Discontinuity, Nonlinearity, and Complexity

Volume 5 Issue 1 March 2016

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

An Interdisciplinary Journal of

Discontinuity, Nonlinearity, and Complexity

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

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

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

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

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 Applied Statistics Lab GE Global Research 1 Research Circle, K1-4A64 Niskayuna, NY 12309 Email: [email protected]

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

Nikolai A. Kudryashov Department of Applied Mathematics Moscow Engineering and Physics Institute (State University), 31 Kashirskoe Shosse 115409 Moscow, Russia Fax: +7 495 324 11 81 Email: [email protected]

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Josep J. Masdemont Department of Matematica Aplicada I Universitat Politecnica de Catalunya (UPC) Diagonal 647 ETSEIB 08028 Barcelona, Spain Email: [email protected]

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

Continued on back materials

An Interdisciplinary Journal of Discontinuity, Nonlinearity, and Complexity Volume 5, Issue 1, March 2016

Editors Valentin Afraimovich Xavier Leoncini Lev Ostrovsky Dimitry Volchenkov

L&H Scientific Publishing, LLC, USA

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Discontinuity, Nonlinearity, and Complexity 5(1) (2016) 1–8

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

Introducing Increasing Returns to Scale and Endogenous Technological Progress in the Structural Dynamic Economic Model SDEM-2 Dmitry V. Kovalevsky† Nansen International Environmental and Remote Sensing Centre, 14th Line 7, office 49, Vasilievsky Island, 199034 St. Petersburg, Russia Saint Petersburg State University, Ulyanovskaya 3, 198504 St. Petersburg, Russia Nansen Environmental and Remote Sensing Center, Thormøhlens gate 47, N-5006 Bergen, Norway Submission Info Communicated by Dimitri Volchenkov Received 23 December 2014 Accepted 1 July 2015 Available online 1 April 2016 Keywords Economic growth Increasing returns to scale Endogenous technological progress Instability

Abstract Two nonlinear modifications of the Structural Dynamic Economic Model SDEM-2 are developed and studied analytically and numerically. In the first model version described in the present paper the production function is assumed to be nonlinear that leads to increasing returns to scale, while the second model version proposed describes endogenous technological progress by treating the technology parameter of the production function as an additional state variable. Dependent on the values of model parameters and on initial conditions, both modifications of SDEM considered demonstrate two different dynamic regimes: either an explosive economic growth or the collapse of the economy. ©2016 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction The Structural Dynamic Economic Model SDEM developed initially in [1] and later updated to a version SDEM2 in our previous work (see the detailed description of SDEM-2 in [2]) describes the evolution of a closed economy driven by a conflict of interests of two key aggregated model actors: entrepreneurs and wage-earners. It should be noted that the basic version of model SDEM-2 is essentially linear, although various nonlinearities can be (and have been) easily introduced in the modelling framework (particularly, the case of nonlinear investment control was considered in [2], Sec. 3.3). In the present paper we develop and study analytically and numerically two new modifications of SDEM where nonlinearities are introduced in the very core of model formulation: namely, a version with a nonlinear production function leading to increasing returns to scale (Sec. 2), and a three-dimensional extension of the model with endogenous technological progress where the technology parameter of the production function grows due to targeted investment in R&D (Sec. 3). The results of our studies are summarized in the concluding Sec. 4. † Corresponding

author. Email address: [email protected], d v [email protected]

ISSN 2164 − 6376, eISSN 2164 − 6414/$-see front materials © 2016 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/DNC.2016.03.001

2

Dmitry V. Kovalevsky / Discontinuity, Nonlinearity, and Complexity 5(1) (2016) 1–8

2 The model SDEM with increasing returns to scale 2.1

Model equations

We take the model SDEM-2 as described in [2] as a basis and make the following modifications: (i) From now on we imply that the only production factor on which the (per capita) output y depends is the (per capita) physical capital k (below referred to as just capital for brevity), so the human capital dynamics equation is dropped out of the dynamic system. (ii) We no longer assume that output is linear in capital. Instead, we imply that the technology parameter ν in the production function y = ν k is capital-dependent: ν = ν (k). Specifically, we assume the power law dependence: ν (k) = ν ∗ kα , ν ∗ = const, α > 0, (1) and since y(k) = ν (k)k,

(2)

we have the resultant production function of the form y = ν ∗ k1+α .

(3)

The fact that the production function (3) exerts faster-than-linear growth in its production factor implies that the economy under study is characterized by increasing returns to scale. The assumption (i) made above implies that the dynamic system describing the economic development now becomes two-dimensional and can be written in the form k˙ = y − (1 − θ ) w − θ d − (λk + λL ) k, w˙ = λw qwtarg − w .

(4) (5)

As in [2], in Eq. (4) w is the wages of wage-earners, d is the dividend of entrepreneurs, θ is the (constant) fraction of entrepreneurs in population, (1 − θ ) is the (constant) fraction of wage-earners in population (0 < θ < 1), λk is the capital depreciation rate, and λL is the (constant) population growth rate. In Eq. (5) wtarg is the (time-dependent) target wage rate (see its calculation below), λw is wage adjustment rate, and q is a parameter describing the negotiating power of entrepreneurs in wage negotiations with trade unions of wageearners (0 < q < 1). Adopting the procedure of target wage rate calculation descibed in detail in [2], Sec. 2.3, to the modified dynamic system (4)–(5), we find that wtarg can be obtained by equating the r.h.s. of Eq. (4) to zero under additional assumption d = 0. This yields wtarg =

y − (λk + λL ) k , 1−θ

(6)

or, after substituting Eq. (3) in the latter equation,

ν ∗ k1+α − (λk + λL ) k . (7) 1−θ We also adopt the same entrepreneur strategy of investment and dividend calculation as in our previous work, wtarg =

θ d = ρd (y − (1 − θ ) w) (cf. Eqs. (7)–(8) in [2]) and assume that ρd = const, 0 < ρd < 1. Putting it all together, we get a two-dimensional nonlinear dynamic system of the from k˙ = (1 − ρd ) ν ∗ k1+α − (1 − θ ) w − (λk + λL ) k, q ∗ 1+α ν k − (λk + λL ) k − w]. w˙ = λw [ 1−θ

(8)

(9) (10)


2.2

3

Stability analysis of fixed points

It can be shown that the dynamic system (9)–(10) has two fixed points on the phase plane: (i) The trivial fixed point (0, 0): k = 0, w = 0. (ii) The non-trivial fixed point (k0 , w0 ) where 1 − q (1 − ρd ) λk + λL (1 − q) (1 − ρd ) ν ∗

(11)

1 q ρd (λk + λL ) k0 . 1 − θ 1 − q 1 − ρd

(12)

k0α = and w0 =

Let us perform the stability analysis of these fixed points. 2.2.1

Stability of trivial fixed point (0, 0)

The dynamics of linearized system in the neighbourhood of trivial fixed point (0, 0) are described by an equation δ k˙ δk = A1 (13) δ w˙ δw where the matrix A1 has the explicit form A1 = −

q 1−θ

λk + λL , (1 − θ ) (1 − ρd ) . (λk + λL ) λw , λw

(14)

The corresponding characteristic equation is quadratic: det(A1 − λ I) = λ 2 − TrA1 · λ + detA1 = 0

(15)

TrA1 = − (λk + λL + λw) < 0,

(16)

detA1 = (1 − q (1 − ρd )) (λk + λL ) λw > 0.

(17)

with the coefficients

The fact that TrA1 < 0 and detA1 > 0 means that real parts of both eigenvalues of the matrix A1 are negative [3]. With the application of a generalization of Lyapunov’s stability theorem ( [4], Sec. 22, Theorem 1) provided in [4], Sec. 26, it can be proven that the trivial fixed point is then asymptotically stable. 2.2.2

Instability of non-trivial fixed point (k0 , w0 )

The dynamics of linearized system in the neighbourhood of non-trivial fixed point (k0 , w0 ) are described by an equation δ k˙ δk = A0 (18) δ w˙ δw where the matrix

A0 =

has the following elements:

a011 , a012 a021 , a022

(19)

a011 = (1 + α )(1 − ρd ) ν ∗ k0α − (λk + λL ) ,

(20)

a012 = − (1 − θ ) (1 − ρd ) ,

(21)

qλw [(1 + α ) ν ∗ k0α − (λk + λL )] , 1−θ

(22)

a021 =

4


a022 = −λw .

(23)

Again the characteristic equation has the same form as Eq. (15) (with A1 replaced by A0 ). A straightforward calculation shows that (24) detA0 = −α (1 − q (1 − ρd )) (λk + λL ) λw < 0. The negative sign of detA0 implies that at least one eigenvalue of A0 has a positive real part. Therefore the non-trivial fixed point is unstable (this fact follows from a generalization of Lyapunov’s instability theorem ( [4], Sec. 22, Theorem 2) provided in [4], Sec. 26). 2.3

Phase plane analysis. Explosive growth regime

The phase plane for the dynamic system (9)–(10) is shown on Fig. 1. According to Eq. (9), a locus where k˙ = 0 is a curve of the form w1 (k) =

1 λk + λL (ν ∗ k1+α − k) 1−θ 1 − ρd

(25)

(a green line labelled dk/dt = 0 on Fig. 1). Analogously, it follows from Eq. (10) that a locus where w˙ = 0 is a curve of the form w2 (k) =

q (ν ∗ k1+α − (λk + λL ) k) 1−θ

(26)

(a blue line labelled dw/dt = 0 on Fig. 1). These two loci intersect in two points: the bold black dot (0, 0) is the stable trivial fixed point (Sec. 2.2.1) while the bold red dot (k0 , w0 ) is the unstable non-trivial fixed point (a saddle point, Sec. 2.2.2). There are also two boundaries on the phase plane. First, the system cannot be located above the red line labelled i = d = 0 on Fig. 1. This line is a boundary at which both investment and dividend are equal to zero. It follows from Eqs. (3) and (8) that the equation of this boundary is w3 (k) =

ν ∗ 1+α k . 1−θ

(27)

Then, the system cannot exist below the horizontal axis as wages w cannot be negative. (We note in this respect that both of the loci (25) and (26) partially lie in this prohibited half-plane.) Numeric simulations with the dynamic system (9)–(10) show that, dependent on the initial conditions, its trajectories either converge to the stable trivial fixed point (a regime of economic collapse) or go to infinity (a regime of explosive growth). In the latter case, we call the growth ‘explosive’ as model state variables become infinite at some finite time. (The same behaviour is revealed for the other version of SDEM developed in the present paper, see Fig. 2 in Sec. 3.2.) It should be noted that other models with regimes of explosive growth have been known before in endogenous growth literature (cf. [5]). 3 The model SDEM with endogenous technological progress 3.1

Model equations. Model equilibria

In this section we modify the model described in Sec. 2.1 by returning to the production function linear in capital (this can be done simply by equating α to zero in the dynamic system (9)–(10)). So we imply again that y(k) = ν k,

(28)

but, unlike has been done in [2], we no longer assume ν to be constant. Instead, ν (t) will now become the third state variable of the dynamic system. It is assumed that the investment flow is divided in two channels now: a


5

Fig. 1 The phase plane for the model SDEM with increasing returns to scale in axes capital k — wages w (the details are explained in Sec. 2.3).

constant fraction (1 − σν ) is invested directly in the ‘extensive’ capital growth, while the remaining fraction σν is invested in endogenous technological progress (i.e. in endogenous growth of the technology parameter). A constraint 0 < σν < 1 is imposed. The full three-dimensional system takes the form k˙ = (1 − σν ) (1 − ρd ) (ν k − (1 − θ ) w) − (λk + λL ) k,

(29)

q (ν − λk − λL ) k − w], 1−θ

(30)

ν˙ = β σν (1 − ρd ) (ν k − (1 − θ ) w) .

(31)

w˙ = λw [

In Eq. (31) β is the (constant) efficiency of investment in technology parameter growth. Finding fixed points of the model (29)–(31) in three-dimensional phase space is straightforward. Indeed, assume that (k0 , w0 , ν0 ) is a fixed point. It follows then from Eq. (31) that

ν0 k0 − (1 − θ ) w0 = 0.

(32)

By substituting Eq. (32) in the r.h.s. of Eq. (29) and equating the latter to zero we immediately find that k0 = 0. Then it follows from Eq. (32) that w0 = 0. Note that ν0 can be arbitrary: the only constraint implied by the economic sense of the model (and not by its mathematical form) is that ν should be positive. So we have a continuum of fixed points (or, in other words, multiple equilibria) of the form (0, 0, ν0 ), 3.2

∀ν0 > 0.

(33)

Stability of model equilibria

We now perform the stability analysis for some fixed point (0, 0, ν0 ) from the continuum (33). The dynamics of linearized system in the neighbourhood of this point are described by an equation ⎞ ⎛ ⎞ ⎛ δ k˙ δk ⎝ δ w˙ ⎠ = B0 ⎝ δ w ⎠ (34) δ ν˙ δν where B is the 3 × 3 matrix of the form

⎞ b11 , b12 , 0 B0 = ⎝ b21 , b22 , 0 ⎠ b31 , b32 , 0 ⎛

(35)

6


of which a 2 × 2 block

b11 , b12 B∗ = b21 , b22 is of interest since the characteristic equation for matrix B0

(36)

det(B0 − λ I) = 0

(37)

−λ det(B∗ − λ I) = 0.

(38)

can be factorized as Explicitly, the elements of the matrix B∗ have the form b11 = (1 − σν ) (1 − ρd ) ν0 − (λk + λL ) ,

(39)

b12 = − (1 − σν ) (1 − ρd ) (1 − θ ) , q (ν0 − λk − λL ) , b21 = λw 1−θ b22 = −λw .

(40) (41) (42)

Analogously to Eq. (15), we have the characteristic equation for matrix B∗ of the form

λ 2 − TrB∗ · λ + det B∗ = 0.

(43)

Tr B∗ = (1 − σν ) (1 − ρd ) ν0 − (λk + λL + λw ) ,

(44)

det B∗ = λw [{1 − q (1 − σν ) (1 − ρd )} (λk + λL ) − (1 − q) (1 − σν ) (1 − ρd ) ν0 ] .

(45)

Explicitly, its coefficients are

Note that, dependent on values of ν0 and model parameters, both Tr B∗ and det B∗ can be either positive or negative. The first condition of stability (46) TrB∗ < 0 requires where

ν1 =

0 < ν0 < ν1

(47)

λk + λL + λw , (1 − σν ) (1 − ρd )

(48)

while the second condition of stability det B∗ > 0

(49)

0 < ν0 < ν2

(50)

requires where

ν2 =

1 − q (1 − σν ) (1 − ρd ) (λk + λL ) . (1 − q) (1 − σν ) (1 − ρd )

(51)

So to ensure stability one must require 0 < ν0 < ν¯

(52)

ν¯ = min(ν1 , ν2 ).

(53)

where Fixed points (0, 0, ν0 ) where ν0 is from the interval (52) are stable, although they are not asymptotically stable( [4], Secs. 33-34). On the contrary, fixed points (0, 0, ν0 ) where ν0 > ν¯ are unstable (like in Sec. 2.2.2 above, this fact again follows from a generalization of Lyapunov’s instability theorem ( [4], Sec. 22, Theorem 2) provided in [4], Sec. 26). Note that there is no obvious relation between ν1 and ν2 : dependent on the values of model parameters, ν1 can be either greater or less than ν2 .


3.3

7

Asymptotic behaviour. Numerical examples

Similarly to the version of SDEM with increasing returns to scale explored in Sec. 2, numerical simulations with the dynamic system (29)–(31) reveal two alternative asymptotic regimes: either a collapse of the economy or explosive growth. In case of economic collapse, the difference is that now we have a continuum of stable fixed points (0, 0, ν ), 0 < ν < ν¯, so k(t) → 0 and w(t) → 0 when t → +∞, but it is not obvious to what particular value ν (t) will converge — it can be found only by numerical simulations. Needless to say, a question about the exact asymptotic value of ν (t) in such a case is of purely theoretical interest only: if k = 0 and w = 0 then the economy has disappeared completely anyway, irrespectively of the value of ν . Some simulation results are provided for illustrative purposes on Fig. 2 for the following values of model parameters: λk =0.05 year−1 , λL =0.01 year−1 , λw =0.3 year−1 , θ =0.05, q = 0.7, β =0.1 good−1 year−1 , σν =0.2, ρd =0.3. The initial conditions are: k0 =0.1 good, w0 =0.015 good/year, ν0 =0.15, 0.17, 0.19, and 0.21 year−1 (four lines for each of these four initial values are shown on each of the Figs. 2a–2c).a According to Eqs. (48), (51), and (53), for the ascribed values of model parameters ν1 =0.643 year−1 , ν2 =0.217 year−1 , so ν2 < ν1 and ν¯ = ν2 =0.217 year−1 . As seen from Fig. 2, for lower initial values of ν (ν0 =0.15 year−1 , black line; ν0 =0.17 year−1 , blue line) the economy collapses: k(t) → 0, w(t) → 0, and ν (t) converges to certain asymptotic values that are less than ν¯ when t → +∞. For ν0 =0.19 year−1 (green line) we have explosive growth, although this is not seen on the figures as the state variables reach infinite values nearly in a millenium (in model year 959). Finally, for ν0 =0.21 year−1 (red line) the singularity point is still within the time span shown on the figures (the state variables reach infinite values in model year 399), so the explosive growth is seen on the graphs. The reader should not be confused by the fact that in the case of numerical simulations provided the threshold value of ν0 separating collapsing trajectories from exploding ones is obviously less than the value ν¯ =0.217 year−1 found above. Indeed, this threshold value depends on k0 and w0 . The value ν¯ is the threshold on the axis k = 0, w = 0 only, separating stable and unstable fixed points, and the two-dimensional surface separating the two basins of attraction in full three-dimensional phase space need not be a plane — instead, it is a curved surface. 4 Conclusions Both modifications of SDEM developed and studied in the present paper demonstrate two very different dynamic regimes: either a super-pessimistic scenario of the collapse of the economy, or a super-optimistic scenario of explosive growth where model state variables (ultimately describing the wealth of the mankind) reach infinite values at finite time. While there is a firm tradition of making pessimistic projections of the first kind, the second scenario is definitely implausible. This means that more stabilizing negative feedbacks should be introduced in the developed model versions to make them more realistic. Among the candidates for external environmental constraints on otherwise unlimited growth extensively discussed in the existing literature are finite stocks of exhaustible natural resources, finite pollution sinks and the climate change problem [6, 7]. Another interesting methodological problem is how to introduce effective controls on policy variables ρd and σν (for simplicity assumed to be constant, i.e. inflexible, in the present paper) already within existing modelling framework to activate internal mechanisms preventing the pessimistic decay scenario. Both lines of research sketched in this concluding section are left for future studies.

a We

use material units (good, good/year etc.) instead of monetary units like $ in our numerical examples. The reader should not be confused by the fact that in the present subsection we use notations k0 , w0 , ν0 for initial conditions, while in Secs. 3.1–3.2 the same notations have been used for the coordinates of a fixed point.

8


Fig. 2 The dynamics of state variables — a) capital k(t), b) wages w(t), c) technology parameter ν (t) — in the model SDEM with endogenous technological progress. On each panel four lines correspond to four different initial conditions for ν : ν0 =0.15 year−1 (black line); ν0 =0.17 year−1 (blue line); ν0 =0.19 year−1 (green line); and ν0 =0.21 year−1 (red line). For the two lower initial values the trajectories collapse, while for the two higher initial values they explode. See the values of other model parameters used in numerical simulations and further discussion in Sec. 3.3.

Acknowledgements The author is indebted to Klaus Hasselmann for helpful comments. This study was supported by the Russian Foundation for Basic Research (Project No. 12-06-00381-a). References [1] Barth, V. (2003), Integrated assessment of climate change using structural dynamic models, Ph.D. Thesis, Max-PlanckInstitut für Meteorologie, Hamburg, 2003. [http://www.mpimet.mpg.de/fileadmin/publikationen/Ex91.pdf] [2] Kovalevsky, D.V. (2014), Balanced growth in the Structural Dynamic Economic Model SDEM-2, Discontinuity, Nonlinearity, and Complexity, 3(3), 237–253. [3] Petrov, Yu.P. (1987), Synthesis of Optimal Control Systems under Incompletely Known Perturbing Forces, Leningrad, LGU Publishing House, 1987 (in Russian). [4] Malkin, I.G. (2004), The Theory of Stability of Motion, 2nd edition, Moscow, Editorial URSS, 2004 (in Russian). [5] Romer, D. (2012), Advanced Macroeconomics, 4th edition, NY, Mc Graw Hill Irwin, 2012. [6] Meadows, D., Randers, J. and Meadows, D. (2004), Limits to Growth. The 30-Year Update, White River Junction, VT: Chelsea Green Publishing Co., 2004. [7] Stern, N. (2007), The Economics of Climate Change: The Stern Review, Cambridge, Cambridge University Press.



The Existence of Optimal Control for Semilinear Distributed Degenerate Systems M. Plekhanova† South Ural State University, Chelyabinsk, Russia Chelyabinsk State University, Chelyabinsk, Russia Abstract

Submission Info

Optimal control problems for a class of semilinear distributed systems unsolved with respect to the times derivative are studied. Two types of initial condition for the system state and various cost functionals are considered in the problems. Abstract results are illustrated by examples of the start control problems for the quasistationary system of phase field equations.

Communicated by Valentin Afraimovich Received 20 January 2015 Accepted 19 March 2015 Available online 1 April 2016 Keywords Optimal control Distributed system Degenerate equation Semilinear distributed systems

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

1 Introduction As part of the work, in particular, consider the distributed control problem described by the initial-boundary value problem for a quasi-stationary phase-field equations. It models the phase transitions of the first kind in the framework of the mesoscopic theory [1]. Physical properties of the medium in the different phases (density, thermal conductivity, heat capacity, etc.) will vary. Therefore, the problem is characterized by physical nonlinearity, making it very difficult to solve it. Nevertheless, the studying is of great importance in many areas of practical application: in metallurgy - in the manufacture of alloys, in construction - in the study of changes in the aggregate state of the moisture contained in the fences when the outdoor temperature fluctuations, and other areas. If the unknown functions w, v are a linear combination of the phase function and the specific temperature, the control problem for the phase-field equations can be considered as w(t0 , x) = w0 (x),

θ θ

v(t0 , x) = v0 (x),

x ∈ Ω,

(1)

∂w (t, x) + (1 − θ )w(t, x) = 0, ∂n

∂v (t, x) + (1 − θ )v(t, x) = 0, (t, x) ∈ ∂ Ω × [t0 , T ], ∂n

† Corresponding

author. Email address: [email protected]


(2)

10

M. Plekhanova / Discontinuity, Nonlinearity, and Complexity 5(1) (2016) 9–18

wt (t, x) = Δw(t, x) − Δv(t, x) + f (t, x, w) + u1 (t, x), (t, x) ∈ Ω × [t0 , T ],

(3)

Δv(t, x) + β v(t, x) + w(t, x) + g(t, x, w) + u2 (t, x) = 0, (t, x) ∈ Ω × [t0 , T ], u1 2L2 (t0 ,T ;L2 (Ω)) + u2 2L2 (t0 ,T ;L2 (Ω)) ≤ R2 , J(x, u) =

1 1 w − w ˜ 2H 1 (t0 ,T ;L2 (Ω)) + v − v ˜ 2H 1 (t0 ,T ;L2 (Ω)) 2 2 C1 C1 + u1 − u˜1 2L2 (t0 ,T ;L2 (Ω)) + u2 − u˜2 2L2 (t0 ,T ;L2 (Ω)) , 2 2

(4) (5) (6)

where control functions u1 , u2 describe the impact of external thermal. Condition (5) takes into account the limited resources. And the main goal is to approach the system state to the desired state w, ˜ v˜ at the lowest cost. This is described with help functional (7). In this paper there are a qualitative investigation of the problem (1)–(7) to determine the conditions of existence of solution. The idea of the study is the reduction of (1)–(7) to the abstract control problem for operator differential equations in Banach spaces. Let X , Y , U be Banach spaces, operators L : X → Y , B : U → Y be linear and continuous (briefly, L ∈ L (X ; Y ), B ∈ L (U ; Y )), ker L = {0}, an operator M be linear, closed and densely defined in X (M ∈ C l(X ; Y )), an operator N : X → Y be nonlinear. Consider an optimal control problem Lx(t) ˙ = Mx(t) + N(t, x(t)) + Bu(t),

(7)

x(t0 ) = x0 ,

(8)

u ∈ U˜∂ ,

(9)

J(x, u) → inf .

(10)

˜ 2H 1 (t0 ,T ;X ) + C2 u − u ˜ 2L2 (t0 ,T ;U ) , U˜∂ is a nonempty closed convex subset of a control space Here J(x, u) = 12 x − x U˜ , vector functions x˜ ∈ H 1 (t0 , T ; X ), u˜ ∈ L2 (t0 , T ; U ) are given, and C is a positive constant. Problem (7)–(10) is an abstract form of optimal control problems for various semilinear distributed systems, that are not resolved with respect to the time derivative. For example, optimal problems for the Navier–Stokes system, systems of Boussinesq equations, etc., may be reduced to the above problem [2, 3]. The classical results on the distributed control problems for differential equations, solved with respect to the time derivative, are presented in [4, 5]. The works [6, 7, 8] are devoted to the solvability of start control problems for semilinear distributed Sobolev type systems, i. e. the systems, described by equations with a degenerate operator at the highest derivative with respect to a specified variable, for example, by equation (7). Feature of the reported results is that the strong (L, p)-radiality of an operator M guarantees the existence of a degenerate strongly continuous resolving semigroup for the equation Lx(t) ˙ = Mx(t). Also assume that the operator N satisfies described below constraints, providing the unique nonlocal strong solvability of Cauchy problem (7), (8). The main results of the work is obtaining theorems on the existence of optimal control problems solutions under various conditions on operators and different initial conditions of problem (7)–(10). 2 Abstract control problem Formulate some useful general results of control theory briefly (see [5]).


11

Assume that Y˜ and V˜ are linear normed spaces, Y˜1 and U˜ are reflexive Banach spaces, and Y˜1 is continuously embedded in Y˜ . Consider an abstract control problem L˜ (y, u) + F˜ (y) = 0,

(11)

u ∈ U˜∂ ,

(12)

J(y, u) → inf .

