi + α ) i=1 Γ(pi + α )

ˆ

η

(η − s) pi +α −1 h(s)ds.

0

Then we have that ˆ 1 ˆ η m (1 − s)α −1 1 γi h(s)ds − ∑ c1 = [ (η − s) pi +α −1 h(s)ds]. Ω 0 Γ(α ) ΩΓ(p + α ) i 0 i=1 Therefore, the unique solution of boundary value problem (4) − (5) is written as u(t) = −

ˆ

(t − s)α −1 1 h(s)ds + Γ(α ) ΩΓ(α )

t

0

ˆ

1

α −1

(1 − s) 0

γi th(s)ds − ∑ i=1 ΩΓ(pi + α ) m

ˆ

η

(η − s) pi +α −1th(s)ds.

0

Hence, by taking into account (3), we have ˆ 1 ˆ η 1 1 m γi (t − s)α −1 α −1 (η − s) pi +α −1th(s)ds h(s)ds + (1 − s) th(s)ds − ∑ u(t) = − Γ( α ) ΩΓ( α ) Ω Γ(p + α ) i 0 0 0 i=1 ˆ 1 ˆ 1 α −1 α −1 (1 − s) t (1 − s) t + h(s)ds − h(s)ds Γ(α ) Γ(α ) 0 0 ˆ t ˆ 1 (t − s)α −1 (1 − s)α −1t h(s)ds − h(s)ds = Γ(α ) Γ(α ) 0 0 ˆ 1 ˆ η m γit pi +α −1 α −1 +∑ (η − s) pi +α −1 h(s)ds) ( (1 − s) h(s)ds − η ΩΓ(p + α ) i 0 0 i=1 ˆ 1 m ˆ 1 γit gi (η , s)h(s)ds g(t, s)h(s)ds + = ∑ 0 i=1 ΩΓ(pi + α ) 0 ˆ 1 G(t, s)h(s)ds. = ˆ

t

0

The proof is completed. Lemma 4 ([26] Schauder fixed point theorem). Let (D, d) be a complete metric space, U be a closed convex subset of D, and T : D → D be the map such that the set Tu : u ∈ U is relatively compact in D. Then the operator T has at least one fixed point u∗ ∈ U : Tu∗ = u∗ .


111

3 Main results In this section, we discuss the existence and uniqueness of solutions for boundary value problem (1) and (2) by the Schauder fixed point theorem and Banach contraction principle. For convenience, we define the Banach space X = u|u ∈ C[−r, 1],C Dβ u ∈ C[−r, 1], 0 < β < 1 . Also, if I is an interval of the real line ℜ, by C(I) and C1 (I) we denote the set of continuous and continuously differentiable functions on I, respectively. Moreover, for u ∈ C(I), we define kukI = max |u(t)| + max |C Dβ u(t)|. t∈I

(11)

t∈I

For u0 = ϕ , in view of the definitions of ut and ϕ , we have u0 = u(θ ) = ϕ (θ ), for θ ∈ [−r, 0]. Thus, we have u(t) = ϕ (t), for t ∈ [−r, 0]. Since f : [0, 1] × Cr × ℜ → ℜ is a continuous function, set f (t, ut ,C Dβ u(t)) := h(t) in Lemma 3. We have by Lemma 3 that a function u is a solution of boundary value problem (1) and (2) if and only if it satisfies ˆ 1  G(t, s) f (s, us ,C Dβ u(s))ds, t ∈ (0, 1), u(t) = 0  ϕ (t), t ∈ [−r, 0]. We define an operator T : X → X as follows: ˆ 1  G(t, s) f (s, us ,C Dβ u(s))ds, Tu(t) = u(t) =  0 ϕ (t),