(13)

Here U˜∂ is a closed convex subset of a control space U˜ , a cost functional J(y, u) is convex, lower semicontinuous, bounded from below on Y˜ × U˜∂ . Linear operator L˜ : Y˜1 × U˜ → V˜ and nonlinear operator F˜ : Y˜1 → V˜ are both continuous. Define a set W˜ of admissible pairs (y, u) ∈ Y˜1 × U˜ of problem (11)–(13), where relations (11), (12) and J(y, u) < ∞ hold. The conditions of nontriviality (i. e., W˜ = 0) / and coercivity of the functional J are assumend to be held; this means that the set {(y, u) ∈ W˜ : J(y, u) ≤ R} is bounded in Y˜1 × U˜ for any R > 0. A solution of problem (11)–(13) is a pair (y, ˆ u) ˆ ∈ W˜ , such that J(y, ˆ u) ˆ =

inf J(y, u).

(y,u)∈W˜

Let Y˜−1 be a linear normed space, such that the embedding Y˜ ⊂ Y˜−1 is continuous and the following conditions are held: (1) the embedding Y˜1 ⊂ Y˜−1 is compact; (2) there exists a dense set S on the space V˜ ∗ , such that the functional y → F˜ (y), v V˜ can be continuously extended from Y˜1 to Y˜−1 for any v ∈ S. Theorem 1. Let all the conditions formulated in this section be satisfied. Then problem (11)–(13) has a solution (y, ˆ u) ˆ ∈ Y˜1 × U˜∂ . 3 The existence of a solution for initial value problems Present the results, used further. Proofs of these results can be found in [9, 11]. Let X and Y be Banach spaces. Denote by L (X ; Y ) the Banach space of linear continuous operators, acting from X to Y . If Y = X , then the notation reduces to L (X ). Let C l(X ; Y ) be the set of linear closed operators with dense domains, acting from X to Y . The set of operators C l(X ; X ) is designated by C l(X ). Assume that L ∈ L (X ; Y ) and M ∈ C l(X ; Y ). Denote ρ L (M) = {μ ∈ C : (μ L − M)−1 ∈ L (Y ; X )}, RLμ (M) = (μ L − M)−1 L, LLμ (M) = L(μ L − M)−1, R+ = {a ∈ R : a > 0}, R+ = {0} ∪ R+ , N0 = {0} ∪ N. Let p ∈ N0 . An operator M is called strongly (L, p)-radial, if (i) ∃a ∈ R (a, +∞) ⊂ ρ L (M); (ii) ∃K > 0 ∀μ ∈ (a, +∞) ∀n ∈ N max{(RLμ (M))n(p+1) L (X ) , (LLμ (M))n(p+1) L (Y ) } ≤

K ; (μ − a)n(p+1)

◦

(iii) there exists a dense subspace Y in Y , such that M(μ L − M)−1 (LLμ (M)) p+1 f Y ≤ for any μ ∈ (a, +∞);

const( f ) (μ − a) p+2

◦

∀ f ∈Y

12


(iv) for any μ ∈ (a, +∞) (RLμ (M)) p+1 (μ L − M)−1 L (Y ;X ) ≤

K . (μ − a) p+2

Remark 1. The equivalence of strong (L, p)-radiality definitions, the definition, used in [9], and more simple one, presented here, is proved in [11]. Denote the kernel kerRL(μ ,p) (M) (ker LL(μ ,p) (M)) by X 0 (Y 0 ) and the closure of the subspace imRL(μ ,p) (M) (imLL(μ ,p) (M)) in the norm of the space X (Y ) by X 1 (Y 1 ). Denote the restriction of the operator M (L) to domMk = X k ∩ domM (X k ), k = 0, 1 by Mk (Lk ). Theorem 2. ([9]). Let an operator M be strongly (L, p)-radial. Then (i) X = X 0 ⊕ X 1 , Y = Y 0 ⊕ Y 1 ; (ii) Lk ∈ L (X k ; Y k ), Mk ∈ C l(X k ; Y k ), k = 0, 1; 1 1 (iii) there exist operators M0−1 ∈ L (Y 0 ; X 0 ) and L−1 1 ∈ L (Y ; X ); −1 0 (iv) the operator G = M0 L0 ∈ L (X ) is nilpotent of degree at most p ; (v) there exists a degenerate strongly continuous semigroup {X t ∈ L (X ) : t ∈ R+ } of the equation Lx˙ = Mx; 1 t (vi) an operator L−1 1 M1 ∈ C l(X ) is an infinitesimal generator of the C0 -continuous semigroup {X1 = t 1 X |X 1 ∈ L (X ) : t ∈ R+ }. Remark 2. The projector along X 0 onto X 1 (along Y 0 onto Y 1 ) has the form P = s- lim (μ RLμ (M)) p+1 (Q =

μ →+∞

s- lim (μ LLμ (M)) p+1 ). μ →+∞

Consider the Cauchy problem for a semilinear Sobolev type equation Lx(t) ˙ = Mx(t) + N(t, x(t)),

t ∈ [t0 , T ],

x(t0 ) = x0 ,

(14) (15)

where N : [t0 , T ] × X → Y is a nonlinear operator. A strong solution of problem (14), (15) on [t0 , T ] is a function x ∈ H 1 (t0 , T ; X ), satisfying condition (15), and almost everywhere on [t0 , T ] equality (14) is valid. Theorem 3. ([10]). Let X be a reflexive Banach space, an operator M be strongly (L, p)-radial, and an operator N : [t0 , T ] × X → Y be Lipschitz continuous in both variables, imN ⊂ Y 1 . Then for any x0 ∈ domM ∩ X 1 , problem (14), (15) has a unique strong solution on [t0 , T ]. Theorem 4. ([10]). Let X be a reflexive Banach space, an operator M be strongly (L, 0)-radial, an operator QN : [t0 , T ] × X → Y be Lipschitz continuous in both variables, for any (t, x) ∈ [t0 , T ] × X the equality N(t, x) = N(t, Px) holds, x0 ∈ domM, (I − P)x0 = −M0−1 (I − Q)N(t0, Px0 ).

(16)

Then problem (14), (15) has a unique strong solution on [t0 , T ]. Remark 3. While a Banach space X is not reflexive, conditions for a classical nonlocal solution x ∈ C1 ([t0 , T ]; X )∩ C([t0 , T ]; D) existence may be obtained by strengthening the conditions on the operator N, hence conditions for a strong solution existence of problems (14), (15) or (14), (17) may be obtained also. (The Banach space D = domM is equipped with the graph norm xD = xX + MxY .) Here the condition of Lipschitz continuity in both variables of the operator QN, used in Theorems 3, 4, needs to be replaced by one of the following two conditions: (A1) A mapping QN : [t0 , T ] × X → Y is continuously differentiable.


13

(A2) An operator L−1 1 QN : [t0 , T ] × D → D is uniformly Lipschitz in D and the mapping QN(·, x) : [t0 , T ] → D is continuous for any x ∈ D . Often there is a system with the initial Showalter condition Px(t0 ) = x0

(17)

in physical applications. Theorems 3 and 4 may be reformulated for this condition as the following theorems, respectively. Theorem 5. ([10]). Let X be a reflexive Banach space, an operator M be strongly (L, p)-radial, an operator N : [t0 , T ] × X → Y be Lipschitz continuous in both variables, imN ⊂ Y 1 . Then problem (14), (17) has a unique strong solution on [t0 , T ] for any x0 ∈ domM1 . Theorem 6. ([10]). Let X be a reflexive Banach space, an operator M be strongly (L, 0)-radial, let an operator QN : [t0 , T ] × X → Y be Lipschitz continuous in both variables, the equality N(t, x) = N(t, Px) holds for any (t, x) ∈ [t0 , T ] × X , x0 ∈ domM1 . Then problem (14), (17) has a unique strong solution on [t0 , T ]. 4 Distributed control problem with a compromise functional Let X , Y , U be Hilbert spaces. As in the previous sections, L ∈ L (X ; Y ), B ∈ L (U ; Y ), M ∈ C l(X ; Y ), N : [t0 , T ] × X → Y . Consider an optimal control problem Lx(t) ˙ = Mx(t) + N(t, x(t)) + Bu(t),

(18)

x(t0 ) = x0 ,

(19)

u ∈ U˜∂ ,

(20)

C 1 ˜ 2H 1 (X ) + u − u ˜ 2L2 (U ) → inf, (21) J(x, u) = x − x 2 2 where x˜ ∈ H 1 (X ), u˜ ∈ L2 (U ) are given functions, C is a positive constant, and a set of admissible controls U˜∂ is a nonempty closed convex subset of L2 (U ). Here and below, use the notion of L2 (t0 , T ; V ) ≡ L2 (V ) and H 1 (t0 , T ; V ) ≡ H 1 (V ). Use the notion of strong solution of Cauchy problem (18), (19), studying optimal control problem (18)– (21). Considering the form of equation (18), find these strong solutions in the Hilbert space Z = {z ∈ H 1 (X ) : L˙z − Mz ∈ L2 (Y )} with norm zZ = zH 1 (X ) + L˙z − MzL2 (Y ) . The completeness of this space is proved, for example, in [12]. Introduce an operator γ0 : H 1 (X ) → X , γ0 x = x(t0 ), that is continuous by the Sobolev theorem on the spaces embedding. The admissible pairs set W˜ for problem (18)–(21) is a set of pairs (x, u), such that u ∈ U˜∂ and x ∈ H 1 (X ). This set of functions is a strong solution of problem (18), (19). Problem (18)–(21) consists in (x, ˆ u) ˆ ∈ W˜ finding, such that J(x, ˆ u) ˆ = inf J(x, u). (x,u)∈W˜

Lemma 7 ([13). ]. Let X be a Banach space, x, yn ∈ C([t0 , T ]; X ), n ∈ N, lim yn − xC([t0 ,T ];X ) = 0. Then {yn (t) : t ∈ [t0 , T ], n ∈ N} is a precompact set on X .

n→∞

Theorem 8. Let M be a strongly (L, p)-radial operator, N : [t0 , T ] × X → Y be Lipschitz continuous in both variables, im N ⊂ Y 1 , im B ⊂ Y 1 , U˜∂ is a nonempty closed convex subset of L2 (U ), x0 ∈ dom M ∩ X 1 . Then there exists a solution (x, ˆ u) ˆ ∈ Z × U˜∂ of problem (18)–(21).

14


Proof. Due to the section conditions, a Hilbert space X is a reflexive Banach space. For fixed u ∈ L2 (U ), ˜ x(t)) = N(t, x(t))+Bu(t). It is obvious, introduce the operator N˜ : [t0 , T ]×X → Y , defined by the equation N(t, ˜ that the operator N : [t0 , T ] × X → Y is also Lipschitz continuous in both variables. Moreover, imN˜ ⊂ Y 1 . Thus, Theorem 3 conditions imply the existence of a strong solution for Cauchy problem (18), (19) with any 1 ˜ pair (u, x0 ) ∈ U∂ × dom M ∩ X . So, the admissible pairs set W˜ is nonempty. Further, apply Theorem 1. Let Y˜ = H 1 (X ), Y˜1 = Z , U˜ = L2 (U ), V˜ = L2 (Y ) × X , F˜ (x(·)) = (−N(·, x(·)), x0 ) and L˜ (x, u) = (Lx˙ − Mx − Bu, γ0 x). The continuity of a linear operator L˜ : Y˜1 × U˜ → V˜ follows from the inequalities (Lx˙ − Mx − Bu, γ0 x)2L2 (Y )×X ≤ 2Lx˙ − Mx2L2 (Y ) + 2Bu2L2 (Y ) + γ0 x2X ≤ C1 (Lx˙ − Mx2L2 (Y ) + x2H 1 (X ) + u2L2 (U ) ) = C1 (x, u)2Z ×U . Here we used the continuity of the operators γ0 and B. Since an operator N is Lipschitz continuous in both variables, for free choice of t ∈ [t0 , T ], x ∈ X , find N(t, x)Y ≤ N(t, x) − N(t0 , 0)Y + N(t0 , 0)Y ≤ l(|T − t0 | + xX ) + N(t0 , 0)Y ≤ K1 (1 + xX ). Prove the functional J coercivity. By the nonequality given above, there is x2Z + u2L2 (U ) = x2H 1 (X ) + Lx˙ − Mx2L2 (Y ) + u2L2 (U ) = x2H 1 (X ) + N(·, x(·)) + Bu2L2 (Y ) + u2L2 (U ) ≤ x2H 1 (X ) + 2N(·, x(·))2L2 (Y ) + 2Bu2L2 (Y ) + u2L2 (U ) ≤ x2H 1 (X ) + 2K12 T + 2K12 x2H 1 (X ) +C1 u2L2 (U ) ≤ C2 J(x, u) +C3 . From xn − x0 Z → 0 it follows that xn − x0 C([0,T ];X ) ; therefore, ˆ T N(t, xn (t)) − N(t, x0 (t))2Y dt → 0 N(·, xn (·)) − N(·, x0 (·))2L2 (Y ) = t0

for n → ∞. By the Lebesgue theorem, the maximum of the norm max N(t, y)2Y exists, due to the precom(t,y)∈K

pactness of the set K = [t0 , T ] × {xn (t) : t ∈ [t0 , T ], n ∈ N0 } (see Lemma 1) and the continuity of the operator N. Thus, the continuity of the operator F˜ is shown. Choose Y˜−1 = L2 (X ) and verify other conditions of Theorem 1. The compactness condition (1) follows from the space H 1 (X ) embedding compactness. Hence, the compactness of the space Z embedding in the space L2 (X ) follows from the Rellish-Kondrashov theorem on Sobolev spaces compact embedding. Consider a dense subspace C([t0 , T ]; Y ) as S ⊂ L2 (Y ) to verify condition (2) from Section 2. Then for v ∈ C([t0 , T ]; Y ), under the Lipschitz continuity condition in the variable x on the operator N, we have N(t, xn (t))−N(t, x(t)),v(t) L2 (Y ) ≤ lvL2 (Y ) xn − xL2 (X ) . This result implies the continuous extendability of the functional F˜ (·), v from Z to L2 (X ). 2 The set of functions u ∈ H 1 (U˜ ), such that (I − P)x0 + M0−1 (I − Q)N(t0 , Px0 ) = −M0−1(I − Q)Bu(t0 ) for x0 ∈ domM, we denote by H∂ (x0 ).


15

Theorem 9. Let M be a strongly (L, 0)-radial operator, N : [t0 , T ] × X → Y be Lipschitz continuous in both variables, the equality N(t, x) = N(t, Px) holds for any (t, x) ∈ [t0 , T ] × X , U˜∂ is a nonempty closed convex / Then problem (18)–(21) has a solution (x, ˆ u) ˆ ∈ Z × U˜∂ . subset of L2 (U˜ ), x0 ∈ domM, U˜∂ ∩ H∂ (x0 ) = 0. ˜ x(t)) = N(t, x(t)) + Bu(t) Proof. Consider the operator N˜ : [t0 , T ] × X → Y , defined by the equation N(t, ˜ ˜ for a fixed u ∈ L2 (U ), as in Theorem 7 proof. The operator N is Lipschitz continuous in both variables t and x, hence, QN˜ also possesses this property. In addition, the condition U˜∂ ∩ H∂ (x0 ) = 0/ ensures condition (16) performing. Thus, by conditions of Theorem 8, the nonemptiness of the admissible pairs set W˜ follows. All other arguments are made in Theorem 7 proof. 2 Obtain the following results for problem (18), (20), (21) with initial Showalter condition (17), based on Theorems 4, 5. Theorem 10. Let M be a strongly (L, p)-radial operator, N : [t0 , T ] × X → Y be Lipschitz continuous in both variables, im N ⊂ Y 1 , im B ⊂ Y 1 , U˜∂ is a nonempty closed convex subset of L2 (U ), x0 ∈ dom M1 . Then, problem (17), (18), (20), (21) has a solution (x, ˆ u) ˆ ∈ Z × U˜∂ . Theorem 11. Let M be a strongly (L, 0)-radial operator, N : [t0 , T ] × X → Y be Lipschitz continuous in both variables, the equality N(t, x) = N(t, Px) holds for any (t, x) ∈ [t0 , T ] × X , U˜∂ is a nonempty closed convex / Then problem (17), (18), (20), (21) has a solution (x, ˆ u) ˆ ∈ subset of L2 (U ), x0 ∈ domM1 , U˜∂ ∩ H 1 (U ) = 0. ˜ Z × U∂ . Note that condition (17) specifies the initial condition not for the whole solution, but only for the projection on the image of the solution semigroup. Therefore, the compatibility condition necessity from Theorem 8, given by means of the set H∂ (x0 ), disappears, and the Showalter problem solution exists, if there exists a sufficiently smooth admissible control (from H 1 (U )). 5 Example Consider the problem w(t0 , x) = w0 (x),

θ θ

v(t0 , x) = v0 (x),

x ∈ Ω,

(22)

∂w (t, x) + (1 − θ )w(t, x) = 0, ∂n

∂v (t, x) + (1 − θ )v(t, x) = 0, (t, x) ∈ ∂ Ω × [t0 , T ] ∂n

(23)

for the system of equations wt (t, x) = Δw(t, x) − Δv(t, x) + f (t, x, w), (t, x) ∈ Ω × [t0 , T ],

(24)

Δv(t, x) + β v(t, x) + w(t, x) + g(t, x, w) = 0, (t, x) ∈ Ω × [t0 , T ]

(25)

in a smooth boundary domain Ω ⊂ Rs , where λ , β , θ ∈ R. Systems of this form occur, for example, in modeling of the first kind phase transitions under the zero relaxation time assumption [1]. Let Hθ2 (Ω) = {z ∈ H 2 (Ω) : θ ∂∂ nz (t, x) + (1 − θ )z(t, x) = 0},

16


X = Y = (L2 (Ω))2 , domM = (Hθ2 (Ω))2 , I O Δ −Δ ∈ C l(X ). L= ∈ L (X ), M = OO I βI +Δ

(26) (27)

Denote Aw = Δw, domA = Hθ2 (Ω) ⊂ L2 (Ω). By {ϕk : k ∈ N} denote the operator A orthonormal eigenfunctions, in the sense of scalar product ·, · on L2 (Ω), numbered in nonincreasing order of the eigenvalues {λk : k ∈ N}, counting multiplicity. / σ (A). Then M is a Theorem 12. ([14]). Let spaces and operators be defined by formulas (26), (27), −β ∈ strongly (L, 0)-radial operator, I A(β + A)−1 I O , Q= , P= O O −(β + A)−1 O X 0 = {0} × L2 (Ω), X 1 = {(w, v) ∈ (L2 (Ω))2 : v = −(β + A)−1 w}, Y 0 = {(w, v) ∈ (L2 (Ω))2 : w = −A(β + A)−1 v}, Y 1 = L2 (Ω) × {0}. Denote by C1,0,1 ([t0 , T ] × Ω × R) the set of functions having all continuous 1-th order partial derivatives with respect to t ∈ [t0 , T ], all continuous 0-th order partial derivatives with respect to x ∈ Ω, all continuous 1-th order partial derivatives with respect to w ∈ R. Theorem 13. Let f , g ∈ C1,0,1 ([t0 , T ]× Ω × R), functions ft , fw , gt , gw are limited on set [t0 , T ]× Ω × R. There are a pair of functions (t f , w f ), (tg , wg ) ∈ [t0 , T ] × L2 (Ω) that f (t f , ·, w f (·)), g(tg , ·, wg (·)) ∈ L2 (Ω). Then problem (22)–(25) has a unique solution (w, v) ∈ H 1 (t0 , T ; (L2 (Ω))2 ) for any w0 , v0 ∈ (Hθ2 (Ω))2 that Δv0 (x) + β v0 (x) + w0 (x) + g(t0 , x, w0 (x)) = 0. Proof. Reduce problem (22)–(25) to problem (14), (15). It is necessary to show the existence and Lipschitz continuous in both variables of the nonlinear operator N : [t0 , T ] × (L2 (Ω))2 → (L2 (Ω))2 , defined by the formula f (t, x, w(x)) N(t, w, v)(x) = . g(t, x, w(x)) The following equality will be true to the formula of Lagrange f (t1 , x, w1 (x)) − f (t2 , x, w2 (x)) = ft (t1 + ξ (t2 − t1 ), x, w1 (x) + ξ (w2 (x) − w1 (x)))(t1 − t2 ) + fw (t1 + ξ (t2 − t1 ), x, w1 (x) + ξ (w2 (x) − w1 (x))) · (w1 (x) − w2 (x)), where ξ ∈ [0, 1]. Therefore | f (t1 , x, w1 (x)) − f (t2 , x, w2 (x))| ≤

sup (t,x,w)∈[t0 ,T ]×Ω×R

| ft (t, x, w)||t1 − t2 | + +|w1 (x) − w2 (x)|

sup (t,x,w)∈[t0 ,T ]×Ω×R

≤ C1 (|t1 − t2 | + |w1 (x) − w2 (x)|), | f (t1 , x, w1 (x)) − f (t2 , x, w2 (x)|2 ≤ C12 (|t1 − t2 |2 + 2|t1 − t2 ||w1 (x) − w2 (x)| + |w1 (x) − w2 (x)|2 ), f (t1 , ·, w1 (·)) − f (t2 , ·, w2 (·)2L2 (Ω) ≤ C12 (mes(Ω)|t1 − t2 |2 + 2|t1 − t2 |w1 − w2 L1 (Ω) + w1 − w2 2L2 (Ω) ) ≤ C2 (|t1 − t2 |2 + 2|t1 − t2 |w1 − w2 L2 (Ω) + w1 − w2 2L2 (Ω) ),

| fw (t, x, w)|


17

f (t1 , ·, w1 (·)) − f (t2 , ·, w2 (·)L2 (Ω) ≤ C3 (|t1 − t2 | + w1 − w2 L2 (Ω) ). Similarly we can prove that g is Lipschitz continuous. For any pair (t, w) ∈ [t0 , T ] × L2 (Ω), we have f (t, ·, w(·))L2 (Ω) ≤ f (t, ·, w(·)) − f (t f , ·, w f (·)L2 (Ω) + f (t f , ·, w f (·))L2 (Ω) ≤ C3 (|t − t f | + w − w f L2 (Ω) ) + f (t f , ·, w f (·))L2 (Ω) . Hence f (t, ·, w(·)) ∈ L2 (Ω), (t, w) ∈ [t0 , T ] × L2 (Ω). Similarly, using a pair of functions (tg , wg ) ∈ [t0 , T ] × L2 (Ω), we can prove that g(t, ·, w(·)) ∈ L2 (Ω) for any (t, w) ∈ [t0 , T ] × L2 (Ω). This means that operator N is defined and it acts from [t0 , T ] × L2 (Ω) to (L2 (Ω))2 . The condition N(t, u, v) = N(t, P(u, v)) follows from the forms of the operator N and the projector P , according to Theorem 11. It remains to refer to Theorem 4. 2 Consider optimal control problem wt (t, x) = Δw(t, x) − Δv(t, x) + f (t, x, w) + u1 (t, x), (t, x) ∈ Ω × [t0 , T ],

(28)

Δv(t, x) + β v(t, x) + w(t, x) + g(t, x, w) + u2 (t, x) = 0, (t, x) ∈ Ω × [t0 , T ], u1 2L2 (t0 ,T ;L2 (Ω)) + u2 2L2 (t0 ,T ;L2 (Ω)) ≤ R2 , J(x, u) =

1 1 w − w ˜ 2H 1 (t0 ,T ;L2 (Ω)) + v − v ˜ 2H 1 (t0 ,T ;L2 (Ω)) 2 2 C1 C1 + u1 − u˜1 2L2 (t0 ,T ;L2 (Ω)) + u2 − u˜2 2L2 (t0 ,T ;L2 (Ω)) , 2 2

(29) (30)

(31)

with initial and boundary conditions (22), (23), where functions w, ˜ v˜ ∈ H 1 (t0 , T ; L2 (Ω)), u˜1 , u˜2 ∈ L2 (t0 , T ; L2 (Ω)) ˜ ˜ are given, C1 > 0, U∂ is a closed convex subset of U = L2 (t0 , T ; (L2 (Ω))2 ). Considering the above arguments, reduce problem (22), (23), (28)–(31) to abstract problem (18)–(21), using the spaces Z˜ = H 1 (t0 , T ; (L2 (Ω))2 ) ∩ L2 (t0 , T ; (H 2 (Ω))2 ), H∂ (w0 , v0 ) = {(u1 , u2 ) ∈ H 1 (t0 , T ; (L2 (Ω))2 ) : Δv0 (x) + β v0 (x) + w0 (x) + g(t0 , x, w0 (x)) = −u2 (t0 , x)}. It’s not difficult to show space Z˜ coincides whith spase Z defined in section 4. Due to Theorem 8, we immediately obtain the following statement. / σ (A), f , g ∈ C1,0,1 ([t0 , T ] × Ω × R), ft , fw , gt , gw are limited on set [t0 , T ] × Ω × R. Theorem 14. Let −β ∈ / There are (t f , w f ), (tg , wg ) ∈ [t0 , T ]×L2 (Ω) that f (t f , ·, w f (·)), [t0 , T ]×Ω×R, w0 , v0 ∈ Hθ2 (Ω), U˜∂ ∩H∂ (w0 , v0 ) = 0. g(tg , ·, wg (·)) ∈ L2 (Ω). Then problem (22), (23), (28)–(31) has a unique solution (w, ˆ v, ˆ u) ˆ ∈ Z × L2 (t0 , T ; (L2 (Ω))2 ).

18


Acknowledgements The work is supported by the grant 14-01-31125 of Russian Foundation for Basic Research and supported by Laboratory of Quantum Topology of Chelyabinsk State University (Russian Federation government grant 14.Z50.31.0020 References [1] Plotnikov, P.I. and Starovoitov, V.N.(1993), Stefan problem with surface tension as the limit of the phase-field model, Differential Equations 29 (3), 461–471. [2] Demidenko, G.V. and Uspenskii, S.V. (2003), Partial Differential Equations and Systems not Solvable with Respect to the Highest-Order Derivative, Marcel Dekker, New York, Basel. [3] Sveshnikov, A.G., Alshin, A.B., Korpusov, M.O., and Pletner, Yu.D. (2007), Linear and Nonlinear Equations of Sobolev Type, Fizmatlit, Moscow [in Russian]. [4] Lions, J.-L. (2009), Optimal Control of Systems Governed By Partial Differential Equations, Springer-Verlag. [5] Fursikov, A.V. (1999), Optimal Control of Distributed Systems. Theory and Applications, Nauchnaya Kniga, Novosibirsk [in Russian]. 350 c. [6] Fedorov, V.E. and Plekhanova, M.V. (2004), Optimal control of linear equations of Sobolev type, Differential Equations 40 (11), 1548–1556. [7] Plekhanova, M.V. and Fedorov, V.E. (2004), An optimal control problem for a class of degenerate equations, J. Comput. Syst. Sci. Int. 43 (5), 698–702. [8] Plekhanova, M.V. and Islamova, A.F. (2011), Solvability of mixed-type optimal control problems for distributed systems, Russian Mathematics 55 (7), 30–39. [9] Fedorov, V.E.(2001), Degenerate strongly continuous semigroups of operators, St. Petersbg. Math. J. 12 (3), 471–489. [10] Fedorov, V.E. and Davydov, P.N. (2010), Global Solvability of Some Semilinear Equations of Sobolev Type, Vestn. Chelyab. Gos. Univ. Seriya Matematika. Fisika. Informatika. 12 (20), 82–89. [11] Fedorov, V.E. (2009), Properties of pseudo-resolvents and conditions for the existence of degenerate semigroups of operators, Vestn. Chelyab. Gos. Univ. Seriya Matematika. Fisika. Informatika. 11 (20), 12–19. [12] Plekhanova, M.V. and Fedorov, V.E. (2007), An Optimality Criterion in a Control Problem for of Sobolev-type Linear Equations, J. Comput. Syst. Sci. Int. 46 (2), 248–254. [13] Fedorov, V.E. and Plekhanova, M.V. (2011), The problem of start control for a class of semilinear distributed systems of Sobolev type, Proceedings of the Steklov Institute of Mathematics 17 (1), 259–267. [14] Fedorov, V.E. and Urazaeva, A.V. (2004) Inverse problem for a class of singular linear operator differential equations, Proceedings of Voronezh Winter, Math. School of S.G.Krein. 161–173.



Spin-transfer Torque and Topological Changes of Magnetic Textures Alberto Verga† Aix-Marseille Université, IM2NP, Campus St J er?me, service 142, 13387 Marseille, France Submission Info Communicated by Xavier Leoncini Received 21 January 2015 Accepted 1 February 2015 Available online 1 April 2016 Keywords Magnetism Skyrmion Topological transition

Abstract The electric manipulation of magnetic textures in nanostructures, important for applications in spintronics, can be realized through the spin-transfer torque mechanism: a spin-polarized current can modify the magnetization of skyrmions and magnetic vortices, and eventually change the topology of the magnetization. The spin-transfer torque and the intrinsic space and time scales of the topological changes are essentially quantum mechanical. We model the interaction between itinerant and fixed spins with a simple tightbinding hamiltonian in a square lattice. The dynamics is described by the Schrödinger equation for the electrons and the Landau-Lifshitz equation for the evolution of the magnetic texture. We investigate the phenomenology of the topological change of a Belavin-Polyakov skyrmion under the action of a spin-polarized current and show that adding an exchange dissipation term, regularizes the transition towards a ferromagnetic state. ©2016 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction The equilibrium magnetization field in ferromagnetic nanodots and in helical metals, often possesses a nontrivial topology. Magnetic vortices in permalloy [8] and skyrmion lattices in transition metal compounds [5] were experimentally observed. These configurations are interesting because a change between states having different topologies (switching of the vortex core, motions and annihilation of skyrmions), can be used as basic states in non-volatile memories and spintronic devices [7]. The basic physical mechanism that can trigger these topological changes, without magnetic fields, is the spin-transfer torque [6]. It describes the interaction between itinerant spins s, produced by a spin-polarized current jjs , and the magnetic moments (fixed spins S ) of the magnetic material, when the underlying magnetization is non-uniform. Within the framework of micromagnetism, to account for the spin-transfer torque one has to extend the usual Landau-Lifshitz equation [4] with terms proportional to the current and the magnetization gradients jjs · ∇SS [10]. However, this approach neglects strong non-adiabatic effects, such as the generation of current inhomogeneities due the scattering of electrons on the magnetization gradients. In order to investigate these effects, we recently proposed a self-consistent model, where electrons obey to quantum dynamics [3]. Here we develop further this model, and investigate the influence of dissipation. In this paper we are interested in the skyrmion-ferromagnetic transition induced by a polarized current. The basic magnetic texture is taken to be a stabilized lattice version of the Belavin-Polyakov [2] skyrmion in a † Corresponding



20

Alberto D. Verga / Discontinuity, Nonlinearity, and Complexity 5(1) (2016) 19–24

Fig. 1 Skyrmions of topological charge Q = 1 and size λ defined by the stereographic projection on the complex plane w = w(z), for (left) w = z/λ , (center) w = (1 + i)z/λ , and (right) w = (1 − i)z/λ . Arrows give the magnetization in the plane, and color, from blue to red, gives its z-component: at the center SS = (0, 0, 1). A constant z-polarized current induces the precession of the skyrmion magnetization and tends to reduce it core size.

periodic system. After a presentation of the basic equations coupling the Schr?dinger equation for the itinerant spins (electrons) to the Landau-Lifshitz equation for the ferromagnetic spins (fixed classical magnetic moments), defined on a square lattice, we present a qualitative model to show the basic mechanism of the transition. The transition from the skyrmion state to the ferromagnetic state implies a change in topology, and thus a violation of the topological charge conservation. This is possible because of the large fluctuations of the electron spin field (it acts a stochastic perturbation to the magnetization field) and because of the discreteness of the system (the lattice constant introduces a microscopic length cutoff). We performed a series of numerical computations to study the phenomenology of the transition and to identify the mechanism of the topological change.

2 Model The motion of an electron in a lattice of step a and size L2 is given by the Heisenberg equation for the two components (for the spin up and down) annihilation operator ci at site i, ic˙i (t) = He (t, Si )ci (t) ,

(1)

where He is the time dependent hamiltonian, He = −ε

∑ eiφ

i, j (t)

i, j

c†i c j − Js ∑ S i · (c†i σ ci ) − B p · ∑ c†i σ ci , i

(2)

i

where the fixed Si and itinerant spins c†i σ ci (σ is the vector of Pauli matrices) are coupled by the exchange constant Js ; ε is the energy to jump from site i = (xi , yi )/a = x i /a to its neighbor j. The system is subject to a constant electric field E in the x-direction, responsible of the phase factor appearing in the kinetic energy term,

φi, j (t) = (i − j) · xˆ

eaEt h¯

(with −e the electron charge). The last term in (2) contains the current polarization effective magnetic fieldBBp in energy units. The magnetic texture follows the dynamics given by the Landau-Lifshitz equation,

∂ S i = S i × ( f i − α S i × f i + Js s i ) − β ∇2 f i , ∂t

(3)


21

2 2 1

0

Q(t)

1

Q(t)

Q(t)

1

0

-1 0.0

0

-1 0.5

1.0

t/t0

×10−

1.5

2.0

-1

0.0

0.5

1.0

t/t0

3

×10−

1.5

2.0

0

1

3

2

3

t/t0

×10−

4

5

6

3

Fig. 2 Topological charge Q as a function of time (in adimensional units) for different values of the exchange dissipation parameter: (left) β = 0.1, (center) β = 0.01, and (right) β = 0.001. The red line is for Q + . The initial skyrmion charge is Q = −1, and the current is polarized in the −z direction. The change from Q = −1 to Q = 0 corresponds to a transition from the skyrmion to the ferromagnetic sate. Decreasing the dissipation strength results in an increase of the time necessary to reach the transition: t = 1236, 1748, 5936, for β = 0.1, 0.01, 0.001, respectively. The electric field is E = 10−3 .

where the effective field,

δ HS , δ Si

(4)

J K D (∇SSi )2 + ∑ Szi2 − ∑ S i · (∇ × S i ). ∑ 2 i 2 i 2 i

(5)

fi = − is derived from the coarse-grained free energy, HS =

J is the Heisenberg exchange constant, K is the anisotropy (easy-plane if positive, and easy-axis if negative), and D is the Dzyaloshinski-Moriya spin-orbit coupling energy. The term in β represents a dissipation of the Si ) [1]. This equation is coupled magnetization related to the exchange energy (proportional to the gradients ∇S to the equation for the electrons through the torque in Js , where the electron spin is computed by the formula s i (t) = c†i (t)σ ci (t) where the braket is for the quantum mean value. We use periodic boundary conditions in order to have a well defined topology. Note that ∇ is a difference operator acting on the lattice sites. In practice, the difference operators are computed in Fourier space and transformed to the lattice space. In units such that ε = a = h¯ = e = 1, typical parameters are: Js = 1, J = 0.4, K = D = 0 (for the Belavin-Polyakov skyrmion), B p = ne = 0.1, L = 128. S | = 1, and the topoWithout the dissipation term, the system (1-5) conserves the magnetization modulus |S logical charge Q = Q(t), ˆ Q=

dxx q(xx ,t), 4π

ˆ Q+ =

dxx |q(xx ,t)|, q = S · ∂x S × ∂y S , 4π

(6)

where the integration is over the lattice; Q+ (t) is a rough measure of the number of vortices. 3 Skyrmion To study the skyrmion-ferromagnetic transition it is convenient to work in the continuous limit, and to transform the magnetization field (which has only two independent components) by the stereographic projection: Sx =

w + w¯ 1 w − w¯ 1 − |w|2 , S = , S = , y z 1 + |w|2 i 1 + |w|2 1 + |w|2

(7)

22


-1

0

1

Fig. 3 Phenomenology of the topological transition. Contours of S z (fixed spins), arrows of (s x , sy ) (itinerant spins), and color density of the topological b-field. The dissipation is (left) β = 0.1, (center) β = 0.01, and (right) β = 0.001. The transition towards the ferromagnetic state is correlated with the appearance of intense b-field structures possessing a topological charge opposite to one of the skyrmion texture.

with

Sx + iSy . (8) 1 + Sz When Sz = 1, w goes to zero, and in the opposite pole, Sz = −1, w goes to infinity. This projection maps the vector field over the unit sphere S = S (xx ,t) to the field over the complex plane w = w(z, z¯,t) (where z = x + iy). The Landau-Lifshitz equation, in absence of dissipation, becomes, w=

i∂t w = −J ∂ ∂¯ w +

2J w¯ ∂ w∂¯ w − 12 s+ + sz w + 12 s− w2 , 1 + |w|2

(9)

(only the exchange field is considered here) where we defined the complex derivative ∂ = ∂ /∂ x − i∂ /∂ y an its complex conjugate ∂¯ . The second line in (9) corresponds to the spin-transfer torque term, where s± = sx ± isy . Equilibrium solutions of (9) are arbitrary analytic functions w = w(z) and ss = 0. The simplest one is the simple zero, w = z/λ , the Belavin-Polyakov skyrmion of charge Q = 1, centered at the origin and of characteristic size λ ∈ R (spin up at the origin and spin down at infinity). In order to investigate how the spin torque perturbs the skyrmion state w = z/λ , we focus on two simple limiting cases: first, small deviations from the skyrmion state by a uniform polarized current ss= (0, 0, sz ); and second, a small circular region around the skyrmion core relevant to track the transition towards the ferromagnetic state |w| → ∞. In the first case, we linearize (9) around the skyrmion state, w = z/λ + f (z, z¯,t): i∂t f = −J ∂ ∂¯ f +

2J z¯

λ 2 + |z|2

sz ∂¯ f + z , λ

(10)

the spin torque appears as a source term in this approximation. An interesting particular solution is readily found: sz t f = f0 (z,t) = −i z . λ The pure imaginary factor has the effect of changing the orientation of the magnetization field around the center of the skyrmion, passing successively in time from left to right chirality (as shown in Fig. 1). A sequence observed in the numerical simulations, in addition to the fact that the effective size of the core reduces: λ → λ / 1 + (szt)2 (even if, for long times, the perturbation analysis ceases its validity).


23

We turn now to the second case, for which we assume that the collapse of the skyrmion core is radially symmetric w = w(r,t), with r the polar radius. We are interested in the asymptotic limit of very large |w|, and propose a self-similar form [9]: r 1 f( ) (11) w(r,t) = α (t∗ − t) (t∗ − t)β to describe the approach of w to infinity (when t → t∗ , t∗ is the collapse time). Equation (9) becomes, J 2J i∂t w(r,t) = − ∂r (r∂r w) + (∂r w)2 + 12 s− w2 r w

(12)

Inserting ansatz (11) into (12) one obtains, two conditions that determine the unknown exponents:

α = 1,

β = 1/2 .

(13)

It is worth noting that the exchange interaction, which is scale invariant, do not permit to select the α exponent; its value is determined by the coupling term with the spin polarized current. A crude estimation of the finite time singularity is t∗ ∼ λ /s0 a, where s0 ∼ ne B p is the typical itinerant spin strength per site. 4 Results We investigate now, the role of dissipation in the transition towards the ferromagnetic state, using the numerical computation of equations (1) and (3). The first effect of the dissipation is to break the invariance of the topological charge. We observe in particular that the exchange dissipation term favorises the transition from the skyrmion state to the ferromagnetic state. Indeed, we plot in Fig. 2 the evolution of the topological charge as a function of the dissipation strength. In addition to the increasing stability of the skyrmion state, it is worth noting that the characteristic transition time sharply increases with dissipation. This is, as expected, a regularization effect. In the limit of vanishing dissipation the transition produces, ultimately, by the change of a single spin. A quantitative explanation of the mechanism behind the transition is based on the behavior of the electron spins. As we can observe in Fig. 3, although the electrons dynamics is almost stochastic (due to multiple scattering and interference effects), their spins organize near the skyrmion core. This can be verified by measuring the topological b-field, defined in a similar way as the density q of topological charge, but substitutingssto S : b = n · ∂x n × ∂y n ,

n = s /|ss | ;

(14)

represented in figure 3 by the color density. We verify that the transition is associated with the nucleation of a well localized electron vortex having a topological charge density opposite to the one of the background skyrmion (the white spots that appear near the skyrmion core at times t = 1100, 1748, 5936 for the three values of the dissipation, respectively). We remark, that in spite of the dissipation, the characteristic length scale of these structures is comparable with a few lattice steps.

5 Conclusion We investigated the transition between a skyrmion state and a ferromagnetic state driven by a spin polarized electron current. We demonstrated that the torque exerted by the itinerant spins modifies the distribution of the magnetization around the skyrmion core, and tends initially to reduce its size. When the system is dominated by the exchange interaction, a self-similar collapse of the core in a finite time produces, changing the topology of the system. In addition, this topological change strongly depends on the dissipation mechanism. In absence of dissipation the collapse time explicitly depends on the lattice cutoff. At variance, in the case where dissipation is effective, the singularity tends to regularize, and the life time of the skyrmion state reduces with increasing

24


dissipation. However, the microscopic mechanism of topological change is in the dissipative case, similar to the nondissipative one (even if the time and length scales may differ). It is related to the appearance of a peculiar electronic structure possessing a net charge opposite to the one in the skyrmion. The synchronization of the fixed spins with this electron vortical structure, leads to the annihilation of the topological charge, which pass from its initial value Q = −1 (skyrmion state) to zero (homogeneous ferromagnetic state). Acknowledgments We thank R. G. Elias for useful discussions.

References [1] Baryakhtar, V.G., Ivanov, B.A., Sukstanskii, A.L., and Melikhov, E. Yu. (1997), Soliton relaxation in magnets, Physical Review B, 56(2). [2] Belavin, A.A. and Polyakov, A.M. (1975), Metastable states of two-dimensional isotropic ferromagnets, JETP Lett., 22, 245–247.. [3] El´ıas, R.G. and Verga, A.D. (2014), Topological changes of two-dimensional magnetic textures, Phys. Rev. B, 89(13), 134405. [4] Landau, L.D. and Lifshitz, E.M. (1935), On the theory of the dispersion of magnetic permeability in ferromagnetic bodies, Phys. Zeitsch. der Sow., 8, 153–169. reprint: Ukr. J. Phys. 2008, Vol. 53, Special Issue, p.14-22. [5] Muhlbauer, S., Binz, B., Jonietz, F., Pfleiderer, C., Rosch, A., Neubauer, A. Georgii, R., and Boni, P. (2009), Skyrmion Lattice in a Chiral Magnet, Science, 323(5916), 915–919. [6] Ralph, D.C. and Stiles, M.D. (2008), Spin transfer torques, Journal of Magnetism and Magnetic Materials, 320(7), 1190–1216. [7] Romming, N., Hanneken, C., Menzel, M., Bickel, J.E., Wolter, B., von Bergmann, K. Kubetzka, A., and Wiesendang, R. (2013), Writing and deleting single magnetic skyrmions, Science, 341(6146), 636–639, 08. [8] Van Waeyenberge, B., Puzic, A., Stoll, H., Chou, K.W., Tyliszczak, T., Hertel, R., Fahnle, M., Bruckl, H., Rott, K., Reiss, G., Neudecker, I., Weiss, D., Back, C.H., and Schutz, G. (2006), Magnetic vortex core reversal by excitation with short bursts of an alternating field, Nature, 444 (7118),461–464, 11. [9] Zakharov, V.E. (1972), Collapse of langmuir waves, Sov. Phys. JETP, 35 (5), 908–914. [10] Zhang, S. and Li, Z. (2004), Roles of nonequilibrium conduction electrons on the magnetization dynamics of ferromagnets, Phys. Rev. Lett., 93, 127204.



Equilibrium States Under Constraint in a Variational Problem on a Surface Panayotis Vyridis†, M.K. Christophe Ndjatchi, Fernando Garc´ıa Flores, Julio César Flores Urbina Department of Physics and Mathematics, National Polytechnic Institute (IPN), Campus Zacatecas(UPIIZ) P.C.098160, Zacatecas, Mexico Submission Info Communicated by Valentin Afraimovich Received 14 May 2015 Accepted 10 March 2015 Available online 1 April 2016

Abstract We study the equilibrium states for an energy functional with a parametric force field on a region of a surface under a constraint of geometrical character. We use an improved method, based in Skrypnik’s variational theories [10]. In local coordinates, equilibrium points satisfy an elliptic boundary value problem. This model can be described as the deformation of the elastic medium and membranes.

Keywords Variational principles Critical points Bifurcation points Elliptic boundary value problem Mean curvature


1 Introduction We consider the functional 1 I[u, λ ] = 2

ˆ

1 ai jkl (x) ∇i u ∇k u dS + 2 S j

l

ˆ

2

Γ

ˆ

|∂i ∂i u| ds − λ

Γ

q(u, x) ds ,

(1)

where λ is a real parameter, S is an open and connected region in a smooth surface M in R3 , with boundary ∂ S consisting of two non-intersecting sufficiently smooth components Γ and Γ1 , where ∇i is the i-th component of the tangent differentiation with respect to the surface M [5]: ∇i =

∂ ∂ − η i (x) η j (x) j , ∂ xi ∂x

i = 1, 2, 3 ,

x ∈ M,

(2)

η (x) , x ∈ R3 is a continuously differentiable vector field identified to the outer normal vector field for every x ∈ M and ∂i is the i-th component of the tangent directional differentiation along the curve ∂ S:

∂i = τ i (x)

d ∂ = τ i (x) τ j (x) j , ds ∂x

i = 1, 2, 3 ,

x ∈ ∂S,

† Corresponding



(3)

26

Panayotis Vyridis et al. / Discontinuity, Nonlinearity, and Complexity 5(1) (2016) 25–32

τ (x) for x ∈ R3 is a continuously differentiable vector field identified for each x ∈ ∂ S to the unitary tangent vector field of the curve ∂ S, and belonging to the tangent plane Tx M for each x ∈ ∂ S. The coefficients ai jkl ∈ L∞ (S) satisfy the symmetry properties ai jkl (x) = akli j (x), and they are positive definite, i.e. ai jkl (x) ξ i j ξ kl ≥ Λ ξ i j ξ i j ,

Λ > 0.

(4)

Finally, we assume that function q is three times differentiable with the following properties q(0, x) = 0 ,

qui (0, x) = 0 ,

x ∈ Γ,

i = 1, 2, 3.

Hence, u is the displacement field, the first two terms of (1), i.e. the functional ˆ ˆ 1 1 j l ai jkl (x) ∇i u ∇k u dS + |∂i ∂i u|2 ds F[u ] = 2 S 2 Γ

(5)

(6)

represents the sum of the elastic energy of the medium, stored by the deformation with the work, done by the outer forces due to the deformation of the shell Γ. Finally the third term of (7) i.e the functional ˆ G[u ] = q(u, x) ds (7) Γ

denotes the stored potential energy of the shell. The medium is fixed up to a part Γ1 of the boundary ∂ S. The aim of this work is the investigation of the critical points u for functional (1) under the existence of the constraint with the property of leaving the boundary Γ invariant on the surface M. The mapping y : Γ −→ M

y(x) = x +u(x) ,

(8)

for small values of u, leaves the boundary Γ invariant on the surface M if y(Γ) ⊂ M .

(9)

A generalized Skrypnik’s variational method [3] approaches the existence of the critical points of (1) as a bifurcation problem in the following sense: Every λ ∈ R, which corresponds to a nonzero critical point u of (1), is a bifurcation point for the Euler - Lagrange equation I [u] = 0 .

(10)

on an appropriate subset of the domain of (1). Assuming additional smoothness of boundary ∂ S, coefficients ai jkl and function q, the integral equation (10) and its linearization can be written in the equivalent form of an elliptic boundary value problem in local coordinates. 2 Existence of bifurcation points Define the space

H0 (S, R3 ) = u ∈ W21 (S, R3 ) , u|Γ ∈ W22 (Γ, R3 ) , u|Γ1 = 0 .

We denote by W21 (S, R3 ) and W22 (Γ, R3 ) the Sobolev spaces of functions defined on S and Γ with values in R3 , respectively. A vector field u ∈ H0 (S, R3 ) is a critical point for the functional (1) if I [u ]v = 0 .

(11)


or equivalently

ˆ S

j

l

ˆ

ai jkl (x) ∇i u ∇k v dS +

ˆ Γ

∂i ∂i u ∂ j ∂ j v ds − λ

Γ

qui (u, x) vi ds = 0 ,

27

(12)

for all v ∈ H0 (S, R3 ). Using methods, described in [1, 3], we obtain that the square root of (6) defines a norm in the space H0 (S, R3 ) equivalent to the standard one. Furthermore by (7) we derive G [u ]v =

ˆ Γ

qui (u, x) vi ds .

(13)

Hence (11) or (12) can be written as (u,v)H0 − λ (G [u ],v)H0 = 0 ,

(14)

for all v ∈ H0 (S, R3 ), where ( , )H0 denotes the inner product in H0 (S, R3 ). Thus the Euler - Lagrange equation (10) can be expressed in the form of an operator as u − λ G [u ] = (Id − λ G )u = 0 ,

(15)

where Id is the identity operator and the operator G defined by (13) in general is not linear. In a small neighborhood of 0 ∈ H0 (S, R3 ), using (8) we define the mapping Φ[u ] = dist x +u(x), M x∈Γ = dist y(x), M x∈Γ .

(16)

Obviously Φ[u ] ∈ W22 (∂ S) and the inclusion (9) is valid if and only if Φ[u ] = 0 .

(17)

Thus the constraint (9) equivalently expressed in the form of the equation (17) restricts the search of the nonzero critical points u of (1) corresponding to the bifurcation points λ in a appropriate subspace of H0 (S, R3 ). Actually a generalized Skrypnik’s theory, described in [4], guarantees the existence of bifurcation points for the constrained problem (15), (17). Theorem 1. Suppose that the functional G is weakly continuous, differentiable, and its differential is Lipschitz continuous with (18) G [u ] = Au + N(u) , where A is a linear self-adjoint and compact operator. For the nonlinear part N the following estimate holds: N(u) ≤ c u p ,

(19)

where c is a positive constant, p > 1 and u belongs to a small neighborhood of 0 ∈ H0 (S, R3 ). Let Φ be a continuous differentiable mapping, which satisfies the conditions Φ[0 ] = 0 ,

Ker Φ [0 ] = H1 = 0

(20)

and there exists a Hilbert space Y1 with W22 (∂ S) ⊂ Y1 such that the mapping Φ : H0 (S, R3 ) −→ Y1 is weakly continuous. Then the number λ = 0 is a bifurcation point for problem (15), (17) if and only if the equation (P Id P − λ PAP)u = 0 ,

u ∈ H0 (S, R3 ) ,

where P : H0 (S, R3 ) −→ H1 is the orthogonal projector, has a nonzero solution.

(21)

28


It is obvious that bifurcation points exist when PAP = 0. Notice that the equation (21) is the linearization at the point 0 of (11) or (12) in the subspace H1 , which implies that every λ ∈ R, corresponding to a non zero critical point u ∈ H1 for the problem (15), (17) if and only if the linear equation I [0, λ ](u,w) = (I [0, λ ]u, w) = 0

(22)

or equivalently ˆ S

j

ˆ

l

ai jkl (x) ∇i u ∇k w dS +

ˆ

Γ

∂i ∂i u ∂ j ∂ j w ds − λ

Γ

qui u j (0, x) ui w j ds = 0 ,

(23)

has a nonzero solution u for all w ∈ H1 . Lemma 2. The functional (7) satisfies the conditions of the theorem 1. Proof. First we note that the functional (7) is differentiable due to the smoothness of function q. The compactness of Sobolev embedding of W22 (Γ, R3 ) into C(Γ, R3 ) [7] implies that the functional (7) is weakly continuous and its derivative G [u ] satisfies the local Lipschitz continuous with G [u ] = Au + N(u) , where operator A : H0 (S, R3 ) −→ H0 (S, R3 ) is defined as ˆ (Au,v)H0 = qui u j (0, x) vi u j ds , Γ

and

ˆ (N(u),v)H0 =

Γ

[qui (u, x) − qui u j (0, x) u j ] vi ds

for all v ∈ H0 (S, R3 ). Obviously, operator A is linear and symmetric. The above compact embedding implies that operator A is compact and there exists a positive constant C > 0 such that N(u)H0 ≤ C u2H0 . Lemma 3. The functional (16) satisfies the conditions of the theorem 1. Furthermore H0 (S, R3 ) = H1 ⊕ H2 , where

H1 = v ∈ H0 (S, R3 ) : v ·nΓ = 0 , H2 = v ∈ H0 (S, R3 ) : (v ·n)vΓ = 0 .