t ∈ (0, 1), t ∈ [−r, 0],

and ˆ 1 |G(t, s)g(s)|ds), l = max ( 0≤t≤1 0 ˆ 1 ∂ l1 = max ( | G(t, s)g(s)|ds), 0≤t≤1 0 ∂ t (1 + α Ω + (Ω + 1)Γ(2 − β )) m 1 γi η pi +α Q= +∑ [1 + ]. ΩΓ(α + 1)Γ(2 − β ) Γ(2 − β ) i=1 ΩΓ(pi + α + 1) Theorem 5. Assume the following: (H1) There exists a nonnegative function g ∈ L[0, 1] such that | f (t, u, v)| ≤ g(t) + a |u|k1 + b |v|k2 for each u ∈ Cr , v ∈ ℜ, where a, b ∈ ℜ are nonnegative constants and 0 < k1 , k2 < 1; or (H2) There exists a nonnegative function g ∈ L[0, 1] such that | f (t, u, v)| ≤ g(t) + a |u|k1 + b |v|k2 for each u ∈ Cr , v ∈ ℜ, where a, b ∈ ℜ are nonnegative constants and k1 , k2 > 1; Then boundary value problem (1) and (2) has a solution.

(12)

112


Proof. Suppose (H1) holds. Choose

ω ≥ max{3(l +

1 1 l1 ), (3aQ) 1−k1 , (3bQ) 1−K2 }, Γ(2 − β )

(13)

and define the cone U = {u ∈ X | kuk ≤ ω , ω > 0}. For any u ∈ U , we have ˆ 1 G(t, s) f (s, us ,C Dβ u(s))ds| |Tu(t)| = | 0 ˆ t ˆ 1 ˆ 1 (t − s)α −1 t(1 − s)α −1 ds + ds |G(t, s)g(s)|ds + (a|ω |k1 + b|ω |k2 )( ≤ Γ(α ) ΩΓ(α ) 0 0 0 ˆ η m γit (η − s) pi +α −1 )ds +∑ ΩΓ(p + α ) i 0 i=1 ≤ l + (a|ω |k1 + b|ω |k2 )(

m tα t γit η pi +α + +∑ ) α Γ(α ) α ΩΓ(α ) i=1 ΩΓ(pi + α ) (pi + α )

≤ l + (a|ω |k1 + b|ω |k2 )(

m tα t γit η pi +α + +∑ ) Γ(α + 1) ΩΓ(α + 1) i=1 ΩΓ(pi + α + 1)

≤ l + (a|ω |k1 + b|ω |k2 )(

m 1 γi η pi +α 1 + +∑ ). Γ(α + 1) ΩΓ(α + 1) i=1 ΩΓ(pi + α + 1)

Also ′

∂ ||G(t, s) f (s, us ,C DB u(s))|ds 0 ∂t ˆ 1 ˆ t ∂ (α − 1)(t − s)α −2 ds. ≤ | G(t, s)g(s)|ds + (a|ω |k1 + b|ω |k2 )( Γ(α ) 0 ∂t 0 ˆ 1 ˆ η m (1 − s)α −1 γi + (η − s) pi +α −1 )ds ds + ∑ Γ( α ) ΩΓ(p + α ) i 0 0 i=1

|Tu (t)| ≤

ˆ

1

|

≤ l1 + (a|ω |k1 + b|ω |k2 )(

m 1 1 γi η pi +α + +∑ ) Γ(α ) α ΩΓ(α ) i=1 ΩΓ(pi + α ) (pi + α )

≤ l1 + (a|ω |k1 + b|ω |k2 )(

m 1 1 γi η pi +α + +∑ ). Γ(α ) α ΩΓ(α ) i=1 ΩΓ(pi + α + 1)

ˆ 1 1 | D u(t)| ≤ (t − s)−β |Tu′ (s)|ds Γ(1 − β ) 0 ˆ t m 1 1 1 γi η pi +α ≤ + +∑ )]ds (t − s)−β [l1 + (a|ω |k1 + b|ω |k2 )( Γ(1 − β ) 0 Γ(α ) α ΩΓ(α ) i=1 ΩΓ(pi + α + 1) ˆ t l1 ≤ (t − s)−β ds Γ(1 − β ) 0 ˆ t m 1 1 γi η pi +α 1 + + +∑ ) (t − s)−β (a|ω |k1 + b|ω |k2 )( Γ(1 − β ) 0 Γ(α ) α ΩΓ(α ) i=1 ΩΓ(pi + α + 1) C

β

≤

m l1t 1−β (a|ω |k1 + b|ω |k2 )t 1−β 1 1 γi η pi +α + ( + +∑ ) Γ(2 − β ) Γ(2 − β ) Γ(α ) α ΩΓ(α ) i=1 ΩΓ(pi + α + 1)

≤

m (a|ω |k1 + b|ω |k2 ) 1 1 l1 γi η pi +α + ( + +∑ ). Γ(2 − β ) Γ(2 − β ) Γ(α ) α ΩΓ(α ) i=1 ΩΓ(pi + α + 1)

(14)


113

In view of (11) and (13) we obtain l1 (1 + α Ω + (Ω + 1)Γ(2 − β )) + (a|ω |k1 + b|ω |k2 )( Γ(2 − β ) ΩΓ(α + 1)Γ(2 − β ) m p + α i 1 γi η +∑ [1 + ]) ΩΓ(p + α + 1) Γ(2 −β) i i=1 ω ≤ + (a|ω |k1 + b|ω |k2 )Q 3 ω ω ω ≤ + + 3 3 3 ≤ ω,

kTu(t)k ≤ l +

(15)

which implies that T : U → U . The continuity of the operator T follows from the continuity of f and G. Now if (H2) holds, we choose 0 < ω ≤ min{3(l +

1 1−k1 1 1−k1 l1 ), ( ) 1 ,( ) 2 }, Γ(2 − β ) 3aQ 3bQ

(16)

and by the same process as above, we obtain l1 (1 + α Ω + (Ω + 1)Γ(2 − β )) + (a|ω |k1 + b|ω |k2 )( Γ(2 − β ) ΩΓ(α + 1)Γ(2 − β ) m p + α γi η i 1 +∑ (1 + )] Γ(2 − β ) i=1 ΩΓ(pi + α + 1) ω ≤ + (a|ω |k1 + b|ω |k2 )Q 3 ω ω ω ≤ + + 3 3 3 ≤ ω,

kTu(t)k ≤ l +

which implies that T : U → U . Now we show that T is a completely continuous operator. Let L = max0≤t≤1 f (t, ut ,C Dβ u(t)) . Then for u ∈ U and t1 ,t2 ∈ [−r, 1] with t1 < t2 , in view of lemma 3, if 0 ≤ t1 < t2 ≤ 1, then ˆ

|Tu(t2 ) − Tu(t1 )| = | ˆ ≤

0

1

ˆ

β

1

G(t1 , s) f (s, us ,C Dβ u(s))ds| G(t2 , s) f (s, us , D u(s))ds − 0 0 ˆ 1 ˆ t2 t1 |G(t2 , s) − G(t1 , s)|Lds + |G(t2 , s) − G(t1 , s)|Lds + |G(t2 , s) − G(t1 , s)|Lds C

t2

η

t1

L γi L (η − s) pi +α −1 (t2 − t1 )ds| | (t2 − t1 )(1 − s)α −1 ds| + ∑ | ΩΓ(α ) 0 ΩΓ(p + α ) i 0 i=1 ˆ t2 ˆ t1 α −1 α −1 (t2 − 1) (t1 − s) +| ds − ds| Γ(α ) Γ(α ) 0 0 m t2α t1α γi η pi +q 1 +∑ ]+| − |]. ≤ L[|t2 − t1 |[ ΩΓ(α + 1) i=1 ΩΓ(pi + α + 1) Γ(α + 1) Γ(α + 1) ≤

ˆ

t1

m

If −r ≤ t1 < t2 ≤ 0, then |Tu(t2 ) − Tu(t1 )| = |φ (t2 ) − φ (t1 )| .