Proof. First, we observe that for u,v ∈ H0 (S, R3 ) the variation of the mapping Φ is Φ[u +v ] − Φ[u ] = Φ [u ]v + B[u ](v,v) , where Φ [u ]v = and

ˆ B[u ](v,v) =

0

1

(1 − t)

∂ ρ (x +u(x)) vi (x) , ∂ xi

∂2 ρ (x +u(x) + tv(x)) vi (x) v j (x) dt . ∂ xi ∂ x j


29

Following the methods described in [8], by the boundness of the embedding of W22 (∂ S) into spaces C(∂ S) and C1 (∂ S) [7] we obtain the estimates:

Φ [u ]vW 2 (∂ S) ≤ C vW 2 + uC1 vC1 + uC2 1 vC + uW 2 vC 2

2

2

≤ C (1 + uW 2 ) v|W 2 2

2

and

B[u ](v,v)W 2 (∂ S) ≤ C vC vW 2 + vC2 1 + (uC + vC + uW 2 + vW 2 )vW 2 2 2 2 2 2 2 , ≤ C 1 + uW 2 + vW 2 vW 2 2

2

2

where C and C are various constants. This means that the mapping Φ is continiously differentiable. In the same manner, we can verify that Φ [u ] depends continiously on u. Now the conclusion comes from the Lyapunov - Schmidt reduction [8]. By (16) it is obvious that Φ[0 ] = 0 , Moreover, by the derivation of the distance function [7] we obtain Φ [0 ]v =v ·nΓ and thus we set

H1 = Ker Φ [0 ] ,

H2 = H1⊥ .

Now a version of the implicit function theorem [8] implies the existence a mapping h of a small neighborhood of 0 ∈ H1 onto a small neighborhood of 0 ∈ H2 such that h(0) = 0 ,

u =v + h(v) , v ∈ H1 ,

h (0) = 0 .

(24)

Finally, Φ is weakly continuous as a maping from W22 (∂ S, Tx M) into Y1 = C(∂ S), due to the compactness of the embedding of W22 (∂ S) into C(∂ S). Using (24) the linearization (23) becomes ˆ ˆ ˆ i l ai jkl (x)∇ j v ∇k w dS + ∂i ∂iv ∂ j ∂ j w ds − λ qui u j (0, x) vi w j ds = 0 , (25) Γ

S

Γ

where v,w ∈ H1 . Thus λ is a bifurcation point corresponding to a critical point u ∈ H0 (S, R3 ) for the problem (15), (17) or, according to (24), equivalently to a critical point v ∈ H1 for the problem (15) if and only if the equation (25) has a nonzero solution v for all w ∈ H1 . Under the additional assumptions of smoothness

∂ S ∈ C∞ ,

ai jkl ∈ C∞ (S) ,

q ∈ C∞ (R3 , ∂ S) ,

using the formula of integration by parts [8] over S ˆ ˆ ˆ ˆ i i u∇i v dS = u v ν ds − Hn u v dS − v ∇i u dS S

and over ∂ S [2]

∂S

S

ˆ

ˆ ∂S

u ∂i v ds = −

∂S

i

(26)

S

i

(K ν + Rη ) u v ds −

ˆ ∂S

v ∂i u ds ,

(27)

where ν (x) a differentiable vector field in R3 , which is the normal vector field of the curve ∂ S for every x ∈ ∂ S, vertical to τ (x), located in the tangent plane of M at x ∈ ∂ S H is the mean curvature of surface M, K is the

30


geodesic curvature and R is the normal curvature of curve ∂ S, located in the surface M [6], the integral equation (12) in local coordinates is reduced to the equivalent elliptic boundary value problem:

x∈S H η l ai jkl (x) ∇ j vi + ∇l ai jkl (x) ∇ j vi = 0, ai jkl (x)τ k ν l ∇ j vi + τ k ∂i ∂i ∂ j ∂ j vk − λ quk (v, x)τ k = 0 , x ∈ Γ ai jkl (x)ν k ν l ∇ j vi + ν k ∂i ∂i ∂ j ∂ j vk − λ quk (v, x)ν k = 0 , x ∈ Γ v = 0,

(28)

x ∈ Γ1 .

and the linearized integral equation (25)

H η l ai jkl (x) ∇ j vi + ∇l ai jkl (x) ∇ j vi = 0,

x∈S

ai jkl (x)τ k ν l ∇ j vi + τ k ∂i ∂i ∂ j ∂ j vk − λ quk ui (0, x)τ k vi = 0 , x ∈ Γ ai jkl (x)ν k ν l ∇ j vi + ν k ∂i ∂i ∂ j ∂ j vk − λ quk ui (0, x)ν k vi = 0 , x ∈ Γ v = 0,

(29)

x ∈ Γ1 .

Notice that the boundary conditions on Γ of the problems (28) and (29) have been obtained by the definition of the space H1 in the lemma 3 due to the constraint (17). 3 The case of the unit sphere We bring as an example the case when the circular region S, where Γ is a circumference of radius ρ is located on the unit sphere M in R3 . In this case the mean curvature of M is H = 1 and we will consider that the coefficients ai jkl are constants, defined by (30) ai jkl (x) = δi j δkl , where δi j stands for the Kronecker’s symbol. On the unit sphere the outer normal vector is defined by ni (x) = xi ,

x∈S

and thus the differential equation of the problem (28) or (29) is converted in the form xk ∇i vi + ∇k ∇i vi = 0 ,

x∈S

(31)

x ∈ R3

(32)

Now we introduce the polar coordinate system r = |x| ,

θi =

xi , r

and thus

∂ ∂ ∂ θ i δki − θ i θ k = θi i , , = ∂r ∂x ∂ xk r Observe that on the unit sphere we have r = 1, therefore xk xk = |x|2 = 1 ,

∇i =

∂ ∂ − θi , i ∂x ∂r

θ i = xi ,

x∈S

θ i ∇i = 0 .

(33)

(34)


31

and the relations (33) become

∂θi = δki − θ i θ k , ∂ xk

∂ ∂ = xi i , ∂r ∂x

∇i =

∂ ∂ − xi , ∂ xi ∂r

xi ∇i = 0 .

(35)

Using the notation

∂u ∂u ∂ 2u , uxi = i , uxi x j = i j , ∂r ∂x ∂x ∂x and the relations (35) the differential equation (31) is equivalent to ur =

xk vixi − vkr − xi virxk + xi xk virr = 0 ,

urx j =

∂ 2u ∂ r∂ x j

x ∈ S.

(36)

Multiplying (36) by xk and using (34), (35) we obtain an equivalent differential equation of first order vixi − xi vir = 0 ,

x ∈ S.

(37)

In order to interpret the boundary conditions of the problem (28) or (29), we choose a coordinate system, which is transformed from the initial one by an appropriate composition of a translation and rotation, picking up the axes x1 and x2 from he tangent plane of the unit sphere M at the point x ∈ S, while the axis x3 comes along the normal vector n(x) of M at x ∈ S. In this coordinate system the tangent unit vectors are written τ (x) =

1 (−x2 , x1 , 0) , ρ

ν (x) =

1 (−x1 , −x2 , 0) , ρ

x ∈ S.

(38)

and the tangential differential operator (3) along ∂ S can be expressed as

∂1 =

1 2 2 ∂ ∂

− x1 x2 2 , (x ) 2 1 ρ ∂x ∂x

∂2 =

1 1 2 ∂ ∂

− (x1 )2 2 . x x 2 1 ρ ∂x ∂x

A straightforward calculation gives

1 ∂i ∂i vk = 2 − x1 vkx1 − x2 vkx2 + (x2 )2 vkx1 x1 − 2x1 x2 vkx1 x2 + (x1 )2 vkx2 x2 . ρ

(39)

(40)

We introduce the rotation vector field [12] in this chosen coordinate system R = (0, 0, R3 ) , where R3 = x1 Another calculation gives |x × R|2 = −xi

∂ ∂ − x2 1 ∂ x2 ∂x

(41)

∂ ∂2 ∂2 ∂2 1 2 1 2 2 2 + (x ) − 2x x + (x ) . ∂ xi ∂ x2 ∂ x2 ∂ x1 ∂ x2 ∂ x1 ∂ x1

(42)

Comparing (40) with (42) we obtain

1 |x × R|2 vk . ρ2 Finally, using (30), (38) and (43) the boundary value problem (28) can be expressed in the form of:

∂i ∂i vk =

xk vixi − vkr − xi virxk + xi xk virr = 0, 1 k τ |x × R|4 vk − λ qvk (u, x)τ k ρ4

= 0,

x∈S x∈Γ

vixi − xi vir + ρ14 ν k |x × R|4 vk − λ quk (v, x)ν k = 0 , x ∈ Γ v = 0,

x ∈ Γ1 .

(43)

32


and analogously the boundary value problem (29) becomes xk vixi − vkr − xi virxk + xi xk virr = 0, 1 k τ |x × R|4 vk − λ qui uk (0, x)vi τ k ρ4

= 0,

x∈S x∈Γ

vixi − xi vir + ρ14 ν k |x × R|4 vk − λ qui uk (0, x)vi ν k = 0 , x ∈ Γ v = 0,

x ∈ Γ1 .

References [1] Vyridis, P. (2001), Bifurcation in a Quasilinear Variational Problem on a One - Dimensional Manifold, J. Math. Sci. (N.Y.) 106(3), 2919. [2] Vyridis, P. (2002), Variational Problem on Equilibrium of an Elastic Medium, Located in an Elastic Shell, J. Math. Sci. (N.Y.) 112(1), 3992. [3] Vyridis, P. (2011), Bifurcation in a Variational problem on a surface with a constraint, Int. J. Nonlinear Anal. Appl. 2 (1), 1–10. [4] Vyridis, P.(2014), Bifurcation in a Variational problem on a surface with a distance constraint, J. of Nonlinear Sci. Appl. 7, 160–167. [5] Giusti, E. (1984), Minimal Surfaces and Functions of Bounded Variation, Monographs in Mathematics, Vol. 80, Birkhäuser, Boston-Basel-Stuttgart. [6] Cartan, H. (1971), Differential Calculus - Differential forms, Herman Paris. [7] Gilbarg, D. and Trüdinger, N.S. (1977) Elliptic Partial Differential Equations of Second Order, Springer-Verlag. [8] Osmolovskii, V.G. (1997), Linear and nonlinear pertubations of operator div, Translations of Mathematical Monographs, Vol. 160. [9] Osmolovskii, V.G. (2000), The Variational Problem on Phase Transitions in Mechanics of Continuum Media , St. Petersburg University Publications. [10] Skrypnik, I.V. (1973), Nonlinear Partial Differential Equations of Higher Order, Kiev. [11] Skrypnik, I.V.(1976), Solvability and properties of solutions to nonlinear elliptic equations, Recent Problems in Mathematics, VINITI, 9, 131–242. [12] Dubrovin, B.A., Fomenko, A.T., and Novikov, S.P. (1990), Modern Geometry - Methods and Applications, Part I, Spinger - Verlag New York Inc.



Stability of Hopfield Neural Networks with Delay and Piecewise Constant Argument M.U. Akhmet†, M. Karacaören Department of Mathematics, Middle East Technical University, 06800, Ankara, Turkey Submission Info Communicated by Valentin Afraimovich Received 19 March 2015 Accepted 23 April 2015 Available online 1 April 2016

Abstract In this paper, by using the concept of differential equations with piecewise constant argument, the model of Hopfield neural networks with constant delay is developed. Sufficient conditions for the existence of an equilibrium as well as its global exponential stability by means of Lyapunov functionals and a linear matrix inequality (LMI) are obtained. An example is given to illustrate our results.

Keywords Hopfield neural networks -models Lyapunov functionals Piecewise constant argument Constant time delay Lyapunov functionals Linear matrix inequality


1 Introduction Hopfield neural network models, which constitute a class of recurrent neural networks were first proposed by J.J. Hopfield [1]. These neural network models have been widely studied by many authors due to the extensive applications on image and signal processing, biology, pattern recognition and optimization problems [2]- [7]. The original Hopfield neural network model can be described by the following differential equations: n

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

(1)

j=1

for i = 1, . . . , n, where ai ≥ 0, x j , bi j and f j denotes the state variable, interconnection strengths from neuron j to neuron i, and activation functions, respectively. In the literature, there are many papers where delays have been introduced to Hopfield neural networks [2]– [7]. Time delays are present due to finite switching speed of the amplifiers and communication time [8]– [11]. It essentially changes the characteristic properties of the neural network systems such that stability, convergence and divergence and encountered in the biological systems and computer sciences [11]– [14]. Up to now, various kinds of delays were introduced to the Hopfield neural network systems such that constant delays, single time delays, time varying delays and distributed delays. The following † Corresponding

author. Email address: [email protected] ISSN 2164 − 6376, eISSN 2164 − 6414/$-see front materials © 2016 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/DNC.2016.03.005

34

M.U. Akhmet, M. Karacaören / Discontinuity, Nonlinearity, and Complexity 5(1) (2016) 33–42

system with constant delays was introduced by Marcus and Westervelt [15] n

n

j=1

j=1

xi (t) = −ai (t)xi (t) + ∑ bi j f j (x j (t)) + ∑ bi j f j (x j (t − τ j )),

(2)

for i = 1, . . . , n. Neural networks with time varying delays were studied deeply in recent years. Exponential stability, asymptotic stability, existence and uniqueness of solutions of them have been analyzed by many authors. Further studies were taken about the following model with time variable delay [16]- [21] n

n

j=1

j=1

xi (t) = −ai (t)xi (t) + ∑ bi j f j (x j (t)) + ∑ bi j f j (x j (t − τ j (t))),

(3)

for i = 1, . . . , n. Recently the differential equations with piecewise constant argument has been studied in many papers [22]– [34]. The main idea of differential equation with piecewise constant argument is combining the continuous and discrete dynamical systems. With this view, it is important for the modeling the biological and computer sciences problems. This type of differential equations has been under investigation since 1980s. Busenberg and Cooke firstly introduced the piecewise constant argument in 1982. Cooke and Wiener, Wiener, Shah and Wiener have studied the type of differential equations [21]– [23]. Neural networks with piecewise constant argument have been introduced into the following form [25]– [35] n

n

j=1

j=1

xi (t) = −ai (t)xi (t) + ∑ bi j f j (x j (t)) + ∑ bi j f j (x j (β (t))),

(4)

for i = 1, . . . , n. Qualitative properties of this neural network system, such that existence and uniqueness of solutions, stability of equilibrium, existence and stability of periodic solutions are investigated. In implementation of neural network models to real world problems, stability of them has a primary importance. So, the stability analysis of neural network systems is crucial. The linear matrix inequalities (LMIs) have been frequently used for the stability analysis of the neural networks as well as they have been used for dynamical systems. Many stability criteria based on LMI have been derived in the literature for different Hopfield neural network models because of the efficiency of this method [35]- [53]. Also this technique has been used in control theory [54]. In this paper we are concerned about a model including both delays and piecewise constant argument. It is the first time that global exponential stability of equilibrium of Hopfield neural networks model with both delays and piecewise constant argument is considered. 2 Model description and preliminaries Let N and R+ be the sets of natural and nonnegative real numbers, respectively, i.e., N = {0, 1, 2, ...}, R+ = [0, ∞), Rn denotes the n dimensional real space. The notation X > 0 (or X < 0) denotes that X is a symmetric and positive definite (or negative definite) matrix. For real symmetric matrices X and Y , the notation X = Y (respectively, X > Y ) means that the matrix X is positive semi-definite (respectively, positive definite). The notations X T and X −1 refer, respectively, the transpose and the inverse of a square matrix X . λmax (X ) and λmin (X ) represent the maximal eigenvalue and minimal eigenvalue of X , respectively. The norm · means n either one-norm: x1 = ∑ |xi |, x ∈ Rn or the induced matrix 2-norm: X 2 = λmax (X T X ). Let θi , and ζi , i=1

denote two fixed real-valued sequences such that θi < θi+1 , θi ≤ ζi ≤ θi+1 for all i ∈ N, with θi → ∞ as i → ∞. Throughout the paper, we assume that there exists a positive constant θ¯ such that θi+1 − θi ≤ θ¯, i ∈ N.


35

In this section, we will consider the description of the following neural network with piecewise argument and constant delay: x (t) = −Ax(t) + Bg(x(t)) +Cg(x(β (t))) + Dg(x(t − τ )) + E,

(5)

where β (t) = θk if t ∈ [θk , θk+1 ), k ∈ N, t ∈ R+ , x = [x1 , . . . , xn ]T ∈ Rn is the neuron state vector, g(x(t)) = [g1 (x1 (t)), . . . , gn (xn (t))]T ∈ Rn is the activation function of neurons, E = [E1 , . . . , En ]T is an external input vector. Additionally, we have A = diag(a1 , . . . , an ) where ai > 0, B = (bi j )n×n , C = (ci j )n×n , D = (di j )n×n , denote the connection weight matrices. (A1) The activation function g satisfies g(0) = 0; (A2) There exists Lipschitz constant L = diag(L1 , . . . , Ln ) > 0, such that |gi (u) − gi (v)| ≤ Li |u − v|, for all u, v ∈ Rn , i = 1, 2, . . . , n; (A3) The activation function g is bounded, i.e. for some constant M > 0, |g(x(t))| < M, for all t ∈ R and x ∈ R; (A4) θ¯ < τ . Consider the equilibrium point, x∗ = (x∗1 , . . . , x∗n )T , of the system (5). Theorem 1. Suppose that the assumptions (A1), (A2) and (A3) are satisfied. If n

ai > Li ∑ (|bi j | + |ci j | + |di j |), i = 1, 2, . . . , n,

(6)

j=1

then system (5) has a unique equilibrium point. Proof. Step 1: Existence:vIf x∗ = (x∗1 , x∗2 , . . . , x∗n ) is an equilibrium point of the system (5), then each x∗i satisfies the following equation: x∗i (t) = =

n n 1 n Ei [ ∑ bi j g j (x∗j ) + ∑ ci j g j (x∗j ) + ∑ di j g j (x∗j )] + ai j=1 ai j=1 j=1 n

1

Ei

∑ [ ai (bi j + ci j + di j )]g j (x∗j ) + ai ,

i = 1, 2, . . . , n.

j=1

Denote H(x∗j ) =

n

1

Ei

∑ [ ai (bi j + ci j + di j )]g j (x∗j ) + ai .

j=1

Thus, x∗ is a fixed point of the map H : Rn → Rn . The i-th component of the function H(x) satisfies the following equation n 1 Ei |H(x∗j )| = | ∑ [ (bi j + ci j + di j )]g j (x∗j ) + | a ai j=1 i

≤ ≤

n

1

Ei

∑ |[ ai (bi j + ci j + di j )]||g j (x∗j )| + | ai |

j=1 n

1

∑ |[ ai (bi j + ci j + di j )]|M +

j=1

|Ei | , ai

36


where x = (x1 , x2 , . . . , xn )T . Then we have n

1

∑ |[ ai (bi j + ci j + di j )|]M + 1≤i≤n

|H(x∗j )| ≤ max

j=1

|Ei | , for i = 1, 2, . . . , n. ai

(7)

H : Rn → Rn is bounded for all x ∈ Rn . Also we can easily say that H is continuous. From Brouwer’s Fixed Point Theorem, H has at least one fixed point. Step 2: Uniqueness: Consider a mapping ⎞ ⎛ f1 (x1 , x2 , . . . , xn ) ⎟ ⎜ .. f (x1 , x2 , . . . , xn ) = ⎝ ⎠, . fn (x1 , x2 , . . . , xn ) x∗ is a fixed point of the map f : Rn → Rn . x∗i = f (x∗1 , x∗2 , . . . , x∗n ) =

1 n { ∑ (bi j + ci j + di j )g j (x∗j ) + Ei }. ai j=1

Suppose that there exists another fixed point denoted y∗ . Then ai (x∗i − y∗i ) =

n

∑ (bi j + ci j + di j )(g j (x∗j ) − g j (y∗j )).

j=1

From conditions (A1)–(A3) and a > 0, n

ai |x∗i − y∗i | − ∑ (|bi j | + |ci j | + |di j |)Lg j |x∗j − y∗j | ≤ 0, i ∈ I.

(8)

j=1

Consequently from (6) we obtain x∗i = y∗i . So there exists a unique equilibrium. The theorem is proved. Now, we will consider the following initial value problem x (t) = −Ax(t) + BG(x(t)) +CG(x(β (t))) + DG(x(t − τ )) + E, x(t) = ϕ (t), σ − τ ≤ t ≤ σ , ,

(9) (10)

where β (t) = θk if t ∈ [θk , θk+1 ), k ∈ N, t ∈ R+ and ϕ (t) is a continuous function. Theorem 2. Suppose that the conditions (A1) − (A4) are hold. Then for every (σ , ϕ ) ∈ R+ × Rn , there exists a unique solution x(t) = x(t, σ , ϕ ) of (9)-(10), such that x(σ ) = ϕ (t) on R+ . Proof. Existence: Equation (9)-(10) can be investigated step by step on intervals [θi , θi+1 ), i ∈ Z. We assume without loss of generality that θi ≤ σ ≤ θi+1 and i = 0. We are looking for the solution x(t), which satisfies the equation x(t) = ϕ (t) for [σ − τ , σ ]. Consider the following cases: (a) Assume that there exists an integer j such that θ j ≤ σ + τ < θ j+1 , j > 1. We will show that there exists a unique solution on the interval [σ , σ + τ ). For t ∈ [σ , θ1 ), x(t) satisfies the following equation x (t) = −Ax(t) + BG(x(t)) +CG(ϕ (σ )) + DG(ϕ (t − τ )) + E.

(11)

Since the equation is quasilinear, with Lipschitzian nonlinear part, the solution exists, unique and is continuable to θ1 . For each i < j, x(t) satisfies the following equation x (t) = −Ax(t) + BG(x(t)) +CG(x(θi )) + DG(x(t − τ )) + E.


37

on the interval [θi , θi+1 ]. Consequently, repeating the discussion for the first interval, one can continue the solution till θ j . Now, consider t ∈ [θ j , σ + τ ). Again, similarly to the previous intervals one can show that the solution exists on [θ j , σ + τ ). (b) Now, assume that σ + τ < θ1 < σ + 2τ . Consider the interval [σ , σ + τ ), then x(t) satisfies the following quasilinear differential equation x (t) = −Ax(t) + BG(x(t)) +CG(ϕ (σ )) + DG(ϕ (t − τ )) + E. It is obvious that the solution exists and is unique on the interval [σ , σ + τ ). Now for t ∈ [σ + τ , θ1 ), x(t) satisfies the following equation; x (t) = −Ax(t) + BG(x(t)) +CG(x(σ + τ )) + DG(x(t − τ )) + E. The above equation is a quasilinear ordinary differential equation, since x(σ + τ ) and x(t − τ ) are known from the previous step. So, there exists a solution on [σ + τ , θ1 ). One can see that by combination of the two cases, (a) and (b) the solution is continuable uniquely on the interval [σ , ∞). The theorem is proved. Definition 1. The equilibrium x = x∗ of (5) is said to be globally exponentially stable if there exist positive constants α1 and α2 such that ||x(t)|| ≤ α1 e−α2t sup ||x(ξ )||. −τ ≤ξ ≤0

By means of the transformation u(t) = x(t) − x∗ , system (5) can be simplified as u (t) = −Au(t) + BG(u(t)) +CG(u(β (t))) + DG(u(t − τ )),

(12)

where G j (u j (t)) = g j (u j (t) + x∗i ) − g j (x∗j ), with g j (0) = 0. It is obvious that the stability of the zero solution of (12) is equivalent to that of the equilibrium x∗ of (5). Therefore, in what follows, we discuss the stability of the zero solution of (12). Lemma 3. Given any real matrices U, W, Z of appropriate dimensions and a scalar ε > 0 such that 0 < W = W T , then the following matrix inequality holds: 1 U T Z + Z T U ≤ ε U T WU + Z T W −1 Z. ε 3 Main results Theorem 4. Suppose that (A1)-(A4) hold true. The equilibrium x∗ of (12) is globally exponentially stable, if there exist matrices P > 0, Q > 0 and two diagonal matrices R > 0, S > 0 such that the following LMI holds; ⎞ ⎛ AP + PA − PBRBT P − L(R−1 + Q + S)L −PC −PD ⎝ (13) S 0 ⎠ > 0. −CT P T 0 Q −D P Proof. Firstly we choose a functional candidate for system (12) as below T

V (ut ) = u (t)Pu(t) +

ˆ

t

t−τ

T

G (u(ξ ))QG(u(ξ ))d ξ +

ˆ

t

β (t)

GT (u(ξ ))SG(u(ξ ))d ξ .