ˆ

114


If −r ≤ t1 < 0 < t2 ≤ 1, then |Tu(t2 ) − Tu(t1 )| = |Tu(t2 ) − Tu(0)| + |Tu(0) − Tu(t1 )| ˆ 1 |G(t2 , s) − G(0, s)|| f (s, us ,C Dβ u(s))|ds + |φ (0) − φ (t1 )| ≤ 0

≤

m 1 L γi η pi +α |t2α | + L|t2 |( +∑ ) + φ (t1 ). Γ(α + 1) ΩΓ(α + 1) i=1 ΩΓ(pi + α + 1)

Hence if 0 ≤ t1 < t2 ≤ 1, we have ˆ t1 1 [(t2 − s)−β − (t1 − s)−β ]Tu′ (s)ds| | Γ(1 − β ) 0 ˆ t2 ˆ t1 1 −β ′ ≤ (t2 − s) Tu (s)ds − | (t1 − s)−β Tu′ (s)ds| Γ(1 − β ) 0 0 ˆ t1 ˆ t1 1 (t1 − s)−β Tu′ (s)ds| | (t2 − s)−β Tu′ (s)ds − + Γ(1 − β ) 0 0 ˆ t2 ˆ t1 1 | (t2 − s)−β |Tu′ (s)|ds + ≤ ((t2 − s)−β − (t1 − s)−β )|Tu′ (s)|ds| Γ(1 − β ) t1 0 ˆ 1 ˆ t2 ∂ 1 −β | G(s, z)|| f (z, uz ,C Dβ u(z))|dz)ds | (t2 − s) ( ≤ Γ(−β ) t1 0 ∂s ˆ 1 ˆ t1 ∂ −β −β | G(s, z)|| f (z, uz ,C Dβ u(z))|dz)ds|. + ((t2 − s) − (t1 − s) )( 0 ∂s 0

|C Dβ Tu(t2 ) −C Dβ Tu(t1 )| =

ˆ

1

| 0

ˆ t ˆ 1 1 ∂ ∂ (t − s)α −1 ∂ G(s, z)|ds = | − [ ]ds + [(1 − s)α −1 ]ds. ∂s ∂ t Γ( α ) ΩΓ( α ) ∂ t 0 0 ˆ η γi ∂ 1 m − ∑ (η − s) pi +α −1tds| Ω i=1 Γ(pi + α ) 0 ∂ t ˆ 1 ˆ t 1 (α − 1)(t − s)α −2 ]ds + [(1 − s)α −1 ]ds [ ≤ Γ( α ) ΩΓ( α ) 0 0 ˆ η γi 1 m + ∑ (η − s) pi +α −1 ds Ω i=1 Γ(pi + α ) 0

|C Dβ Tu(t2 ) −C Dβ Tu(t1 )| ≤

≤

t α −1 1 1 γi η pi +α 1 m + [ ]+ ∑ Γ(α ) ΩΓ(α ) α Ω i=1 Γ(pi + α + 1)

≤

1 1 1 m γi η pi +α + + ∑ . Γ(α ) ΩΓ(α + 1) Ω i=1 Γ(pi + α + 1)

1 1 1 1 m γi η pi +α [ + + ∑ ] Γ(1 − β ) Γ(α ) ΩΓ(α + 1) Ω i=1 Γ(pi + α + 1) ˆ t1 ˆ t2 −β [(t2 − s)−β − (t1 − s)−β ]ds| | (t2 − s) ds + t1

0

1−β

1−β

t t L 1 1 1 m γi η pi +α ≤ [ + + ∑ ]| 2 − 1 | Γ(1 − β ) Γ(α ) ΩΓ(α + 1) Ω i=1 Γ(pi + α + 1) 1 − β 1 − β ≤

1 1 1 m γi η pi +α L 1−β 1−β [ + + ∑ ]|t2 − t1 |. Γ(2 − β ) Γ(α ) ΩΓ(α + 1) Ω i=1 Γ(pi + α + 1)