38


Then we will find the time derivative of V (ut ) along the trajectories of system (12) ˙ + GT (u(t))QG(u(t)) V˙ (ut ) = u˙T (t)Pu(t) + uT (t)Pu(t) −GT (u(t − τ ))QG(u(t − τ )) + GT (u(t))SG(u(t)) −GT (u(β (t)))SG(u(β (t))) = [−Au(t) + BG(u(t)) +CG(u(β (t))) + DG(u(t − τ ))]T Pu(t) +uT (t)P[−Au(t) + BG(u(t)) +CG(u(β (t))) + DG(u(t − τ ))] +GT (u(t))QG(u(t)) − GT (u(t − τ ))QG(u(t − τ )) + GT (u(t))SG(u(t)) −GT (u(β (t)))SG(u(β (t))) = −uT (t)(AT P + PA)u(t) + GT (u(t))BT Pu(t) + GT (u(β (t)))CT Pu(t) T

T

(14)

T

+G(u(t − τ ))D Pu(t) + u (t)PBG(u(t)) + u (t)PCG(u(β (t))) +uT (t)PDG(u(t − τ )) + GT (u(t))QG(u(t)) − GT (u(t − τ ))QG(u(t − τ )) +GT (u(t))SG(u(t)) − GT (u(β (t)))SG(u(β (t))). It follows from Lemma (3), uT (t)PBG(u(t)) + GT (u(t))BT Pu(t) ≤ uT (t)PBRBT Pu(t) + GT (u(t))R−1 G(u(t)). (15) Substituting (15) into (14), we have V˙ (u(t), G(u(β (t))), G(u(t − τ ))) ≤ uT (t)(−AP − PA + PBRBT P + +L(R−1 + Q + S)L)u(t) + +GT (u(β (t)))CT Pu(t) + uT (t)PCG(u(β (t))) + +GT (u(t − τ ))DT Pu(t) + uT (t)PDG(u(t − τ )) + +GT (u(β (t)))SG(u(β (t))) +GT (u(t − τ ))QG(u(t − τ )). Then we obtain V˙ (ut ) ≤ −η (t)Ση T (t),

where η (t) = uT (t) GT (u(β (t))) GT (u(t − τ )) , and ⎞ ⎛ AP + PA − PBRBT P − L(R−1 + Q + S)L −PC −PD S 0 ⎠. −CT P Σ=⎝ T 0 Q −D P

(16)

Now we will prove the global exponential stability of the solution. Note that = max {Li } for i = 1, . . . , n 1≤i≤n

and 2

V (ut ) ≤ λmax (P)||u(t)|| + λmax (Q)

2

ˆ

t

t−τ

2

||u(ξ )|| d ξ + λmax (Q)

2

ˆ

t

β (t)

||u(ξ )||2 d ξ . (17)

From (13) and (16), one can see that there exists a scalar m > 0 such that ⎞ ⎛ AP + PA − PBRBT P − L(R−1 + Q + S)L − mI −PC −PD ⎝ S 0 ⎠ > 0. −CT P T 0 Q −D P


39

Then we can obtain easily the following equation for any scalar c > 0, d ct (e V (ut )) = ect [c(V (ut )) + V˙ (ut )] dt ct

2

e [(cλmax (P) − m)||u(t)|| + cλmax (Q) ˆ t 2 +λmax (Q) ||u(ξ )||2 d ξ ]

≤

2

ˆ

t−τ

β (t)

ct

2

e [(cλmax (P) − m)||u(t)|| + 2cλmax (Q)

≤

By intergating two sides from 0 to T > 0, we obtain

ˆ

cT

t

2

ˆ

||u(ξ )||2 d ξ

t

t−τ

||u(ξ )||2 d ξ ].

T

ect ||u(t)||2 dt e V (uT ) −V (u0 ) ≤ (cλmax (P) − m) 0 ˆ Tˆ t ect ||u(ξ )||2 d ξ dt. +2cλmax (Q) t−τ

0

One can see that easily

ˆ 0

T

ˆ

t

t−τ

ect ||u(ξ )||2 d ξ dt ≤ τ

ˆ

T

−τ

ec(t+τ ) ||u(t)||2 dt

ˆ

cτ

≤ τe

−τ cτ

+τ e Then we obtain

0

||u(t)||2 dt

ˆ

0

T

ect ||u(t)||2 dt.

ˆ T ect ||u(t)||2 dt ecT V (uT ) ≤ (cλmax (P) − m + 2cλmax(Q)2 τ ecτ ) 0 ˆ 0 ||u(t)||2 dt +V (u0 ) +2cλmax (Q)2 τ ecτ −τ

By choosing a scalar c > 0 such that m = cλmax (P) + 2cλmax (Q)2 τ ecτ , we have ˆ 0 cT 2 cτ ||u(t)||2 dt +V (u0 ) e V (uT ) ≤ 2cλmax (Q) τ e

(18)

We know from definition of the V (ut ), V (u0 ) satisfies the following inequality ˆ 0 2 2 ||u(ξ )||2 d ξ V (u0 ) ≤ λmax (P)||u0 || + λmax (Q)

(19)

−τ

−τ

Substituting (19) into (18), we have ecT V (uT ) ≤ (2cλmax (Q)2 τ 2 ecτ + λmax (P) + λmax (Q)2 τ ) sup ||u(ξ )||2 . −τ ≤ξ ≤0

Also we know from (17), λmin (P)||u(T )||2 ≤ V (uT ). Consequently, we have 2cλmax (Q)2 τ 2 ecτ + λmax (P) + λmax (Q)2 τ −cT /2 e sup ||u(ξ )||2 . ||uT )|| ≤ λmin (P) −τ ≤ξ ≤0 The theorem is proved.

40


4 An illustrative example Consider the following Hopfield neural network system with piecewise constant argument.

0.01 0.02 tanh(x1 (t)) 0.1 0 x1 (t) + x (t) = − 0.03 0.01 0 0.1 x2 (t) tanh(x2 (t))

0.01 0.02 tanh(x1 (β (t))) + . 0.02 0.03 tanh(x2 (β (t)))

1

(20)

2

1.8

0.9

1.6 0.8

x2

x1

1.4 0.7

1.2 0.6 1 0.5

0.4

0.8

0

5

10

t

15

20

25

0

5

10

t

15

20

25

Fig. 1 Time response of x1 (t) and x2 (t) with piecewise constant arguments

Here the coefficients of the delay terms in the main model has been chosen zero. If L1 = 0.1 and L2 = 0.1, it can be shown easily that (20) satisfies the condition of Theorem 2.1. So there exists a unique equilibrium of (20) such that x∗ = [0.4372, 0.6623]T . For

1.5 1 20 30 20 P= ,Q = ,R = ,S = , 1 1.5 02 03 02 the condition of the Theorem 4.1 is satisfied. So, the equilibrium of the system (20) is globally exponentially stable. 5 Conclusion In this paper, the Hopfield neural network with piecewise constant argument and constant delay has been studied. Up to now, various kinds of delays were introduced to the Hopfield neural network systems such that constant delays, single time delays, time varying delays and distributed delays. But it is the first time that Hopfield neural networks model with both piecewise constant argument and constant delay is considered. This combination provided a more realistic approximation to the real life problems. An LMI method has been used to obtain the global exponential stability of equilibrium point of the system. An example is given to illustrate our results. References [1] Hopfield, J.J.(1984), Neurons with graded response have collective computational properties like those of two-stage neurons, Proc. Nat. Acad. Sci. Biol. 81, 3088–3092.


41

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Vibrational Resonance in a System with a Signum Nonlinearity K. Abirami1 , S. Rajasekar1†, M.A.F. Sanjuan2 1 School

of Physics, Bharathidasan University, Tiruchirapalli 620 024, Tamilnadu, India de F´ısica, Universidad Rey Juan Carlos, Tulipán s/n, 28933 Móstoles, Madrid, Spain

2 Departamento

Submission Info Communicated by Albert C.J. Luo Received 9 May 2015 Accepted 28 May 2015 Available online 1 April 2016 Keywords Vibrational resonance Signum nonlinearity Biharmonic force

Abstract We present our investigation on vibrational resonance in a system with a signum nonlinearity. We construct an exact analytical solution of the system in the presence of an external biharmonic force with two frequencies ω and Ω, Ω ω and use it for the computation of the response amplitude Q at the low-frequency ω . We analyse the effect of the strength of the signum nonlinearity on vibrational resonance for the cases of the potential with a single-well, a double-well and a single-well with a double-hump. An interesting feature of vibrational resonance in the system is that Q does not decay to zero for g (the amplitude of the high-frequency force) → ∞. We compare the features of the vibrational resonance of these two systems, since the potential of the system with the signum nonlinearity and that of the Duffing oscillator show ssimilar forms. The strength of the nonlinearity in these two systems is found to give rise distinct effects on resonance. ©2016 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Nonlinear systems are capable of showing a rich variety of fascinating dynamics. Nonlinearity can arise from the dissipative force or from the restoring force. In the ubiquitous Duffing oscillator [1, 2], the dissipative force is linear while the restoring force has a cubic nonlinearity. In the van der Pol oscillator, the dissipative force is nonlinear and the restoring force is linear [2]. There is an another class of nonlinear systems where the nonlinear function is piecewise linear [3–5]. An extensively studied piecewise linear system is the Chua’s circuit [3–6], in which the nonlinear circuit element is Chua’s diode and its characteristic function is a piecewise linear function. An interesting feature of nonlinear systems with a piecewise linear function as the only nonlinearity is that, in general, it is possible to construct an exact analytical solution even for chaotic systems. The dynamics of many piecewise linear systems with more than two linear regions have been investigated in detail [7–12]. A simple nonlinearity can be a piecewise linear function, for example the signum function sgn(x), where sgn(x) = 1 for x > 0 and −1 for x < 0 and |x|. Such nonlinearities are found to arise in certain real systems [13–22]. The potential V (x) = 12 ω02 x2 + β |x| admits all the four forms of the potential V (x) = 12 ω02 x2 + 14 β x4 of the Duffing oscillator, an ubiquitous and a prototype nonlinear oscillator. A periodically driven system with a signum nonlinearity is found to show period doubling route to chaos and an onion-like chaotic attractor [19]. † Corresponding

author. Email address: [email protected], [email protected], [email protected] ISSN 2164 − 6376, eISSN 2164 − 6414/$-see front materials © 2016 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/DNC.2016.03.006

44

K. Abirami, S. Rajasekar, M.A.F. Sanjuan / Discontinuity, Nonlinearity, and Complexity 5(1) (2016) 43–58

The prime goal of the present paper is to construct an exact analytical solution of a system with the signum nonlinearity sgn(x) and driven by a biharmonic force and investigate the vibrational resonance phenomenon through its analytical solution. Vibrational resonance [23] is a nonlinear resonant behaviour induced at the low-frequency ω of the input periodic signal by a high-frequency (Ω ω ) periodic force. The occurrence of vibrational resonance has been studied in certain nonlinear continuous time dynamical systems [23–37] and maps [38, 39] with smooth nonlinearities. For systems with polynomial type potentials, using the theoretical method proposed by Blekhman and Landa [26], an analytical approximate expression for response amplitude Q at the low-frequency ω can be obtained. Using this expression for Q the features of vibrational resonance have been studied in certain nonlinear systems. The paper is organized as follows. In sec. 2 we construct the exact analytical solution of the system with the signum nonlinearity sgn(x) and driven by a biharmonic force F(t) = f cos ω t + g cos Ωt, Ω ω . We present the solution separately for the underdamped, critically damped and overdamped cases. We analyse the contribution of various terms in the analytical solution on the response amplitude Q(ω ). Then consider the system with single-well potential in sec. 3. We show the occurrence of vibrational resonance by varying the control parameter g and explain the mechanism of resonance in terms of the resonant frequency of the system. Next, we point out the influence of strength β of the nonlinear term on the value of g at which resonance occurs (denoted as gVR ), the value of Q at resonance (denoted as Qmax ) and the value of Q in the limit of g → ∞ (denoted as QL ). The vibrational resonance of the system will be compared with the single-well Duffing oscillator. Section 4 is devoted to the signum nonlinear system with a double-hump single-well potential. The parameter β has a strong influence on vibrational resonance. Depending upon the value of β the system displays no resonance, a single resonance and a double resonance. We distinguish the vibrational resonance of the signum nonlinear system with that of the Duffing oscillator with double-hump single-well potential. The features of vibrational resonance in the signum nonlinear system with double-well potential is presented in sec. 5. Section 6 contains the conclusions.

2 The Biharmonically Driven Oscillator with a Signum Nonlinearity The equation of motion of a biharmonically driven system with the signum nonlinearity is given by x¨ + d x˙ + ω02 x + β sgn(x) = f cos ω t + g cos Ωt,

Ωω

(1)

where d is the damping coefficient, ω02 is the natural frequency of the system, β is the strength of the signum nonlinearity and sgn(x) is the signum function. In the system (1) f cos ω t and g cos Ωt are two input signals with high-frequency Ω and low-frequency ω and g and f are the strengths or amplitudes of these signals, respectively. The potential of the system (1) in absence of the damping and external periodic forces is 1 VS = ω02 x2 + β |x| . 2

(2)

The potential of the well-known Duffing oscillator is 1 1 VD = ω02 x2 + β x4 . 2 4

(3)

The shapes of the potentials VS and VD depend on the parameters ω02 and β . Table 1 summarizes the conditions on ω02 and β for each of the four shapes of the potentials VS and VD . Note that VS is of a double-well form for ω02 > 0, β < 0 while for this choice of the parameters VD is of a single-well with a double-hump shape. For ω02 < 0 and β > 0 the shape of VS is of a single-well with a double-hump while that of VD is of a doublewell. The point is that VS mimics all the four forms of the potential VD . Figure 1 depicts the four forms of the potential functions VS and VD . The inverted potential is physically uninteresting, since x(t) → ±∞ for all


45

Table 1 Sign of ω02 and β and the corresponding shape of the Duffing oscillator and the signum oscillator. In this table CC and DC refer to continuous curve and dashed curve, respectively. Sign of

Shape of the Potential of

ω02

β

Duffing oscillator

Signum oscillator

+

+

Single-well (DC in Fig. 1(a))

Single-well (CC in Fig. 1(a))

−

+

Double-well (DC in Fig. 1(b))

Single-well with a double-hump (CC in Fig. 1(b))

+

−

Single-well with a double-hump (DC in Fig. 1(c))

Double-well (CC in Fig. 1(c))

−

−

Inverted single-well (DC in Fig. 1(d))

Inverted single-well (CC in Fig. 1(d))

V (x)

6

(a) ω02 = 1, β = 1

1

(b) ω02 = −1, β = 1

4 2

0

0

V (x)

0.4

(c) ω02 = 1, β = −1

0

(d) ω02 = −1, β = −1

0 -1

-0.4

-2 -2

0

x

2

-2

0

x

2

Fig. 1 Shapes of the potentials VS and VD for some selective values of ω 02 and β . In all the subplots, the continuous and the dashed curves represent VS and VD , respectively.

initial conditions of the periodically driven system. We construct the exact analytical solution and analyse the vibrational resonance separately for the other three forms of the potential VS . Also, we compare the resonance dynamics of the system (1) with that of the Duffing oscillator.

2.1

Analytical Solution

In order to construct the general solution of Eq. (1), we rewrite it with x = x+ for x ≥ 0 and x = x− for x < 0 as x¨+ + d x˙+ + ω02 x+ = f cos ω t + g cos Ωt − β , for x+ ≥ 0

(4a)

x¨− + d x˙− + ω02 x−

(4b)

= f cos ω t + g cos Ωt + β , for x− < 0.

As the Eqs. (4a) and (4b) are linear in x+ and x− , respectively, and are constant coefficients equations, their solutions can be written separately for the regions x+ ≥ 0 and x− < 0 and matching the two solutions at x = 0. The solutions are written separately for the three cases d2 > 4ω02 , d 2 = 4ω02 and d 2 < 4ω02 .

46


Case 1: Overdamping – d2 > 4ω02 For the overdamping case, we have obtained the solution as x± (t) = A± eλ+t + B± eλ− t + F1 sin ω t + F2 cos ω t + G1 sin Ωt + G2 cos Ωt ∓

β , ω02

(5a)

where A± =

B± =

e−λ+ t0 [x˙0 − λ− x0 + (λ− F1 + ω F2 ) sin ω t0 + (λ− F2 − ω F1 ) cos ω t0 λ+ − λ− β + (λ− G1 + ΩG2 ) sin Ωt0 + (λ− G2 − ΩG1 ) cos Ωt0 ∓ λ− 2 ] , ω0

(5b)

e−λ− t0 [x˙0 − λ+ x0 − (λ+ F1 + ω F2 ) sin ω t0 − (λ+ F2 − ω F1 ) cos ω t0 λ+ − λ− β − (λ+ G1 + ΩG2 ) sin Ωt0 − (λ+ G2 − ΩG1 ) cos Ωt0 ± λ+ 2 ] , ω0

(5c)

with 1 λ± = [−d ± d 2 − 4ω02 ] , 2 f ω02 − ω 2 f dω , F2 = − , F1 = 2 2 ω02 − ω 2 + d 2 ω 2 ω02 − ω 2 + d 2 ω 2 g ω02 − Ω2 gdΩ , G2 = − . G1 = 2 2 ω02 − Ω2 + d 2 Ω2 ω02 − Ω2 + d 2 Ω2

(5d) (5e) (5f)

In the above solution, the constants A± and B± are to be evaluated at the initial time t = t0 and reevaluated at ˙ = x˙0 = y0 . With t0 = 0 other times at which x = 0. The initial conditions are t = t0 = 0 with x(0) = x0 > 0, x(0) we calculate A+ and B+ . Then, we calculate x(t) = x+ (t) from Eq. (5a) with time step Δt until x(t) < 0. Now, using x(t − 2Δt) > 0 and x(t − Δt) > 0, we compute the value of t = tc (by extrapolation) for which x(tc ) = 0. Next, we compute x(t ˙ c ) by extrapolation. Then, we calculate A− and B− from Eqs. (5b) and (5c), respectively with t0 = tc , and proceed to calculate x(t) = x− (t) until x(t) > 0 and update t0 , A+ and B+ and so on. Case 2: Critical Damping – d2 = 4ω02 For d 2 = 4ω02 , the solution of (1) is given by x± = A± e−dt/2 + tB± e−dt/2 + F1 sin ω t + F2 cos ω t + G1 sin Ωt + G2 cos Ωt ∓

β , ω02

(6)

where A± and B± are to be determined by the values of x± and x˙± at the time t = t0 = 0 and at other times at which x = 0. Case 3: Underdamping – d 2 < 4ω02 When d 2 < 4ω02 , x± (t) are given by t + B±e−dt/2 sin ω t + F1 sin ω t + F2 cos ω t + G1 sin Ωt + G2 cos Ωt ∓ x± (t) = A± e−dt/2 cos ω

β , ω02

(7a)


47

where A± =

B± =

= with ω 2.2

β edt0 /2 d t0 + ω cos ω t0 )[x0 − F1 sin ω t0 − F2 cos ω t0 − G1 sin Ωt0 − G2 cos Ωt0 ∓ 2 ] , {(− sin ω ω 2 ω0 (7b) − sin ω t0 x˙0 − F1 ω cos ω t0 + F2 ω sin ω t0 − G1 Ω cos Ωt0 + G2 Ω sin Ωt0 ]} , edt0 /2 d β t0 − ω sin ω t0 )[x0 − F1 sin ω t0 − F2 cos ω t0 − G1 sin Ωt0 − G2 cos Ωt0 ∓ 2 ] {( cos ω ω 2 ω0 − cos ω t0 [x˙0 − F1 ω cos ω t0 + F2 ω sin ω t0 − G1 Ω cos Ωt0 + G2 Ω sin Ωt0 ]}

(7c)

|d 2 − 4ω02 |, and F1 , F2 , G1 and G2 are given by Eqs. (5e) and (5f).

Contribution to the Response Amplitude by Different Parts of the Solution

We are interested in the effect of the high-frequency driving force on the amplitude of the low-frequency component of the output signal x(t). We treat the amplitude of the high-frequency force as the control parameter. Before reporting the high-frequency induced resonance, we identify the contribution to the response amplitude Q by the various parts of the solution and the term in the solution which gives a dominant contribution to Q. The response amplitude Q at the low-frequency ω is defined through Q = Q2c + Q2s / f where ˆ MT 2 x(t) sin ω t dt , Qs = MT 0 ˆ MT 2 x(t) cos ω t dt . Qc = MT 0

(8a) (8b)

In Eqs. (8) T = 2π /ω and M = 1000. For convenience we rewrite the solutions x± (t) as exp

imp

x± (t) = xω + xω ± + xrem ,

(9a)

where exp

xω = F1 sin ω t + F2 cos ω t , ⎧ λ+ t λ− t 2 2 ⎪ ⎨A± e + B±e , for d > 4ω0 imp t + B±e−dt/2 sin ω t , for d 2 < 4ω02 xω ± = A± e−dt/2 cos ω ⎪ ⎩ A± e−dt/2 + B± e−dt/2 , for d 2 = 4ω02 xrem = G1 sin Ωt + G2 cos Ωt ∓

β . ω02

(9b) (9c)

(9d)

imp In xexp ω , the frequency ω is explicitly present in the argument of the sinusoidal terms. In xω ± , the frequency ω appears only in the expressions of A± and B± , which are updated only when x(t) = 0. xrem is independent of ω . imp We calculate the response amplitudes for x(t) = xexp ω , xω ± and xrem at the frequency ω and denote them as Qexp , x(t) = x± (t), that is with all the Qimp and Qrem , respectively. We call the response amplitude computed with three parts of the solution, as Q. Obviously Qrem = 0 while Qexp = F12 + F22 f is independent of the control parameter g. Figure 2 presents the variation of Q, Qimp and Qexp with the control parameter g for the single-well, the double-well and the single-well with a double-hump potentials. In this figure, we observe that in general Q = Qexp + Qimp. However, the variation of Qimp is similar to the variation of Q. The contribution to Q comes from all the terms present in the solution, though some of the terms do not contain ω .


3

(a)

2

(b)

2

Q

Q

Q

1

Q

200

0

400

g

Qexp Qimp

3

Qexp

Qexp 0

(c)

Q

Qimp

Qimp

1 0

6

Q

48

0

250

500

g

0

0

200

g

400

Fig. 2 Variation of Q, Q imp and Qexp with g for the system (1) with (a) single-well potential, (b) single-well with a double-hump potential and (c) double-well potential. The values of the parameters are d = 0.5, f = 0.05, Ω = 20ω and (a) ω02 = 1, β = 0.5 and ω = 2, (b) ω 02 = −1, β = 2 and ω = 2 and (c) ω 02 = 1, β = −1 and ω = 0.5.

8

(a)

β = 0.05

6

0.5

β = 0.05 0

(b)

β = 0.5

ωr2

Q

1

0

200

g

400

β = 0.5

ω2 0

200

g

400

Fig. 3 (a) Variation of Q computed from the analytical solution (7) (continuous curve) and computed from numerical solution (solid circles) with the control parameter g for two values of β for the system (1). The values of the other parameters are ω02 = 1, d = 0.5, f = 0.05, ω = 2 and Ω = 20ω . (b) ω r2 given by Eq. (11) versus g for two values of β . The horizondal dashed line represents the value of ω r2 ≈ ω 2 for which Q becomes a maximum while ω r2 becomes a minimum.

3 The System (1) with Single-Well Potential We consider the system (1) with a single-well potential and with the following choice of parameters d = 0.5, f = 0.05, ω02 = 1, β > 0, ω = 2 and Ω = 20ω . Here d2 < 4ω02 and the analytical solution of the system is given by Eqs. (7). Figure 3(a) depicts the variation of Q(ω ) computed from the analytical solution (continuous curve) and from the numerical solution obtained by numerically integrating Eq. (1) with a fourth-order Runge–Kutta method. Then, it is used in Eqs. (8) (solid circle) as a function of the control parameter g for two values of β . The values of both the Q’s are almost the same. For β = 0.5, the response amplitude Q becomes a maximum with the value 1 at g = 174. There is no resonance for β = 0.05. In order to identify the mechanism of occurrence and nonoccurrence of resonance found for β = 0.5, we consider the resonant frequency ωr . For a system driven by the biharmonic force, one can assume that the response of the system essentially consists of a slow motion X(t, ω ) and the fast motion ψ (t, Ωt) and write x(t) = X + ψ . For nonlinear oscillators with polynomial potentials applying a theoretical approach, an approximate equation of motion for the slow variable X and an approximate analytical expression for Q can be obtained


49

1

Q

d = 0.5 0.5

0

d=2 d = 2.5

0

200

400

g

Fig. 4 Q versus g of the system (1) for d = 0.5 (underdamped case), d = 2 (critical damped case) and d = 2.5 (overdamped case) with ω02 = 1, β = 0.5, f = 0.05, ω = 2 and Ω = 20ω .

[23, 26]. The general form of the analytical expression of Q is given by Q=

1 (ωr2 − ω 2 )2 + d 2 ω 2

,

(10)

where ωr is the resonant frequency of oscillation of the linearized equation of motion of the slow variable X. ωr is generally a function of the parameters ω02 , β , g and Ω, that is ωr2 = F(ω02 , β , g, Ω) [23, 26]. The form of F depends on the form of the nonlinearity present in the system. Assuming the response amplitude Q in the form of Eq. (10), we write 1 ωr2 = ω 2 + − d2ω 2 (11) Q2 and compute ωr2 corresponding to the Fig. 3(a). The result is presented in Fig. 3(b). In this figure the horizontal dashed line represents the value of ωr2 = ω 2 . For β = 0.5 as g increases from a small value, the quantity ωr2 decreases from a large value > ω2 . Consequently, Q increases from a small value. At a value of g = 174, ωr2 ≈ ω 2 and according to Eq. (10) Q ≈ 1/d ω = 1. In Fig. 3(a) at resonance Q ≈ 1. This implies that for the system (1) one can introduce a resonant frequency (its analytical expression for the system (1) is not known) associated with the slow variable and resonance occurs when either ωr matches with the frequency ω of the low-frequency input signal or the quantity ωr2 − ω 2 becomes a nonzero minimum. We denote the value of g at which resonance occurs as gVR . For g > gVR , Q decreases but not decays to zero. In Fig. 3(b) ωr2 matches with ω 2 at only one value of g and Q becomes maximum at one value of g. For β = 0.05 ωr2 = ω 2 (refer to Fig. 3(b)) when g is varied and there is no resonance. In the above, we considered the choice d = 0.5 and ω02 = 1. For this choice d2 = 0.25 < ω02 = 1 (underdamped case) and the analytical solution used is given by Eqs. (7). In Fig. 4, we plot Q versus g for the overdamped case with d = 2.5 and the critical damping case with d = 2 together with d = 0.5 (underdamped case). The analytical solutions used for d = 0.5, 2 and 2.5 are given by the Eqs. (5), (6) and (7), respectively. According to Eq. (10) the effect of damping is to change the value of Q, however, it does not affect the value of gVR at which the resonance occurs. Moreover, for d = 0.5, 2 and 2.5 the value of Q at resonance are expected to be 1, 0.25 and 0.2, respectively. Same results are found in Fig. 4. Next, we present the effect of β on vibrational resonance. Q is calculated for g ∈ [0, 500] and β ∈ [0, 2]. Figure 5 shows the response amplitude profile as a function of g and β . For β ≤ 0.05, there is no resonance and

50


(b)

(a) 1

1

Q 0.5

Q0.5

0 0.1

β

0 2

0.05

200

100

β

1

g

00

250

500

g

00

(c)

2

Q1 0 2

1.5

ω

1

0.50

1000

2000

g

400

(a)

1

Qmax

gVR

Qmax , QL

Fig. 5 Variation of Q with g for selected values of β ∈ [0, 2] of the system (1) with ω 02 = 1, d = 0.5, f = 0.05, ω = 2 and Ω = 20ω .

0.5

(b)

200

QL 0 0

0.5

β

1

0 0

0.5

β

1

Fig. 6 Plot of (a) Q max and QL and (b) g VR as a function of β for the system (1) with ω 02 = 1, d = 0.5, f = 0.05, ω = 2 and Ω = 20ω .

the value of Q at g = 0 is > 0. A single resonance occurs for β > 0.05. In Fig. 5 we can clearly observe that for β > 0.05, Q(g = 0) = 0. For β > 0.05 in the absence of the high-frequency force g cos Ωt and driven by the lowfrequency force f cos ω t the long time evolution of the system is not oscillatory. The system exhibits amplitude death and x(t) → 0 as t → ∞. A periodic oscillation with period 2π /ω is induced by the high-frequency driving force. Another important observation is that Q does not decay to zero after passing through the resonance. Q attains a nonzero saturation value for sufficiently large values of g. Such a response dynamics is previously found in pendulum and Morse oscillator systems [33, 40]. In many oscillators Q is found to decay to zero for large values of g. We denote the limiting value of Q (in the limit of g → ∞) as QL . The variations of Qmax (the value of Q at resonance), QL and gVR with β are shown in Fig. 6. Except for very small values of β , Qmax ≈ 1 (= 1/d ω )


β

2

4

1

2 ΔQ

0

0

2500

g

51

0 5000

Fig. 7 Color code plot of ΔQ (= Q S /QD ) as a function of the parameters β and g for the single-well potential case of the system ( efe1) and the Duffing oscillator.

while QL and gVR increase linearly with β . It is noteworthy to compare the dependence of Q on the parameters of the system (1) and the Duffing oscillator with a single-well potential. For the Duffing oscillator, a theoretical treatment of vibrational resonance analysis gives [26] 1 3β g2 , ωr2 = ω02 + , (12a) QD = 4Ω4 (ωr2 − ω 2 )2 + d 2 ω 2 and

gVR

2Ω4 2 = ω − ω02 3β

1/2 .

(12b)

From Eqs. (12) for the Duffing oscillator system we infer the following: (i) Vibrational resonance occurs only for ω2 > ω02 . (ii) There is always one resonance for β > 0 and ω2 > ω02 . (iii) QD → 0 as g → ∞. (iv) gVR decreases with increase in β . (v) QD,max = 1/d ω . For ω 2 < ω02 there is no resonance and Q decreases monotonically from a nonzero value with increase in the value of g and approaches 0 in the limit of g → ∞. For the system (1) the following results are observed: (i) For each fixed value of ω with ω2 > ω02 , resonance occurs for β values above a critical value. (ii) QS (Q of the system (1)) does not decay to zero as g → ∞, but approaches a limiting value. (iii) gVR increases with β (as shown in Fig. 6(b)). (iv) QS,max ≈ 1/d ω except for small values of β (refer Fig. 6(a)).

52


Q

2 1 0 2

β

800

1 400 0 0

g

Fig. 8 The effect of β on Q computed from the analytical solution for the system (1) with the double-hump potential. The values of the parameters are d = 0.5, ω 02 = −1, f = 0.05, ω = 1 and Ω = 20ω .

(v) For ω 2 < ω02 , as g increases from a small value, the response amplitude Q increases from a small value and then attains a saturation displaying a sigmoidal type variation (see Fig. 5(c)). In order to compare the values of Q of the two systems, we compute the quantity ΔQ = QS /QD . Figure 7 presents the dependence of ΔQ on β and g. 4 Single-Well with a Double-Hump Potential The shape of the potential of the system (1) for ω02 < 0 and β > 0 is the single-well with a double-hump as shown in Fig. 1(b). For our analysis of the vibrational resonance in (1), we fix ω02 = −1, β > 0, d = 0.5, f = 0.05, ω = 1 and Ω = 20ω . Figure 8 depicts the effect of β on Q computed from the analytical solution. The effect of β is clearly seen in this figure. There are three notable effects of β : (i) Q(ω ) = 0 when g = 0, even though the system is driven by a periodic force. (ii) For β < βc1 = 0.104, there is no resonance. (iii) For βc1 < β < βc2 = 0.39 Q, it displays only one resonance. There are two resonances for β > βc2 . (iv) For each fixed value of β , as g increases, the maximum and minimum values of x(t) increase. Above a critical value of g, x(t) crosses the barriers at x± = ±β /|ω02 |. Since VS (x) → −∞ for x → ±∞, the solution x(t) → ±∞. The solution x(t) is now unbounded and hence the response amplitude curve terminates at a value of g. We denote the value of g below which x(t) is bounded as gb . This critical value of g increases for increasing values of β . In order to describe the occurrence of resonance in the system (1) for the double-hump single-well potential case, we plot in Fig. 9 Q and ωr2 versus g for β = 1. Q is maximum at g = 141 with Qmax = 2 and at g = 352 with Qmax = 1.93. When β = 0.5, Q is maximum at only one value of g(= 67) with Qmax = 2. Furthermore, x(t) becomes unbounded for g > gb = 170. When g increases from a small value, then ωr2 exhibits a finite number of oscillations. Moreover, Q becomes a maximum (minimum) when ωr2 attains locally a minimum (maximum) (1) value. At g = gVR = 147, ωr2 becomes a minimum with the value ω2 and hence Q becomes a maximum with (1) (1) the value Qmax = 1/d ω = 2. The second resonance occurs at g = gVR = 352. At this value of g, the quantity (2) ωr2 is locally a minimum, but with a value = ω2 and hence Qmax = 2. That is, when the resonance is due to the matching of ωr2 with ω 2 of the low-frequency force then Qmax = 1/d ω . If the resonance is due to the local minimization of ωr2 without matching with ω 2 then Qmax < 1/d ω . (1) (2) , g(2) , Qmax and Qmax are computed for β ∈ [βc1 , 2]. Figure 10 depicts the variation of these The values of g(1) VR VR and g(2) vary linearly with β , but with different rate. The width between g(1) and quantities with β . Both g(1) VR VR VR


2.5

ωr2

Q

2

53

1

1.5

ω2 0

(a) 0

200

400

g

0.5

(b) 0

200

400

g

Fig. 9 (a) Variation of Q with the control parameter g of the system (1) with the double-hump single-well potential. Here d = 0.5, ω02 = −1, β = 1, f = 0.05, ω = 1 and Ω = 20ω . (b) ω r2 versus g corrsponding to the subplot (a).

700

2

Qmax

gVR

(2) gVR (1) gVR

350

1

2

β

(1)

Q(2) max

1.8

1.6

(a)

0 0

Q(1) max

(b) 0

3

1

β

2

3

(2)

Fig. 10 Depedence of g (1) , g(2) , Qmax and Qmax with the parameter β for the system (1) with a double-hump single-well VR VR potential. (1)

g(2) increases with β . In Fig. 10(a), Qmax ≈ 2 implying that the first resonance is due to ωr2 ≈ ω 2 . For β > βc2 , VR (2)

(2)

the response amplitude Qmax increases and approaches Qmax for β 1. Next, we compare the vibrational resonance of the system (1) with that of the Duffing oscillator. For ω02 < 0 and β > 0, the potential VS is a single-well with a double-hump while the shape of VD is a double-well. VD is a double-hump single-well form for ω02 < 0 and β > 0. For the system (1), we choose ω02 < 0 and β > 0, while for the Duffing oscillator we choose ω02 > 0 and β < 0. For the Duffing oscillator an approximate theoretically obtained response amplitude is given by Eq. (12a), where ωr2 = ω 2 − β g2 /2Ω2 . Matching of ωr2 with ω 2 yields gVR =

2Ω4 ω02 − ω 2 , 3|β |

ω 2 < ω02 .

(13)

Equation (13) implies that at most only one resonance is possible in the Duffing oscillator, while in the system (1) for a range of values of β two resonances take place (see Figs. 8 and 10). In both systems Qmax at the first resonance is always 1/d ω . The potentials VS and VD can be compared in terms of the depth of the potential well and the location x∗± at which the potentials are locally maximum. The x∗± are the x-components of the unstable equilibrium points of the system (1) and the Duffing oscillator in absence of the driving force. The depths of the potential wells of VS


V (x)

0.6

0.6

VS VD

(a)

0.4

V (x)

54

0.2 0 -2

0

0.4 0.2 0 -2

2

x

VS VD

(b)

0

2

x

Fig. 11 Plot of VS and VD with (a) x∗S = x∗D and (b) d S = dD . In (a) ω02 = −1, β = 1 for VS and ω02 = 1, β = −1 for VD . In (b) ω02 = 1, β = −1 for VS and ω02 = 1, β = −2.

(b)

3

4 0.2

ΔQ

2

0.1 1

0

50

g

100

0

dS = dD

x∗±,S = x∗±,D

(a)

0.2

3

ΔQ

2

0.1

1 0

50

g

100

0

Fig. 12 Color code represeentation of ΔQ (= Q S /QD ) as a function of g and (a) x ∗±,S = x∗±,D and (b) d S = dD .

and VD , denoted as dS and dD , respectively, are given by 2 4 β ω dS = 2 , dD = 0 . 4β 2ω0

(14a)

The x∗ of the two systems are β ∗ x±,S = ± 2 , ω 0

ω2 x±,D = ± 0 . β

(14b)

Figures 11(a) and (b) show VS and VD with x∗±,S = x∗±,D and dS = dD , respectively. We computed QS and QD as a function of g and (i) x∗±,S = x±,D and (ii) dS = dD . Then ΔQ = QS /QD is calculated. Figure 12 presents the result.

5 Double-Well Potential The potential of the system (1) becomes of a double-well form for ω02 > 0 and β < 0. We fix the values of the parameters as d = 0.5, ω02 = 1, f = 0.05, ω = 0.5 and Ω = 20ω . For d = 0.5 and ω02 = 1 the system is underdamped in absence of an external periodic force. The analytical solution of the forced system (1) for the underdamped case is given by Eq. (7). Figures 13(a) and (b) present the dependence of the response amplitude Q(ω ) as a function of g for a range of fixed values of β for ω = 0.5 and ω = 1.5, respectively. In Fig. 13(a) we


(a) Q3

(b) 0.8 Q 0.6

1 -2

β

55

0.4 -2 -1

500 250

g

00

-1

β

2000 1000

0 0

g

Fig. 13 Q as a function of g and β for the double-well potential case of the system (1). The value of the other parameters are d = 0.5, f = 0.05, ω 02 = 1, Ω = 20ω . (a) ω = 0.5 and (b) ω = 1.5.

4

90

(a) 0 -2

-1

β

Qmax , QL

gVR

180

Qmax

3

QL

2

(b) 1 -2

0

-1

β

0

Fig. 14 (a) g VR versus β and (b) Q max and QL versus β for the double-well potential case of the system (1).

observe that for each fixed value of β < 0, the response amplitude Q is a constant for g values less than a critical value gc . This critical value varies with β . As g passes through gc , the response amplitude increases, reaches a maximum, decreases with further increase in g and then approaches a constant value. In Fig. 13(a), we can clearly notice the effect of β on the response profile. The variations of gVR , Qmax and QL with the parameter β are presented in Fig. 14. gVR increases linearly with increases in |β |. Except for β values near zero, Qmax becomes 4 (= 1/d ω ) for β < 0. QL remains at the value of 1.25. In order to compare the response amplitudes of the signum nonlinearity system and the Duffing oscillator, we consider the location of the local minima and the depth of the potential wells. They are given by the Eqs. (14a) and (14b). Figure 15 presents the dependence of ΔQ (= QS /QD ), where QS and QD are the response amplitude of the system (1) and the Duffing oscillator, as a function of the parameter g and the cases of dS = dD and x∗±,S = x∗±,D . For the double-well potential Duffing oscillator (ω02 < 0, β > 0) the approximate analytical expression for Q is given by Eq. (12a) with

ωr2 = −|ω02 | + 3β x∗ 2 ,

c1 x∗ = ± − , 0, β

c1 = −|ω02 | +

3β g2 . 2Ω4

(15)

Resonance occurs at

(1)

gVR

Ω4 = 2|ω02 | − ω 2 3β

1/2 ,

2|ω02 | > ω 2

(16a)


2

1

0

0

250

g

500

6 5 4 3 ΔQ 2 1 0

(b) dS = dD

(a) x∗±,S = x∗±,D

56

2

1

0

0

250

g

500

6 5 4 3 ΔQ 2 1 0

Fig. 15 Color code representation of ΔQ (= Q S /QD ) as a function of g and (a) x ∗±,S = x∗±,D and (b) d S = dD .

and

1/2 2Ω4 2 2 . (16b) |ω0 | + ω gVR = 3β and g(2) are obtained from ωr2 = ω 2 with x∗ = ± −c1 /β and 0, respectively. Two resonances occur for g(1) VR VR 2|ω02 | > ω 2 while one resonance for 2|ω02 | < ω 2 . In Fig. 13(a) Q is plotted for the system (1), where ω02 = 1 and ω = 0.5 so that 2|ω02 | > ω 2 . We observe only one resonance. In contrast to this for 2|ω02 | > ω 2 the Duffing oscillator displays two resonances. For 2|ω02 | < ω 2 there is no resonance for the signum nonlinear system as shown in Fig. 13(b) for ω02 = 1 and ω = 1.5. For this case the Duffing oscillator admits one resonance. and g(2) decrease with increase in the value of β . In contrast to this, In the Duffing oscillator both g(1) VR VR as shown in Fig. 14(a) for the system (1), gVR increases with increase in the value of |β |. In both systems, at resonance Qmax = 1/d ω except for |β | 1 in the system (1). Further, in the system (1) QL = 0 and becomes independent of β (Fig. 14(b)) while in the Duffing oscillator Q → 0 as g → ∞. (2)

6 Conclusions In the present paper the phenomenon of vibrational resonance is analysed in a piecewise linear system. An important and interesting significance of this system is that it possesses an exact analytical solution in the presence of an external biharmonic force. As the potentials of the system (1) and the Duffing oscillator have similar shapes, we have compared the features of vibrational resonance in these systems. Even though the exact analytical solution of the system (1) is known, determination of an analytical expression for its response amplitude is not feasible. On the other hand, without determining the analytical solution, an analytical expression for the response amplitude is obtained for the Duffing oscillator through a theoretical approach. In the Duffing oscillator, the resonance is always due to the matching of the resonant frequency ωr with the frequency ω of the external driving force. In the system (1) resonance is due to either ωr = ω or local minimization of ωr . In the Duffing oscillator Q → 0 as g → ∞, while in the system (1) QL = 0. In the system (1) gVR always increases with increase in the value of the parameter |β | for all the three physically interesting types of potential. In contrast to this, in the Duffing oscillator gVR always decreases with increases in |β |. For a certain range of values of ω02 and β , the system (1) exhibits a sigmoid type variation of Q with the control parameter g. Such a type of response is not realized in the Duffing oscillator system. When g = 0 and f = 0, the response amplitude Q(ω ) is always nonzero in the Duffing oscillator. That is, the output of the system is oscillating and contains the frequency ω . In contrast to this, in the system (1) for a wide range of values of the parameters of the system, oscillation death occurs for f = 0 and g = 0 (see Figs. 5(b) and 8). The addition of a high-frequency force induces oscillations containing the frequency ω . A detailed investigation of vibrational resonance in other piecewise linear systems


57

may bring out some more other common features of resonance in piecewise linear systems over the nonlinear systems such as the Duffing oscillator and quintic oscillator.

Acknowledgments KA acknowledges the support from University Grants Commission (UGC), India in the form of UGC-Rajiv Gandhi National Fellowship. MAFS acknowledges financial support from the Spanish Ministry of Economy and Competitiveness under Project No. FIS2013-40653-P.

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Controllability of Nonlinear Fractional Delay Integrodifferential Systems R. Joice Nirmala†, K. Balachandran Department of Mathematics, Bharathiar University, Coimbatore 641 046, India Submission Info Communicated by Albert C. J. Luo Received 13 April 2015 Accepted 1 July 2015 Available online 1 April 2016

Abstract In this paper we establish the sufficient conditions for controllability of nonlinear fractional delay integrodifferential systems. The results are obtained by using the solution representation of fractional delay differential equations and the application of Schauder’s fixed point theorem. Examples are provided to illustrate the results.

Keywords Fractional delay differential equation Controllability Mittag-Leffler function Laplace transform Fixed point theorem


1 Introduction Many authors considered the system which is governed by a principle of causality; that is, the future state of the system is independent of the past states and is determined solely by the present. If it is also assumed that the system is governed by an equation involving the state and rate of change of the state, then, generally, one is considering either ordinary or partial differential equations. However under closer scrutiny, it becomes apparent that the principle of causality is often only a first approximation to the true situation and that a more realistic model would include some of the past states of the system. Also, in some problems, it is meaningless not to have dependence on the past. The simplest type of past dependence in a differential equation is that in which the past dependence is through the state variable and not through the derivative of the state variable the so called retarded functional differential equations or retarded differential difference equations. Delay differential equations were initially introduced in the 18th century by Laplace and Condorcet. The principal difficulty in studying delay differential equations lies in its special transcendental character. The delay operator can be expressed in the form of an infinite series. Delay equations always lead to an infinite spectrum of frequencies. The determination of this spectrum requires a corresponding determination of zeros of certain analytic functions. Another approach to the delay problem is to look at the entire delay spectrum. Delay differential equations are often solved using numerical methods, asymptotic solutions, and graphical tools. Several attempts have been † Corresponding

author. Email address: [email protected] ISSN 2164 − 6376, eISSN 2164 − 6414/$-see front materials © 2016 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/DNC.2016.03.007

60

R. Joice Nirmala, K. Balachandran / Discontinuity, Nonlinearity, and Complexity 5(1) (2016) 59–73

made to find an analytical solution for delay differential equations by solving its characteristic equation under different conditions. A related study on delay differential equations has been initiated by Bellman and Cooke [1], Halaney [2], Hale [3], Oguztoreli [4] and Smith [5]. The variation of parameter formula for delay differential equation was given by Hale [3]. Shu [6] obtained the explicit representations of solutions of linear delay systems. Yi, Nelson and Ulsoy [7–9] obtained the variation of parameter formula for delay differential equations with state delay by using Lambert W function and they established the observability and controllability of delay differential equations. This equation has many applications in control problems, growth of single species, predator-prey model, distribution of albumin in blood stream in humans, the spread of measles in metropolitan area and water distribution in an irrigation main canal pool. Controllability is one of the qualitative properties of a dynamical system and is of particular importance in control theory. Controllability generally means that it is possible to steer a dynamical system from an arbitrary initial state to an arbitrary final state using the control. It plays a major role in both finite and infinite dimensional spaces that is systems represented by ordinary differential equations and partial differential equations respectively. Many articles and monographs on control theory and its applications were published during the early years. Controllability of nonlinear delay system was discussed by Dauer and Gahl [10], the controllability of perturbed nonlinear system was studied by Balachandran and Dauer [11]. Kalmka [12, 13] has made significant contributions on the controllability of linear dynamical systems and he discussed the use of Schauder’s fixed point theorem for nonlinear control problems. Further, also see [14], Kalmka [15, 16] studied controllability and relative controllabilty of nonlinear systems with delay in control. Wiess [17] studied the controllability of delayed differential systems. Manzanilla, Marmol and Vanegas [18] investigated the controllability of differential equation with delayed and advanced arguments. Recently fractional differential equations have attracted the attention of many researchers. The detailed descriptions of fractional differential equations are given in [19–25]. Machoda et.al [26–28] describes the fractional dynamics of DNA and fractional digital control systems. Balachandran, Park and Trujillo [29] discussed the controllability of nonlinear fractional dynamical systems. The finite time controllability of fractional order system was given by Adams and Hartley [30]. Bettayeb and Djennoune [31] obtained results on the controllability and observability of fractional dynamical systems. The fractional delay differential equations have found many applications in control theory, agriculture, chaos and bioengineering. Si-Ammour et.al. [32] obtained the sliding mode control for linear fractional systems with input and state delays. Fractional order controller robust to time delay for water distribution in an irrigation main canal pool is studied by Feliu et.al. [33]. Wang and Yu [34] studied the chaos in the fractional order logistic delay systems. Chen et.al. [35] worked on the robust controllabilty of interval fractional order linear time invariant systems. The controllability and observability for fractional control system is discussed by Shamardan and Moubarak [36]. The main difficulty arising in the investigation of controllability of nonlinear fractional delay dynamical system is the lack of general methods for solving nonlinear differential and functional differential difference equations. Fixed point technique is the most powerful method to obtain the controllability results for nonlinear dynamical systems. It has been employed successfully in controllability problems of dynamical systems with time-delay such as time varying delay, multiple delay and distributed delay. The problem of controllability of nonlinear systems and integrodifferential systems including delay has been studied by many researchers [37–43]. In this paper, we establish the controllability of nonlinear fractional delay integrodifferential systems by means of Schauder’s fixed point theorem. Examples are provided to illustrate the theory. 2 Preliminaries In this section we introduce some well known fractional operators and special functions, along with some properties of special functions, [20, 21, 23, 24, 44].


61

Definition 1. Let f be a real or complex valued function of the variable t > 0 and s is a real or complex parameter. The Laplace transform of f is ˆ ∞ e−st f (t)dt, for Re(s) > 0. F(s) = 0

Definition 2. The Caputo fractional derivative of order α > 0, n − 1 < α ≤ n, is defined as ˆ t 1 C α D0 f (t) = (t − s)n−α −1 f (n) (s)ds, Γ(n − α ) 0 where the function f (t) has absolutely continuous derivative upto order n − 1. The Laplace transform of Caputo derivative is Lt [Dtα x(t)](s) = sα L[x(t)](s) −

n−1

∑ xk (0)sα −1−k , n − 1 < α ≤ n.

k=0

The Mittag-Leffler functions of various types are defined as Eα (z) = Eα ,1 (z) = Eα ,β (z) =

∞

∞

zk , z ∈ C, Re(α ) > 0, ∑ k=o Γ(α k + 1) zk

∑ Γ(α k + β ) ,

z, β ∈ C, Re(α ) > 0,

(2)

k=0

∞

γ

Eα ,β (−λ t α ) =

(γ )k (−λ )k

∑ k!Γ(α k + β ) t α k ,

transform of Mittag-Leffler functions (1), (2) and (3) are defined as L[Eα ,1 (±λ t α )](s) =

(3)

k=0

where (γ )n is a Pochhamer symbol which is defined as γ (γ + 1) · · · (γ + n − 1) and (γ )n =

Γ(γ + n) . The Laplace Γ(γ )

sα −1 , Re(α ) > 0, (sα ± λ )

L[t β −1 Eα ,β (±λ t α )](s) = γ

(1)

L[t β −1 Eα ,β (±λ t α )](s) =

(4)

sα −β , Re(α ) > 0, Re(β ) > 0, (sα ± λ )

(5)

sαγ −β , Re(s) > 0, Re(β ) > 0, |λ s−α | < 1. (sα ± λ )γ

(6)

Definition 3. An important function occurring in electrical systems is the delayed unit step function 1, t ≥ a, ua (t) = 0, t < a. The Laplace transformation is L[ua (t)](s) =

e−as , s > 0. s

If F(s) = L[ f (t)](s) for Re(s)>0, then F(s − a) = L[eat f (t)](s),

62


and L[ua (t) f (t − a)](s) = e−as F(s), a ≥ 0, and also we have L−1 [e−as F(s)](t) = ua (t) f (t − a).

(7)

Consider the linear fractional delay differential equation of the form C

Dα x(t) = Ax(t) + Bx(t − 1) + f (t), 0 < α ≤ 1,

(8)

x(t) = φ (t), −1 < t ≤ 0, where x ∈ Rn , A and B are n × n matrices, x ∈ Rn , φ (t) is a continuous function on [−1, 0] and f is a real valued continuous function in Rn . By taking Laplace transform on both sides of (8) we get ˆ ∞ ˆ ∞ ˆ ∞ x(t)e−st dt + B e−st x(t − 1)dt + e−st f (t)dt, sα X (s) − sα −1 φ (0) = A 0

0

0

and by simple calculations we have α

α −1

s X (s) − s

ˆ

0

φ (0) = AX (s) + Be e−sτ x(τ )dτ −1 ˆ ∞ ˆ ∞ +Be−s e−sτ x(τ )dτ + e−st f (t)dt, −s

0

α

α −1

s X (s) − s

−s

φ (0) = AX (s) + Be

0

ˆ

0

−1

e−sτ φ (τ )dτ + Be−s X (s) + F(s),

then, α

−s

α −1

(s I − A − Be )X (s) = s where F(s) =

´∞ 0

−s

φ (0) + Be

ˆ

0

−1

e−sτ φ (τ )dτ + F(s),

e−st f (t)dt. Thus e−s sα −1 ] φ (0) + B[ ] X (s) = [ α s I − A − Be−s sα I − A − Be−s F(s) +[ α ]. s I − A − Be−s

ˆ

0 −1

e−sτ φ (τ )dτ

Taking inverse Laplace transform we get sα −1 ](t)φ (0) sα I − A − Be−s ˆ 0 1 −1 −1 −s ](t) ∗ L [e e−sτ φ (τ )dτ ](t) +BL [ α s I − A − Be−s −1 1 ](t). +L−1 [F(s)](t) ∗ L−1 [ α s I − A − Be−s

x(t) = L−1 [

Let Xα (t) = L−1 [sα −1 (sα I − A − Be−s)−1 ](t),


63

and by using the convolution of Laplace transform L−1 [

1 sα −1 1 −1 ](t) = L [ ](t) ∗ L−1 [ α −1 ](t) sα I − A − Be−s sα I − A − Be−s s t α −2 = Xα (t) ∗ Γ(α − 1) ˆ t (t − s)α −2 = Xα (s)ds 0 Γ(α − 1) = t α −1 Xα ,α (t),

where Xα ,α (t) = t

1−α

ˆ 0

t

(t − s)α −2 Xα (s)ds. Γ(α − 1)

Define ω (t) : [−1, ∞) → [0, 1] by ω (t) = 0 for t ≥ 0 and ω (t) = 1 for t < 0. The function φ (t) is extended to (−1, ∞) by defining φ (t) = φ (0) for t ≥ 0, then ˆ 0 ˆ 1 −s −st −s e φ (t)dt = e e−s(−1+λ ) φ (−1 + λ )dλ e −1 ˆ ∞ 0 = e−sλ φ (−1 + λ )ω (−1 + λ )dλ 0

= L[φ (−1 + .)ω (−1 + .)](t). Thus, ˆ t x(t) = Xα (t)φ (0) + B (t − s)α −1 Xα ,α (t − s)φ (−1 + s)ω (−1 + s)ds 0 ˆ t + (t − s)α −1 Xα ,α (t − s) f (s)ds, 0

and so ˆ

1

(t − s)α −1 Xα ,α (t − s)φ (−1 + s)ds x(t) = Xα (t)φ (0) + B 0 ˆ t + (t − s)α −1 Xα ,α (t − s) f (s)ds. 0

Hence ˆ 0 (t − s − 1)α −1 Xα ,α (t − s − 1)φ (s)ds x(t) = Xα (t)φ (0) + B −1 ˆ t + (t − s)α −1 Xα ,α (t − s) f (s)ds.