115

If −r ≤ t1 < t2 ≤ 0, in views of the definition of φ , we have |C Dβ Tu(t2 ) −C Dβ Tu(t1 )| = |C Dβ φ (t2 ) −C Dβ φ (t1 )|. If −r ≤ t1 < 0 < t2 ≤ 1, then |C Dβ Tu(t2 ) −C Dβ Tu(t1 )| = |C Dβ Tu(t2 ) −C Dβ Tu(0)| + |C Dβ Tu(0) −C Dβ Tu(t1 )| ˆ t2 1 (t2 − s)−β Tu′ (s)| + 0 | = Γ(1 − β ) 0 ˆ t2 ˆ 1 ∂ 1 (t2 − s)−β ( | | G(s, z)|| f (z, uz ,C Dβ u(z))|dz)ds| ≤ Γ(1 − β ) 0 0 ∂s γi η pi +α L 1 1 1 m 1−β ≤ [ + + ∑ ]t2 . Γ(2 − β ) Γ(α ) ΩΓ(α + 1) Ω i=1 Γ(pi + α + 1) Hence, if 0 ≤ t1 < t2 ≤ 1, we have kTu(t2 ) − Tu(t1 )k ≤ L[|t2 − t1 |[ 1−β

m t2α t1α γi η pi +q 1 +∑ ]+| − |] ΩΓ(α + 1) i=1 ΩΓ(pi + α + 1) Γ(α + 1) Γ(α + 1) 1−β

L|t − t1 | 1 1 1 m γi η pi +α + 2 [ + + ∑ ]. Γ(2 − β ) Γ(α ) ΩΓ(α + 1) Ω i=1 Γ(pi + α + 1) If −r ≤ t1 < t2 ≤ 0, we have kTu(t2 ) − Tu(t1 )k = kφ (t2 ) − φ (t1 )k . If −r ≤ t1 < 0 < t2 ≤ 1, then kTu(t2 ) − Tu(t1 )k ≤ L[|t2 |[

m |t2α | 1 γi η pi +q +∑ ]+ ] ΩΓ(α + 1) i=1 ΩΓ(pi + α + 1) Γ(α + 1) 1−β

+

L|t2 | 1 1 1 m γi η pi +α [ + + ∑ ]. Γ(2 − β ) Γ(α ) ΩΓ(α + 1) Ω i=1 Γ(pi + α + 1)

In any case, it implies that kTu(t2 ) − Tu(t1 )k → 0 as t2 → t1 . i.e., for any ε > 0, there exists δ > 0, independent on t1 ,t2 and u, such that |Tu(t2 ) − Tu(t1 )| ≤ ε , whenever |t2 − t1 | < δ . Therefore T : X → X is completely continuous. The proof is complete. For convenience, we denote M= N=

1 1 m γi η pi +α 1 + + ∑ , Γ(α + 1) ΩΓ(α + 1) Ω i=1 Γ(pi + α + 1)

1 1 1 1 m γi η pi +α [ + + ∑ ]. Γ(2 − β ) Γ(α ) ΩΓ(α + 1) Ω i=1 Γ(pi + α + 1)

Theorem 6. Assume that (H3) There exist a constant P > 0 such that | f (t, µ , ν ) − f (t, µ¯ , ν¯ )| ≤ P (|µ − µ¯ | + |ν − ν¯ |) each µ , µ¯ ∈ Cr , ν , ν¯ ∈ F. If P < (M + N)−1 , then boundary value problem (1) and (2) has a unique solution.

116


Proof. Consider the operator T : X → X defined by (12). Clearly, the fixed point of the operator T is the solution of boundary value problem (1) and (2). We will use the Banach contraction principle to prove that T has a fixed point. We first show that T is a contraction. For each t ∈ [0, 1], ˆ t ¯ ds |G(t, s)| f (s, us ,C Dβ u(s)) − f (s, u¯s ,C Dβ u(s)) |Tu(t) − T u(t)| ¯ ≤ 0 ˆ t ˆ (t − s)α −1 1 1 (1 − s)α −1t ≤ P[ku − uk ¯ + kv − vk][ ¯ ds + ds Γ(α ) Ω 0 Γ(α ) 0 ˆ η γi 1 m (η − s) pi +α −1tds] + ∑ Ω i=1 α Γ(α ) 0 ≤ P[ku − uk ¯ + kv − vk][ ¯

tα t 1 m γit η pi +α + + ∑ ] α Γ(α ) Ωα Γ(α ) Ω i=1 α Γ(pi + α + 1)