(9)

0

Now, consider the linear fractional delay dynamical system with control of the form C

Dα x(t) = Ax(t) + Bx(t − 1) +Cu(t), t ∈ J = [0, T ], x(t) = φ (t), −1 < t ≤ 0,

(10)

64


where x ∈ Rn , u ∈ Rm , A and B are n × n matrices, C is n × m matrix. The solution of (10) is given by ˆ 0 (t − s − 1)α −1 Xα ,α (t − s − 1)φ (s)ds x(t) = Xα (t)φ (0) + B −1 ˆ t + (t − s)α −1 Xα ,α (t − s)Cu(s)ds.

(11)

0

3 Linear Systems In this section we consider the linear system and established the necessary and sufficient conditions for the controllability of the system (10). Definition 4. System (10) is said to be completely controllable on J if, for every initial function φ (t) on [−1, 0] and x1 ∈ Rn , there exists a continuous control function u on J such that the solution x(t) of (10) satisfies x(T ) = x1 . Now we rewrite the solution (11) as ˆ xL (t) = xL (t; φ ) +

t 0

(t − s)α −1 Xα ,α (t − s)Cu(s)ds, for t ∈ J,

(12)

where ˆ xL (t; φ ) = Xα (t)φ (0) + B

0 −1

(t − s − 1)α −1 Xα ,α (t − s − 1)φ (s)ds.

Define the controllability Grammian W as ˆ T (Xα ,α (T − s)C)(Xα ,α (T − s)C)∗ ds, W=

(13)

0

where ∗ denotes the matrix transpose. Theorem 1. The system (10) is completely controllable on J if and only if controllability Grammian W is nonsingular. Proof. Assume W is nonsingular. Let φ (t) be continuous on [−1, 0] and let x1 ∈ Rn . Let u be the control function given by u(t) = (T − t)1−α (Xα ,α (T − t)C)∗W −1 (x1 − xL (T ; φ )), for t ∈ J.

(14)

Substituting t = T in the solution (12) we get ˆ xL (T ) = xL (T ; φ ) +

T 0

(T − s)α −1 Xα ,α (T − s)Cu(s)ds,

and using (14) in (15) we have ˆ xL (T ) = xL (T ; φ ) +

0

T

(T − s)α −1 (Xα ,α (T − s)C)(T − s)1−α

×(Xα ,α (T − s)C)∗W −1 (x1 − xL (T ; φ ))ds = x1 .

(15)


65

Now, assume that W is singular. Then, there exists a vector y = 0, such that y∗Wy = 0. It follows that ˆ

T 0

y∗ (Xα ,α (T − s)C)(y∗ (Xα ,α (T − s)C))∗ ds = 0.

Therefore, y∗ (Xα ,α (T − s)C) = 0 for s ∈ J. Consider the zero initial function φ = 0 and the final point x1 = y. Since the system is controllable there exists a control u(t) on J that steers the response to x1 = y at t = T, that is xL (T ) = y. From φ = 0, xL (T, φ ) = 0 and y∗ y = 0 for y = 0. On the other hand ˆ y = xL (T ) =

T

0

(T − s)α −1 Xα ,α (T − s)Cu(s)ds,

and hence ∗

y y=

ˆ 0

T

y∗ (T − s)α −1 Xα ,α (T − s)Cu(s)ds = 0.

This is a contradiction for y = 0. Hence W is nonsingular. 4 The Nonlinear Integrodifferential Systems Consider the nonlinear fractional delay integrodifferential system of the form ˆ t C α D x(t) = Ax(t) + Bx(t − 1) +Cu(t) + f t, x(t − 1), g(t, s, x(s − 1))ds , t ∈ J, 0

x(t) = φ (t), −1 < t ≤ 0,

(16)

where x ∈ Rn , u ∈ Rm , A and B are n × n matrices, C is n × m matrix with n > m and the nonlinear functions f : J × Rn × Rn → Rn and g : J × J × Rn → Rn are continuous. For each z ∈ Cn (J), the linear system ˆ t C α D x(t) = Ax(t) + Bx(t − 1) +Cu(t) + f t, z(t − 1), g(t, s, z(s − 1) ds), (17) 0

x(t) = φ (t), −1 < t ≤ 0 has the solution of the form

x(t) = Xα (t)φ (0) ˆ t ˆ 0 α −1 (t − s − 1) Xα ,α (t − s − 1)φ (s)ds + (t − s)α −1 Xα ,α (t − s)Cu(s)ds +B −1 0 ˆ s ˆ t g(s, τ , z(τ − 1))dτ ds. + (t − s)α −1 Xα ,α (t − s) f s, z(s − 1), 0

(18)

0

Assume the following condition: (H1) f : J × Rn × Rn → Rn and g : J × J × Rn → Rn are continuous and there exists a positive constant M > 0 such that ˆ t f t, z(t − 1), g(t, s, z(s − 1) ds) ≤ M, (19) 0

for all t, s ∈ J and z ∈ Cn (J).

66


Theorem 2. If the linear system (10) is completely controllable and the hypothesis (H1) holds, then the nonlinear fractional delay integrodifferential system (16) is completely controllable. Proof. Define the operator Φ : Cn (J) → Cn (J) by Φ(z)(t) = Xα (t)φ (0) ˆ t ˆ 0 (t − s − 1)α −1 Xα ,α (t − s − 1)φ (s)ds + (t − s)α −1 Xα ,α (t − s)Cu(s)ds +B −1 0 ˆ s ˆ t g(s, τ , z(τ − 1))dτ ds. + (t − s)α −1 Xα ,α (t − s) f s, z(s − 1), 0

(20)

0

where the control u(t) is defined as u(t) = (T − t)1−α (Xα ,α (T − t)C)∗W −1 [x1 − xL (T ; φ ) ˆ s ˆ T (T − s)α −1 Xα ,α (T − s) f s, z(s − 1), g(s, τ , z(τ − 1))dτ dt]. − 0

(21)

0

Let a1 =

sup |Xα ,α (t − s)|, a2 = sup |xL (t; φ )|.

0≤s≤t≤T

t∈J

Then |u(t)| ≤ T 1−α ||C∗ ||a1 ||W −1 ||[|x1 | + a2 + a1

Tα M]. α

α

Take T 1−α ||C∗ ||a1 ||W −1 ||[|x1 | + a2 + a1 Tα M] = K. Define the closed convex subset Q(r) = {z ∈ Cn (J) : ||z|| ≤ r}, where r = a2 + a1

Tα Tα ||C||K + a1 M. α α

From this, it is easy to see that Φ maps Q(r) into itself. Now we need to show the map Φ has a fixed point. Since f is continuous, it follows that Φ is continuous and hence it is completely continuous by Arzela-Ascoli theorem. Hence, by Schauder’s fixed point theorem, there exists a fixed point z ∈ Q(r) such that Φz = z = x. By substituting (21) into (20) we get x(T ) = x1 . Hence the system is completely controllable on J. Next, consider the nonlinear fractional delay integrodifferential system of the form ˆ t C α D x(t) = Ax(t) + Bx(t − 1) +Cu(t) + f t, x(t − 1), g(t, s, x(s − 1))ds, u(t) , t ∈ J, 0

x(t) = φ (t), −1 < t ≤ 0,

(22)

where x ∈ Rn , u ∈ Rm , A and B are n × n matrices, C is n × m matrix with n > m and the nonlinear functions f : J × Rn × Rn × Rm → Rn and g : J × J × Rn → Rn are continuous. The solution of (22) is as follows ˆ 0 (t − s − 1)α −1 Xα ,α (t − s − 1)φ (s)ds (23) x(t) = Xα (t)φ (0) + B −1 ˆ s ˆ t g(s, τ , x(τ − 1))dτ , u(s) ds. + (t − s)α −1 Xα ,α (t − s) Cu(s) + f s, x(s − 1), 0

0


67

We are now ready to obtain our main result on the controllability ofńonlinear fractional delay system. For this t we will take p = (y, z, u) and let |p| = |y| + |z| + |u|, where z(t) = 0 g(t, s, x(s − 1))ds. Assume the following condition: (H2) g : J × J × Rn → Rn is continuous and satisfies the following condition ˆ ˆ t t g(t, s, x(s − 1))ds ≤ sup |a(t, s)|ds ||x||, 0

(24)

0

´t and also sup( 0 |a(t, s)|ds) < 1. Theorem 3. Let the continuous function f satisfy the condition | f (t, p)| = 0, |p| |p|→∞

(25)

lim

uniformly in t ∈ J. Suppose that the system (10) is completely controllable on J and the hypothesis (H2) is satisfied. Then the system (22) is completely controllable on J. Proof. Let φ (t) be continuous on [−1, 0] and let x1 ∈ Rn . Let Q be the Banach space of all continuous functions (x, u) : [−1, T ] × [0, T ] → Rn × Rm , with norm ||(x, u)|| = ||x|| + ||u||, where ||x|| = sup|x(t)| for t ∈ [−1, T ], and ||u|| = sup|u(t)| for t ∈ [0, T ]. Define Φ : Q → Q by T (x, u) = (y, v), where v(t) = (T − t)1−α (Xα ,α (T − t)C)∗W −1 [x1 − xL (T, φ ) ˆ t ˆ T α −1 (T − t) Xα ,α (T − t) f (t, x(t − 1), g(t, s, x(s − 1))ds, u(t))dt], − 0

0

ˆ

t

y(t) = xL (t; φ ) + (t − s)α −1 Xα ,α (t − s)Cv(s)ds 0 ˆ t ˆ t g(s, τ , x(τ − 1))dτ , u(s))ds, + (t − s)α −1 Xα ,α (t − s) f (s, x(s − 1), 0

0

for t ∈ J and y(t) = φ (t), t ∈ [−1, 0]. Let a1 =

sup |Xα ,α (t − s)C|, a2 = |W −1 |,

0≤s≤t≤T

a3 = sup |xL (t; φ )| + |x1 |, a4 = sup |Xα ,α (t − s)|, t∈J

b = max(α

t,s∈J

−1 α

T a1 , 1), c1 = 8ba1 a2 a4 T α α −1 , c2 = 8a4 T α α −1 ,

d1 = 8ba1 a2 a3 , d2 = 8a3 , c = max{c1 , c2 }, d = max{d1 , d2 }, ˆ s g(s, τ , x(τ − 1))dτ , u(s) | f | = sup f s, x(s − 1), s∈J

0

68


then, |v(t)| ≤ a1 a2 (a3 + a4

Tα | f |) α

d1 c1 + |f| 8b 8b 1 ≤ (d + c| f |), 8b

≤

(26)

for t ∈ J, and Tα Tα ||v||a1 + a4 |f| α α c d ≤ + b||v|| + | f |. 8 8

|y(t)| ≤ a3 +

(27)

By the Proposition 3.1 in [45] f satisfies the following condition : for each pair of positive constants c and d, there exists a positive constant r such that, if |(x, u)| ≤ r, then c| f (t, p)| + d ≤ r,

(28)

for all t ∈ J. It follows that, for given c and d, if r is a constant such that the implication in condition (28) is satisfied, then any r1 such that r < r1 will also satisfy the implication in (28). Now, take c and d as given above, and let r be chosen so that the implication in (28) is satisfied and r sup |φ (t)| ≤ . 4 −1≤t≤0 Therefore, if ||x|| ≤

r r and ||u|| ≤ , 4 4

then |y| + |z| + |u| ≤ r, for s ∈ J. It follows that d + c sup | f s, x(s − 1), 0 g(s, τ , x(τ − 1))dτ , u(s) | ≤ r, for s ∈ J. Therefore, |v(t)| ≤ r/8b for all t ∈ J and hence ||v(t)|| ≤ r/8b. It follows that |y(t)| ≤ r/8 + r/8 for all t ∈ J and hence that ||y(t)|| ≤ r/4. Note that, if

´s

Q(r) = {(x, u) ∈ Q/ ||x|| ≤ r/4 and ||u|| ≤ r/4},

(29)

then, Φ : Q(r) → Q(r). Our objective is to show that Φ has a fixed point. Since f is continuous, it follows that Φ is continuous. Let Q0 be a bounded subset of Q. Consider a sequence {(y j , v j )} contained in Φ(Q0 ), where we let (y j , v j ) = Φ(x j , u j ), ´s for some (x j , u j ) ∈ Q0 , for j = 1, 2, . . .. Since f is continuous | f s, x j (s − 1), 0 g(s, τ , x j (τ − 1))dτ , u j (s) | is uniformly bounded for all s ∈ J, and j = 1, 2, 3, . . . . It follows that {(y j , v j )} is a bounded sequence in Q. Hence {v j (t)} is equicontinuous and a uniformly bounded sequence on [0, T ]. Since {y j (t)} is a uniformly bounded and equicontinuous sequence on [−1, T ], an application of Ascoli’s theorem yield a further subsequence of {(y j , v j )} which converges in Q to some (y0 , v0 ). It follows that Φ(Q0 ) is sequentially compact, hence, the closure is sequentially compact. Thus, Φ is completely continuous. Since Q(r) is closed, bounded and convex, the Schauder’s fixed point theorem implies that Φ has a fixed point (x, u) ∈ Q(r).


69

It follows that, ˆ

t

x(t) = xL (t; φ ) + (t − s)α −1 Xα ,α (t − s)Cu(s)ds 0 ˆ s ˆ t α −1 g(s, τ , x(τ − 1))dτ , u(s) ds, + (t − s) Xα ,α (t − s) f s, x(s − 1), 0

(30)

0

for t ∈ J and x(t) = φ (t) for t ∈ [−1, 0]. Hence, x(t) is a solution of the system and ˆ

T (T − s)α −1 [Xα ,α (T − s)C](T − s)1−α [Xα ,α (T − s)C]∗ dsW −1 x1 − xL (T ; φ ) x(T ) = xL (T ; φ ) + 0 ˆ s ˆ T (T − s)α −1 Xα ,α (T − s) f s, x(s − 1), g(s, τ , x(τ − 1))dτ , u(s) − 0 0 ˆ T ˆ s + (T − s)α −1 Xα ,α (T − s) f s, x(s − 1), g(s, τ , x(τ − 1))dτ , u(s) .

= x1 .

0

0

Hence the system (22) is completely controllable. 5 Examples Example 1. Consider the linear fractional delay dynamical system C

1

D 2 x(t) = Ax(t) + Bx(t − 1) +Cu(t),

(31)

x(t) = Φ(t), −1 < t ≤ 0, 1 01 00 0 x1 (t) and Φ(t) = with initial condition x(0) = where A = ,B= ,C= , x(t) = 1 x2 (t) 10 10 1 1 1 x1 (1) x1 (0) and final condition x(1) = . Here C D 2 x1 (t) = x2 (t) and C D 2 x2 (t) = x1 (t) + x1 (t − 1) + u(t) x2 (0) x2 (1) By taking Laplace transform on both sides of the equation, we get

1 2

s X (s) − s

1 2 −1

ˆ x(0) = A

∞

−st

x(t)e

0

ˆ dt + B

∞

−st

e

0

ˆ x(t − 1)dt +C

∞

e−st u(t)dt,

0

By simple calculation we have 1 2

− 12

−s

(s I − A − Be )X (s) = s

−s

ˆ

+ Be

0 −1

e−sτ dτ +CU (s),

and the inverse Laplace transform yields 1

−1

x(t) = L (

s− 2 1

s 2 I − A − Be−s

−1

)(t) + BL (

+CL−1 (U (s))(t) ∗ L−1 (

1 1 2

1 1

s 2 I − A − Be−s

s I − A − Be−s

)(t).

−1

−s

)(t) ∗ L (e

ˆ

0 −1

e−sτ dτ )(t)

70


Now 1

L−1 (

s− 2 1

s 2 I − A − Be−s

=

1

∞

∑ BnL−1(

)(t) =

n=0 ∞

s− 2 e−ns 1

(s 2 I − A)n+1

∑ un (t)Bn(t − n)

n=0

=

1 2n

)(t) 1

E n+1 (A(t − n) 2 ) 1 1 , n+1 2 2

1

1

(A(t − n) 2 ), t ≥ n, (t − n) 2 n E n+1 1 1 , n+1 2 2

t < n,

0,

(32)

and L (

1

1

−1

s

1 2

s− 2

−1

I − A − Be−s

)(t) = L (

)(t) ∗ L−1 (

1

)(t) s ˆ t 3 1 1 (t − s)− 2 (s − n) 2 n Bn E n+1 (A(s − n) 2 )ds = 1 1 1 , n+1 2 2 n Γ(− 2 ) 1 2

I − A − Be−s

1

1 s− 2

1

1

(A(t − n) 2 ). = Bn (t − n) 2 n− 2 E n+1 1 1 , (n+1)

(33)

2 2

Define ω (t) : [−1, ∞) → [0, 1] where ω (t) = 0 for t ≥ 0 and ω (t) = 1 for t < 0. The function φ (t) is extended to [−1, ∞) by defining φ (t) = φ (0) for t ≥ 0, then ˆ 0 ˆ 1 −s −st −s e dt = e e−s(−1+λ ) dλ e −1 0 ˆ ∞ = e−sλ ω (−1 + λ )dλ = L[ω (−1 + .)](t). 0

Hence, by using the results (32) and (33), the solution of (31) takes the form x(t) =

[t]

∑ Bn(t − n)

n=0

[t]

+B ∑ B

n

ˆ

n=0 [t]

+C ∑ Bn n=0

=

1 2n

2 2

t−n 0

ˆ

t−n

[t]

n=0 [t]

+C ∑ Bn n=0

1

1

ˆ

1 2n

0

−1

ˆ

1

1

2

1

E n+1 (A(t − n) 2 ) 1 1 , n+1 2 2

1

1

1

(t − s − n − 1) 2 n− 2 E n+1 (A(t − s − n − 1) 2 )ds 1 1 , (n+1)

t−n 0

1

(t − s − n) 2 n− 2 E n+1 (A(t − s − n) 2 )u(s)ds 1 1 , n+ 1 2 2

0

[t]

+B ∑ Bn

1

(t − s − n) 2 n− 2 E n+1 (A(t − s − n) 2 )ω (−1 + s)ds 1 1 , (n+1) 2 2

∑ Bn(t − n)

n=0

1

E n+1 (A(t − n) 2 ) 1 1 , n+1

2 2

1

1

1

(t − s − n) 2 n− 2 E n+1 (A(t − s − n) 2 )u(s)ds, 1 1 , n+ 1 2 2

2

where [·] is the greatest integer function. Now consider the controllability on [0, 1]. Here [t] = 0 therefore the solution of (31) on [0, 1] is ˆ 0 1 1 1 2 (t − s − 1)− 2 E 1 , 1 (A(t − s − 1) 2 )ds x(t) = E 1 (At ) + B 2 2 2 −1 ˆ t 1 1 (34) + (t − s)− 2 E 1 , 1 (A(t − s) 2 )Cu(s)ds, 0

2 2


and, on further simplification, 1

1

ˆ

1

x(t) = E 1 (At 2 ) + Bt 2 E 1 , 3 (At 2 ) + 2

2 2

1

0

t

1

1

(t − s)− 2 E 1 , 1 (A(t − s) 2 )Cu(s)ds. 2 2

71

(35)

1

where X 1 (t) = E 1 (At 2 ) and X 1 , 1 (t) = E 1 , 1 (At 2 ). The Mittag-Leffler matrix function is given by 2 2 2 2 2 2 1 E1,1 (t) t 2 E1, 3 (t) 1 2 , E 1 ,1 (At 2 ) = 1 2 t 2 E1, 3 (t) E1,1 (t) 2

and

1 2

E 1 , 1 (A(1 − s) ) = 2 2

− 12

Take S1 = (1 − s)

1

(1 − s)− 2 E1, 1 (1 − s) E1,1 (1 − s)

2

E1,1 (1 − s) 1

(1 − s)− 2 E1, 1 (1 − s)

.

2

E1, 1 (1 − s) and S2 = E1,1 (1 − s). Hence 2

1 S1 S2 2 . E 1 , 1 (A(1 − s) ) = 2 2 S2 S1

(36)

By simple matrix calculation, we have the controllability Grammian matrix as ˆ 1 1 1 (E 1 , 1 (A(1 − s) 2 )C)(E 1 , 1 (A(1 − s) 2 )C)∗ ds, W= 2 2 2 2 0 ˆ 1 S1 S2 S22 ds, = S12 S1 S2 0 which is positive definite. Hence the system (31) is completely controllable on [0, 1]. Example 2. Consider the following nonlinear fractional delay integrodifferential system ˆ t C 21 D x(t) = Ax(t) + Bx(t − 1) +Cu(t) + f t, x(t − 1), g(t, s, x(s − 1))ds ,t ∈ [0, 1], 0

x(t) = Φ(t), −1 < t ≤ 0,

(37)

where A, B, C and Φ(t) are as above and the function f is taken as ⎞ ⎛ ˆ t ´ t −x 0(s−1) 1 ⎠. ds g(t, s, x(s − 1))ds) = ⎝ f (t, x(t − 1), 0e 0 1 + x21 (t − 1) + x22 (t − 1) It is easy to see that f is uniformly bounded on J. Since the linear system is controllable and f satisfies the hypothesis (H1), then, by Theorem 2 the system (37) is controllable on [0, 1]. Example 3. Consider the following nonlinear fractional delay integrodifferential system ˆ t C 21 D x(t) = Ax(t) + Bx(t − 1) +Cu(t) + f t, x(t − 1), g(t, s, x(s − 1))ds, u(t) ,t ∈ [0, 1], 0

x(t) = Φ(t), −1 < t ≤ 0,

(38)

where A, B, C and Φ(t) are as above and the function f is taken as ⎞ ⎛ ˆ t 0 ´t −x1 (s−1) ds ⎠. g(t, s, x(s − 1) ds, u(t)) = ⎝ f t, x(t − 1), 0 sin(s)e 0 1 + x21 (t − 1) + x22 (t − 1) + u2 (t) Since the linear system is controllable and g satisfies the hypothesis (H2), then, by Theorem 3 the system (38) is controllable on [0, 1].

72


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Existence of Stationary Solutions for some Systems of Integro-Differential Equations Vitali Vougalter1†, Vitaly Volpert2 1 Department 2

of Mathematics, University of Toronto, Toronto, Ontario, M5S 2E4, Canada Institute Camille Jordan, UMR 5208 CNRS, University Lyon 1, Villeurbanne, 69622, France Abstract

Submission Info

The article deals with the existence of solutions of a system of nonlocal reaction-diffusion equations which appears in population dynamics. The proof relies on a fixed point technique. Solvability conditions for elliptic operators in unbounded domains which fail to satisfy the Fredholm property are being used.

Communicated by Valentin Afraimovich Received 3 September 2015 Accepted 17 September 2015 Available online 1 April 2016 Keywords Nonlinear diffusion equations Non Fredholm operators Sobolev spaces


1 Introduction In the present article we establish the existence of stationary solutions of the system of N ≥ 2 nonlocal reactiondiffusion equations

∂ us = Ds Δus + ∂t

ˆ Rd

Ks (x − y)gs (u(y,t))dy + fs (x),

1 ≤ s ≤ N,

(1)

which appears in cell population dynamics. The space variable x is correspondent to the cell genotype, us (x,t) are densities for the various groups of cells as functions of their genotype and time and u(x,t) = (u1 (x,t), u2 (x,t), . . . , uN (x,t))T . The right side of (1) describes the evolution of cell densities caused by cell proliferation, mutations and cell influx. In this context, the diffusion terms are correspondent to the change of genotype by means of small random mutations, while the integral terms describe large mutations. Here gs (u) are the rates of cell birth dependent upon u (density dependent proliferation), and the functions Ks (x − y) show the proportions of newly born cells changing their genotype from y to x. Let us assume that they depend on the distance between the genotypes. Finally, the last term in the right side of (1) describes the influxes of cells for different genotypes. Note that the single equation analogous to (1) has been studied recently in [23] and the case of the superdiffusion has been treated in [24]. † Corresponding

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


76

Vitali Vougalter, Vitaly Volpert / Discontinuity, Nonlinearity, and Complexity 5(1) (2016) 75–84

We will investigate the existence of solutions of the system of equations ˆ Ks (x − y)gs (u(y))dy + fs (x) = 0, 1 ≤ s ≤ N. Δus + Rd

(2)

We set here Ds = 1. Otherwise we can divide each equation by the corresponding diffusion coefficient. Let us consider the situation when the linear part of this operator fails to satisfy the Fredholm property and conventional methods of nonlinear analysis may not be applied. We will use solvability conditions for non Fredholm operators along with the method of contraction mappings. Let us consider the problem −Δu +V (x)u − au = f (3) with u ∈ E = H 2 (Rd ) and f ∈ F = L2 (Rd ), d ∈ N, a is a constant and the scalar potential function V (x) either vanishes or tends to 0 at infinity. When a ≥ 0, the essential spectrum of the operator A : E → F correspondent to the left side of problem (3) contains the origin. Consequently, this operator fails to satisfy the Fredholm property. Its image is not closed, for d > 1 the dimension of its kernel and the codimension of its image are not finite. The present work is devoted to the studies of some properties of the operators of this kind. Note that elliptic problems which contain non Fredholm operators were treated actively in recent years. Approaches in weighted Sobolev and Hölder spaces were developed in [2–4, 6, 7]. The Schrödinger type operators without Fredholm property were studied using the methods of the spectral and the scattering theory in [12, 14–16, 18]. The Laplacian operator with drift from the point of view of the non Fredholm operators was studied in [17] and linearized Cahn-Hilliard equations in [19] and [21]. Nonlinear non Fredholm elliptic problems were treated in [20] and [22]. Significant applications to the theory of reaction-diffusion equations were developed in [9, 10]. Operators without Fredholm property arise also when studying wave systems with an infinite number of localized traveling waves (see [1]). In particular, when a = 0 the operator A is Fredholm in some properly chosen weighted spaces (see [2], [3, 4, 6, 7]). But the situation when a = 0 is significantly different such that the approach developed in these works cannot be applied. Let us set Ks (x) = εs Ks (x) with εs ≥ 0, such that

ε := max1≤s≤N εs and suppose that the following assumption holds. Assumption 1. Let 1 ≤ s ≤ N, such that fs (x) : R5 → R, fs (x) ∈ L1 (R5 ) and ∇ fs (x) ∈ L2 (R5 ). Moreover, fs (x) is nontrivial for some s. Assume also that Ks (x) : R5 → R, such that Ks (x) ∈ L1 (R5 ) and ∇Ks (x) ∈ L2 (R5 ). Furthermore, N

K 2 :=

∑ Ks (x)2L (R ) > 0 1

5

s=1

and Q2 :=

N

∑ ∇Ks (x)2L (R ) > 0. 2

5

s=1

Note that as distinct from the preceding work [23] dealing with a single integro-differential equation, we assume here for the technical reason the square integrability of the gradients of kernels involved in the nonlocal terms of our system of equations. The way we choose the space dimension is related to the solvability conditions for linear elliptic problems in unbounded domains (see [22]). There are certain solvability conditions for d < 5 but solvability conditions are not required for d ≥ 5 (see the Appendix). Let us study here only the case of d = 5. We will not consider the problem in dimensions d > 5 to avoid extra technicalities since the proof will rely on similar ideas and no orthogonality conditions for the solvability of equations (8) are required analogously to d = 5 (see Lemma 7


77

of [22]). From the perspective of applications, the space dimension is not limited to d = 3 because the space variable is correspondent to cell genotype and not the usual physical space. By virtue of the Sobolev inequality (see e.g. p.183 of [11]) under the assumption above we have fs (x) ∈ L2 (R5 ),

1 ≤ s ≤ N.