≤ P[ku − uk ¯ + kv − vk][ ¯

1 1 1 m γi η pi +α + + ∑ ] α Γ(α ) Ωα Γ(α ) Ω i=1 α Γ(pi + α + 1)

≤ P[ku − uk ¯ + kv − vk]M. ¯

(17)

By a similar method, we get ˆ t 1 (t − s)−β (Tu′ (s) − T u¯′ (s))ds| ¯ =| | D Tu(t) − D T u(t)| Γ(1 − β ) 0 ˆ t ˆ 1 1 ∂ −β (t − s) ( ≤ | G(s, z)|| f (z, uz ,C Dβ u(z)) Γ(1 − β ) 0 ∂ s 0 β

C

β

C

¯ − f (z, u¯z ,C Dβ u(z))|dz)ds ˆ t ˆ 1 P[ku − uk ¯ + kv − vk] ¯ ∂ −β ≤ (t − s) ( | G(s, z)|dz)ds. Γ(1 − β ) 0 0 ∂s

(18)

ˆ t P[ku − uk ¯ + kv − vk] ¯ 1 1 ¯ ≤ | D Tu(t) − D T u(t)| + (t − s)−β [ Γ(1 − β ) Γ( α ) ΩΓ( α + 1) 0 m p + α 1 γi η i + ∑ ]ds Ω i=1 Γ(pi + α + 1) C

β

C

β

≤

1 1 1 m γi η pi +α P[ku − uk ¯ + kv − vk] ¯ [ + + ∑ ] (1 − β )Γ(1 − β ) Γ(α ) ΩΓ(α + 1) Ω i=1 Γ(pi + α + 1)

≤ P[ku − uk ¯ + kv − vk]N. ¯

(19)

Clearly, for each t ∈ [−r, 0], we have |Tu(t) − T u(t)| ¯ = 0. Therefore, by (13) and (19), we get kTu − T vk ≤ P[ku − uk ¯ + kv − vk]M ¯ + P[ku − uk ¯ + kv − vk]N ¯ ≤ P[ku − uk ¯ + kv − vk](M ¯ + N) ≤ ku − uk ¯ + kv − vk. ¯ and T is a contraction. As a consequence of the Banach contraction principle, we get that T has a fixed point which is a solution of boundary value problem(1) and (2). Example 1. Consider boundary value problems of the following fractional functional differential equations: |ut | + C Dβ u(t) C α , (20) D u(t) = (6 + 9et )(1 + ut + C Dβ u(t) )


117

 

u(0) = 0, √ √ √ u(1) = 1 (I 12 u)( 1 ) + 4 (I 32 u)( 1 ) + π (I 52 u)( π ) + 1 (I 72 u)( 1 ) + 3 (I 29 u)( 1 ), 20 2 15 2 12 20 2 2 10 2

(21)

where C Dα ,C Dβ denote Caputo fractional derivatives, 1 ≤ α ≤ 2, 0 ≤ β ≤ 1,t ∈ (0, 1). Choose m = 5, η = 1/2, α = 3/2, β =√1/2, p1 = 1/2, p2 = 3/2, p3 = 5/2, p4 = 7/2, p5 = 9/2, γ1 = 1/20, γ − √ √ 2 = 4/15, γ3 = pi/12, γ4 pi/20, γ5 = 3/10 and |ut | + C Dβ u(t) C β , f t, ut , D u(t) = (6 + 9et )(1 + |ut | + C Dβ u(t) ) set f (t, µν , ) =

|µ +| + |ν | . (6 + 9et )(1 + |µ | + |ν |)

Let µ , µ¯ ∈ Cr , ν , ν¯ ∈ ℜ. Then for each t ∈ [0, 1], 1 |µ | + |ν | |µ¯ | + |ν¯ | | − | t 6 + 9e 1 + |µ | + |ν | 1 + |µ¯ | + |ν¯ | |µ + µ¯ | − |ν + ν¯ | = t (6 + 9e )(1 + |µ | + |ν |)(1 + |µ¯ | + |ν¯ |) 1 (|µ + µ¯ | − |ν + ν¯ |) ≤ (6 + 9et ) 1 (|µ + µ¯ | − |ν + ν¯ |) . ≤ 15

| f (t, µ , ν ) − f (t, µ¯ , ν¯ )| =

For each t ∈ [0, 1], 1 |µ | + |ν | |µ¯ | + |ν¯ | | − | t 6 + 9e 1 + |µ | + |ν | 1 + |µ¯ | + |ν¯ | |µ + µ¯ | − |ν + ν¯ | = t (6 + 9e )(1 + |µ | + |ν |)(1 + |µ¯ | + |ν¯ |) 1 (|µ + µ¯ | − |ν + ν¯ |) ≤ (6 + 9et ) 1 ≤ (|µ + µ¯ | − |ν + ν¯ |) . 6

| f (t, µ , ν ) − f (t, µ¯ , ν¯ )| =

Thus the condition (H3 ) holds with p =

1 15 .

For the given γi , η and pi , we have

γi η pi +α ≈ 0.5856, i=1 Γ(pi + α + 1) m

Ω = 1− ∑ M=

1 1 1 m γi η pi +α + + ∑ ≈ 3.4953, Γ(α + 1) ΩΓ(α + 1) Ω i=1 Γ(pi + α + 1)

N=

γi η pi +α 1 1 1 1 m [ + + ∑ ] ≈ 2.4256. Γ(2 − β ) Γ(α ) ΩΓ(α + 1) Ω i=1 Γ(pi + α + 1)

1 < 0.690 < (M + N)−1 . It implies that p = 15 Then by Therem 6, boundary value problem (20) and (21) has a unique solution.

118


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Aims and Scope The interdisciplinary journal publishes original and new results on recent developments, discoveries and progresses on Discontinuity, Nonlinearity and Complexity in physical and social sciences. The aim of the journal is to stimulate more research interest for exploration of discontinuity, complexity, nonlinearity and chaos in complex systems. The manuscripts in dynamical systems with nonlinearity and chaos are solicited, which includes mathematical theories and methods, physical principles and laws, and computational techniques. The journal provides a place to researchers for the rapid exchange of ideas and techniques in discontinuity, complexity, nonlinearity and chaos in physical and social sciences. Topics of interest include but not limited to • • • • • • • • • • • • • •

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Continued from inside front cover Didier Bénisti CEA, DAM, DIF 91297 Arpajon Cedex France Fax: +33 169 267 106 Email: [email protected]

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An Interdisciplinary Journal of Discontinuity, Nonlinearity, and Complexity Volume 7, Issue 1

March 2018

Contents Existence and Uniqueness of Solutions for a Coupled System of Higher Order Fractional Differential Equations with Integral Boundary Conditions P. Duraisamy, T. Nandha Gopal……...………………………………………........

1－14

Almost Periodicity in Chaos Marat Akhmet, Mehmet Onur Fen…………………….……..……………………..

15－29

Backlund Transformation and Quasi-Integrable Deformation of Mixed Fermi-PastaUlam and Frenkel-Kontorova Models Kumar Abhinav, A Ghose Choudhury, Partha Guha..……………………...……...

31－41

A New Comparison Theorem and Stability Analysis of Fractional Order CohenGrossberg Neural Networks Xiaolei Liu, Jian Yuan, Gang Zhou, Wenfei Zhao…….....…..........…....…………..

43－53

Existence of Solutions of Stochastic Fractional Integrodifferential Equations P. Umamaheswari, K. Balachandran, N. Annapoorani,,……..……......…...…..….

55－65

Evolution Towards the Steady State in a Hopf Bifurcation: A Scaling Investigation Marek Barski, Adam Stawiarski, Piotr Pająk..…………….....…..……….…….....

67－79

Integrability of a time dependent coupled harmonic oscillator in higher dimensions Ram Mehar Singh, S. B. Bhardwaj, Kushal Sharma, Richa Rani, Fakir Chand...

81－94

Approximation of random fixed point theorems Salahuddin………………….………………………………….……………..…….

95－105

Three-point multi-term fractional integral boundary value problems of fractional functional differential equations with delay K. Sathiyanathan, V. Krishnaveni, M. Sivabalan………………...…………..…….

107－118

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