We consider the Sobolev space H 3 (R5 , RN ) of vector functions 3

{u(x) : R5 → RN | us (x) ∈ L2 (R5 ), (−Δ) 2 us ∈ L2 (R5 ), 1 ≤ s ≤ N} equipped with the norm u2H 3 (R5 ,RN ) :=

N

N

∑ us 2H (R ) = ∑ {us 2L (R ) + (−Δ) 3

5

2

s=1

5

s=1

Also, u2L2 (R5 ,RN ) :=

3 2

us 2L2 (R5 ) }.

(4)

N

∑ us 2L (R ) . 2

5

s=1

3

The operator (−Δ) 2 is defined by virtue of the spectral calculus. By means of the Sobolev embedding we have φ L∞ (R5 ) ≤ ce φ H 3 (R5 ) .

(5)

Here ce > 0 is the constant of the embedding. The hat symbol will stand for the standard Fourier transform, namely ˆ 1 φ(p) = φ (x)e−ipx dx. (6) 5 5 (2π ) 2 R This enables us to express the Sobolev norm of a function as ˆ 2 (1 + |p|6 )|φ(p)|2 d p. φ H 3 (R5 ) = R5

(7)

When the nonnegative parameters εs vanish, we arrive at the standard Poisson equations −Δus = fs (x),

1 ≤ s ≤ N.

(8)

Assumption 1 via Lemma 7 of [22] implies that problem (8) admits a unique solution u0,s (x) ∈ H 2 (R5 ) and no orthogonality relations are required. Clearly, ∇(−Δus ) = ∇ fs (x) ∈ L2 (R5 ). Thus, for the unique solution of the linear problem (8) we arrive at u0,s (x) ∈ H 3 (R5 ), such that u0 (x) = (u0,1 (x), u0,2 (x), ..., u0,N (x))T ∈ H 3 (R5 , RN ). We look for the resulting solution of the nonlinear problem (2) as u(x) = u0 (x) + u p (x), where

u p (x) = (u p,1 (x), u p,2 (x), ..., u p,N (x))T .

(9)

78


Evidently, we obtain the perturbative system of equations ˆ Ks (x − y)gs (u0 (y) + u p (y))dy, −Δu p,s = εs R5

1 ≤ s ≤ N.

(10)

0 < ρ ≤ 1.

(11)

Let us define a closed ball in the Sobolev space Bρ := {u(x) ∈ H 3 (R5 , RN ) | uH 3 (R5 ,RN ) ≤ ρ },

We seek the solution of (10) as the fixed point of the auxiliary nonlinear system of equations ˆ Ks (x − y)gs (u0 (y) + v(y))dy, 1 ≤ s ≤ N −Δus = εs R5

(12)

in ball (11). For a given vector function v(y) it is an equation with respect to u(x). The left side of (12) contains the operator without Fredholm property −Δ : H 2 (R5 ) → L2 (R5 ), due to the fact that its essential spectrum fills the nonnegative semi-axis [0, +∞) and therefore, such operator has no bounded inverse. The analogous situation appeared in [20] and [22] but as distinct from the present work, the equations treated there required orthogonality conditions. The fixed point technique was used in [13] to evaluate the perturbation to the standing solitary wave of the Nonlinear Schrödinger (NLS) equation when either the external potential or the nonlinear term in the NLS were perturbed but the Schrödinger type operator involved in the nonlinear equation possessed the Fredholm property (see Assumption 1 of [13], also [8]). Let us define a closed ball in the space of N dimensions (13) I := {z ∈ RN | |z| ≤ ce u0 H 3 (R5 ,RN ) + ce }. For technical purposes we will use following quantities with 1 ≤ s, j ≤ N N ∂g s a2,s, j := supz∈I ∇ , a2,s := ∑ a22,s, j , a2 := max1≤s≤N a2,s . ∂zj j=1 Also, a1,s := supz∈I |∇gs (z)|,

a1 := max1≤s≤N a1,s .

We make the following assumption about the nonlinear parts of the system of equations (2). Assumption 2. Let 1 ≤ s ≤ N, such that gs (z) : RN → R with gs (z) ∈ C2 (RN ). We also assume that gs (0) = 0, ∇gs (0) = 0 and a2 > 0. Evidently, a1 defined above is positive as well, otherwise all the functions gs (z) will be constants in the ball I and a2 will vanish. For instance, gs (z) = z2 , z ∈ RN clearly satisfies our assumption above. Let us introduce the operator Tg such that u = Tg v, where u is a solution of the system of equations (12). Our main proposition is as follows. Theorem 3. Let Assumptions 1 and 2 hold. Then system (12) defines the map Tg : Bρ → Bρ , which is a strict contraction for all 0 < ε < ε ∗ for some ε ∗ > 0. The unique fixed point u p (x) of this map Tg is the only solution of the system of equations (10) in Bρ . Clearly, the resulting solution of system (2) given by (9) will be nontrivial due to the fact that the source terms fs (x) are nontrivial for some s = 1, ..., N and all gs (z) vanish at the origin due to our assumptions. We will make use of the elementary technical lemma below.

β with R ∈ (0, +∞) and the constants α , β > 0. It achieves the minimal value at R4 4β 1 5 4 1 R∗ = ( ) 5 , which is given by ϕ (R∗ ) = 4 α 5 β 5 . α 45 Let us proceed to the proof of our main result.

Lemma 4. Let ϕ (R) := α R +


79

2 The existence of the perturbed solution Proof of Theorem 3. Let us choose arbitrarily v(x) ∈ Bρ and denote the terms involved in the integral expressions in right side of system (12) as Gs (x) := gs (u0 (x) + v(x)). We apply the standard Fourier transform (6) to both sides of the system of equations (12). This yields 5

us (p) = εs (2π ) 2

K s (p)Gs (p) , p2

1 ≤ s ≤ N.

Hence for the norm we obtain us 2L2 (R5 )

=

(2π )5 εs2

Clearly, for any φ (x) ∈ L1 (R5 ) φ(p)L∞ (R5 ) ≤

ˆ

2 2 |K s (p)| |Gs (p)| d p. |p|4

R5

1 5

(2π ) 2

φ (x)L1 (R5 ) .

(14)

(15)

Note that as distinct from articles [20] and [22] containing results in lower dimensions, in the present work we do not try to control the norms K(p) s . 2 p ∞ 5 L (R )

Let us estimate the right side of (14) by virtue of (15) with R ∈ (0, +∞) as ˆ ˆ 2 2 2 2 |K |K s (p)| |Gs (p)| s (p)| |Gs (p)| 5 2 5 2 d p + (2 π ) ε dp (2π ) εs s |p|4 |p|4 |p|≤R |p|>R ≤ εs2 Ks 2L1 (R5 ) {

1 1 Gs (x)2L1 (R5 ) |S5 |R + 4 Gs (x)2L2 (R5 ) }. 5 (2π ) R

(16)

Here and further down S5 stands for the unit sphere in the space of five dimensions centered at the origin and |S5 | for its Lebesgue measure (see e.g. p.6 of [11]). Since v(x) ∈ Bρ , we arrive at u0 + vL2 (R5 ,RN ) ≤ u0 H 3 (R5 ,RN ) + 1 and by virtue of the Sobolev embedding (5) |u0 + v| ≤ ce u0 H 3 (R5 ,RN ) + ce . Let us use the formula

ˆ Gs (x) =

1

0

∇gs (t(u0 (x) + v(x))).(u0 (x) + v(x))dt,

1 ≤ s ≤ N.

Here and below the dot symbol denotes the scalar product of two vectors in RN . With the ball I defined in (13), we easily obtain |Gs (x)| ≤ supz∈I |∇gs (z)||u0 (x) + v(x)| ≤ a1 |u0 (x) + v(x)|. Thus Gs (x)L2 (R5 ) ≤ a1 u0 + vL2 (R5 ,RN ) ≤ a1 (u0 H 3 (R5 ,RN ) + 1). Clearly, for t ∈ [0, 1] and 1 ≤ j ≤ N

∂ gs (t(u0 (x) + v(x))) = ∂zj

ˆ

t 0

∇

∂ gs (τ (u0 (x) + v(x))).(u0 (x) + v(x))d τ . ∂zj

80

This yields


∂g ∂g s s (t(u0 (x) + v(x))) ≤ supz∈I ∇ |u0 (x) + v(x)| = a2,s, j |u0 (x) + v(x)|, ∂zj ∂zj

such that by means of the Schwarz inequality N

|Gs (x)| ≤ |u0 (x) + v(x)| ∑ a2,s, j |u0, j (x) + v j (x)| ≤ a2 |u0 (x) + v(x)|2 j=1

and

Gs (x)L1 (R5 ) ≤ a2 u0 + v2L2 (R5 ,RN ) ≤ a2 (u0 H 3 (R5 ,RN ) + 1)2 .

(17)

This enables us to derive the upper bound for the right side of (16) as

ε 2 Ks (x)2L1 (R5 ) (u0 H 3 (R5 ,RN ) + 1)2

a2 a21

2 2 (u + 1) |S |R + 3 5 N 0 H (R ,R ) 5 (2π )5 R4

with R ∈ (0, +∞). By virtue of Lemma 4 we derive the minimal value of the expression above. Therefore, 4

u2L2 (R5 ,RN )

2 |S5 | 5 85 33 5 5 ≤ε K a (u 3 (R5 ,RN ) + 1) 5 a 0 H 4 . 2 1 (2π )4 45

2

Clearly, (12) implies

2

(18)

ˆ ∇(−Δus ) = εs ∇

R5

Ks (x − y)Gs (y)dy,

1 ≤ s ≤ N,

such that by means of (15) along with (17) ∇(−Δus )2L2 (R5 ) ≤ εs2 Gs (x)2L1 (R5 ) ∇Ks (x)2L2 (R5 ) ≤ ε 2 a22 (u0 H 3 (R5 ,RN ) + 1)4 ∇Ks (x)2L2 (R5 ) . Hence

N

∑ (−Δ)

s=1

3 2

us 2L2 (R5 ) ≤ ε 2 a22 (u0 H 3 (R5 ,RN ) + 1)4 Q2 .

(19)

By virtue of the definition of the norm (4) along with upper bounds (18) and (19) we arrive at 4 4 2 |S5 | 5 25 5K 2 2 5 5 2 a + a uH 3 (R5 ,RN ) ≤ ε (u0 H 3 (R5 ,RN ) + 1) a2 4 2Q ≤ρ (2π )4 1 4 5 for all values of ε > 0 small enough, such that u(x) ∈ Bρ as well. If for a certain v(x) ∈ Bρ there exist two solutions u1,2 (x) ∈ Bρ of system (12), each component us (x) of their difference u(x) := u1 (x) − u2 (x) ∈ L2 (R5 , RN ) satisfies Laplace’s equation. Since there are no nontrivial square integrable harmonic functions, u(x) = 0 a.e. in R5 . Thus, system (12) defines a map Tg : Bρ → Bρ for all ε > 0 sufficiently small. Thus our goal is to prove that this map is a strict contraction. We choose arbitrarily v1,2 (x) ∈ Bρ . Via the argument above u1,2 = Tg v1,2 ∈ Bρ as well. System (12) gives us ˆ Ks (x − y)gs (u0 (y) + v1 (y))dy, 1 ≤ s ≤ N, (20) −Δu1,s = εs ˆ −Δu2,s = εs

R5

R5

Ks (x − y)gs (u0 (y) + v2 (y))dy,

1 ≤ s ≤ N.

Let us define G1,s (x) := gs (u0 (x) + v1 (x)),

G2,s (x) := gs (u0 (x) + v2 (x)).

(21)


We apply the standard Fourier transform (6) to both sides of systems (20) and (21), which yields K s (p)G1,s (p) , p2

5

u 1,s (p) = εs (2π ) 2

5

u 2,s (p) = εs (2π ) 2

K s (p)G2,s (p) p2

and express the norm u1,s − u2,s 2L2 (R5 )

=

εs2 (2π )5

ˆ R5

2 2 |K s (p)| |G1,s (p) − G2,s (p)| d p. |p|4

Evidently, it can be estimated from above using (15) by

ε

2

|S5 | Ks (x)2L1 (R5 ) { G1,s (x) − G2,s (x)2L1 (R5 ) R + (2π )5

G1,s (x) − G2,s (x)2L2 (R5 ) R4

}

with R ∈ (0, +∞). For 1 ≤ s ≤ N, let us make use of the formula ˆ 1 ∇gs (u0 (x) + tv1 (x) + (1 − t)v2 (x)).(v1 (x) − v2 (x))dt. G1,s (x) − G2,s (x) = 0

Obviously, for v1,2 (x) ∈ Bρ and t ∈ [0, 1] we have v2 (x) + t(v1 (x) − v2 (x))H 3 (R5 ,RN ) ≤ tv1 (x)H 3 (R5 ,RN ) + (1 − t)v2 (x)H 3 (R5 ,RN ) ≤ ρ , such that v2 (x) + t(v1 (x) − v2 (x)) ∈ Bρ as well. We obtain |G1,s (x) − G2,s (x)| ≤ supz∈I |∇gs (z)||v1 (x) − v2 (x)| = a1,s |v1 (x) − v2 (x)|, such that G1,s (x) − G2,s (x)L2 (R5 ) ≤ a1,s v1 (x) − v2 (x)H 3 (R5 ,RN ) . Apparently, for 1 ≤ j ≤ N

∂ gs (u0 (x) + tv1 (x) + (1 − t)v2 (x)) = ∂zj

ˆ

1

∂ gs (τ [u0 (x) + tv1 (x) + (1 − t)v2 (x)]) ∂zj 0 (u0 (x) + tv1 (x) + (1 − t)v2 (x))d τ , ∇

such that ∂g s (u0 (x) + tv1 (x) + (1 − t)v2 (x)) ∂zj ∂g s ≤ supz∈I ∇ {|u0 (x)| + t|v1 (x)| + (1 − t)|v2 (x)|} ∂zj with t ∈ [0, 1]. Therefore, by means of the Schwarz inequality |G1,s (x) − G2,s (x)| ≤

N

1

1

∑ a2,s, j {|u0 (x)| + 2 |v1(x)| + 2 |v2 (x)|}|v1, j (x) − v2, j (x)|

j=1

1 1 ≤ a2,s {|u0 (x)| + |v1 (x)| + |v2 (x)|}|v1 (x) − v2 (x)|. 2 2 The Schwarz inequality yields the upper bound for G1,s (x) − G2,s (x)L1 (R5 ) as 1 1 a2,s {u0 (x)L2 (R5 ,RN ) + v1 (x)L2 (R5 ,RN ) + v2 (x)L2 (R5 ,RN ) }× 2 2

81

82


×v1 (x) − v2 (x)L2 (R5 ,RN ) ≤ a2 {u0 (x)H 3 (R5 ,RN ) + 1}v1 (x) − v2 (x)H 3 (R5 ,RN ) . This enables us to estimate from above u1 (x) − u2 (x)2L2 (R5 ,RN ) by

ε 2 K 2 v1 (x) − v2 (x)2H 3 (R5 ,RN ) {

a2 2 a1 2 2 (u + 1) |S |R + }. 3 5 N 0 H (R ,R ) 5 (2π )5 R4

We use Lemma 4 to minimize the expression above over R ∈ (0, +∞) to prove that u1 (x) − u2 (x)2L2 (R5 ,RN ) has an upper bound given by 2

ε K

2

v1 − v2 2H 3 (R5 ,RN )

8

a25 4 2 (u0 H 3 (R5 ,RN ) + 1)2 |S5 | 5 a15 . 4 4 4 5 (2π ) 5

(22)

By means of (20) and (21) ˆ ∇(−Δ)(u1,s (x) − u2,s (x)) = εs ∇

R5

Ks (x − y)[G1,s (y) − G2,s (y)]dy.

Hence via (15) ∇(−Δ)(u1,s (x) − u2,s (x))2L2 (R5 ) ≤ ε 2 G1,s (x) − G2,s (x)2L1 (R5 ) ∇Ks (x)2L2 (R5 ) . 3

As a results, the norm (−Δ) 2 (u1,s (x) − u2,s (x))2L2 (R5 ) is bounded above by

ε 2 a22 (u0 H 3 (R5 ,RN ) + 1)2 v1 (x) − v2 (x)2H 3 (R5 ,RN ) ∇Ks (x)2L2 (R5 ) , such that N

∑ (−Δ)

s=1

3 2

(u1,s (x) − u2,s (x))2L2 (R5 )

≤ ε 2 a22 (u0 H 3 (R5 ,RN ) + 1)2 Q2 v1 (x) − v2 (x)2H 3 (R5 ,RN ) .

(23)

Inequalities (22) and (23) imply that the norm u1 − u2 H 3 (R5 ,RN ) has an upper bound given by 4 5

ε (u0 H 3 (R5 ,RN ) + 1)a2 [

5K 4

45

2

2

2 1 a15 4 5 + Q2 a 5 ] 2 v − v 3 |S | 5 1 2 H (R5 ,RN ) . 2 (2π )4

Therefore, the map Tg : Bρ → Bρ defined by system (12) is a strict contraction for all values of ε > 0 sufficiently small. Its unique fixed point u p (x) is the only solution of system (10) in Bρ and the resulting u(x) ∈ H 3 (R5 , RN ) given by (9) is a solution of the system of equations (2). 3 Discussion We will finish this work with a short discussion of biological interpretations of the results presented above. All tissues and organs in a biological organism are characterized by cell distribution with respect to their genotype. Without mutations all cells would have identical genotype. Due to mutations, the genotype changes and represents some distribution around its principal value. Stationary solutions of this equation give stationary cell distribution with respect to the genotype. Existence of such stationary distributions is an important property of biological organisms which allows their existence as steady state systems.


83

Existence of stationary solutions is proved in the spaces of integrable functions decaying at infinity. Biologically this means that cell distribution with respect to the genotype decays as the distance from the principal genotype increases. The results of the work show what conditions should be imposed on cell proliferation, mutations and influx in order to get such distributions. In the context of population dynamics, this result applies also to biological species where individuals are distributed around some average genotype. In this case existence of stationary corresponds to the existence of biological species [5]. References [1] Alfimov, G.L., Medvedeva, E.V., and Pelinovsky, D.E. (2014), Wave Systems with an Infinite Number of Localized Traveling Waves, Phys. Rev. Lett., 112, 054103, 5pp. [2] Amrouche, C., Girault, V., and Giroire, J. (1997), Dirichlet and Neumann exterior problems for the n-dimensional Laplace operator. An approach in weighted Sobolev spaces, J. Math, Pures Appl., 76, 55–81. [3] Amrouche, C. and Bonzom, F. (2008), Mixed exterior Laplace’s problem, J. Math. Anal. Appl., 338, 124–140. [4] Benkirane, N. (1988), Propriété d’indice en théorie Holderienne pour des opérateurs elliptiques dans Rn , CRAS, 307, Série I, 577–580. [5] Bessonov, N., Reinberg, N., and Volpert, V. (2014), Mathematics of Darwins Diagram, Math. Model. Nat. Phenom., 9(3), 5–25. [6] Bolley, P. and Pham, T.L. (1993), Propriété d’indice en théorie Holderienne pour des opérateurs différentiels elliptiques dans Rn , J. Math. Pures Appl., 72, 105–119. [7] Bolley, P. and Pham, T.L. (2001), Propriété d’indice en théorie Hölderienne pour le problème extérieur de Dirichlet, Comm. Partial Differential Equations, 26(1–2), 315–334. [8] Cuccagna, S., Pelinovsky, D., and Vougalter, V. (2005),Spectra of positive and negative energies in the linearized NLS problem, Comm. Pure Appl. Math., 58(1), 1–29. [9] Ducrot, A., Marion, M., and Volpert, V. (2005), Systemes de réaction-diffusion sans propriété de Fredholm, CRAS, 340, 659–664. [10] Ducrot, A., Marion, M., and Volpert, V.(2008), Reaction-diffusion problems with non Fredholm operators, Advances Diff. Equations, 13(11–12), 1151–1192. [11] Lieb, E and Loss, M.(1997), Analysis. Graduate Studies in Mathematics, 14, American Mathematical Society, Providence. [12] Volpert, V. (2011), Elliptic partial differential equations. Volume 1. Fredholm theory of elliptic problems in unbounded domains, Birkhauser. [13] Vougalter, V.(2010), On threshold eigenvalues and resonances for the linearized NLS equation, Math. Model. Nat. Phenom., 5(4), 448–469. [14] Volpert, V., Kazmierczak, B., Massot, M., and Peradzynski, Z. (2002), Solvability conditions for elliptic problems with non-Fredholm operators, Appl. Math., 29(2), 219–238. [15] Vougalter, V. and Volpert,V. (2011), Solvability conditions for some non-Fredholm operators, Proc. Edinb. Math. Soc. (2), 54 (1), 249–271. [16] Vougalter, V. and Volpert, V. (2010), On the solvability conditions for some non Fredholm operators, Int. J. Pure Appl. Math., 60(2), 169–191. [17] Vougalter, V. and Volpert, V. (2012), On the solvability conditions for the diffusion equation with convection terms, Commun. Pure Appl. Anal., 11(1), 365–373. [18] Vougalter, V. and Volpert,V. (2010), Solvability relations for some non Fredholm operators,Int. Electron. J. Pure Appl. Math. , 2(1), 75–83. [19] Volpert, V. and Vougalter, V. (2011), On the solvability conditions for a linearized Cahn-Hilliard equation, Rend. Istit. Mat. Univ. Trieste, 43, 1–9. [20] Vougalter, V. and Volpert, V. (2011), On the existence of stationary solutions for some non-Fredholm integrodifferential equations, Doc. Math., 16, 561–580. [21] Vougalter, V. and Volpert,V. (2012), Solvability conditions for a linearized Cahn-Hilliard equation of sixth order, Math. Model. Nat. Phenom., 7(2), 146–154. [22] Vougalter, V. and Volpert,V. (2012), Solvability conditions for some linear and nonlinear non-Fredholm elliptic problems, Anal. Math. Phys., 2(4), 473–496. [23] Vougalter, V. and Volpert,V. (2015), Existence of stationary solutions for some nonlocal reaction-diffusion equations, Dyn. Partial Differ. Equ., 12(1), 43–51. [24] Vougalter, V. and Volpert,V. (2015), Existence of stationary solutions for some integro-differential equations with superdiffusion. Preprint.

84


Appendix In the present article we used solvability conditions for linear elliptic problems in Rd derived in [22]. Let us state them below for the convenience of the readers. We study the existence of solutions of the linear problem −Δφ − ωφ = −h(x), ω ≥ 0

(1)

in the space H 2 (Rd ), d ∈ N equipped with the standard norm u2H 2 (Rd ) := u2L2 (Rd ) + Δu2L2 (Rd ) .

(2)

The right side of (1) is assumed to be square integrable. Lemma 5. Let h(x) ∈ L2 (R). The the following assertions hold: a) When ω > 0 and xh(x) ∈ L1 (R) problem (1) admits a unique solution in H 2 (R) if and only if

√ e±i ω x = 0. h(x), √ 2π L2 (R)

(3)

b) When ω = 0 and x2 h(x) ∈ L1 (R) problem (1) admits a unique solution in H 2 (R) if and only if (h(x), 1)L2 (R) = 0, (h(x), x)L2 (R) = 0.

(4)

Lemma 6. Let h(x) ∈ L2 (Rd ), d ≥ 2. The the following assertions hold: a) When ω > 0 and xh(x) ∈ L1 (Rd ) problem (1) admits a unique solution in H 2 (Rd ) if and only if

eipx d = 0, p ∈ S√ h(x), d ω a.e., d ≥ 2. 2 (2π ) L2 (Rd )

(5)

b) When ω = 0 and |x|2 h(x) ∈ L1 (R2 ) problem (1) admits a unique solution in H 2 (R2 ) if and only if (h(x), 1)L2 (R2 ) = 0, (h(x), xk )L2 (R2 ) = 0, 1 ≤ k ≤ 2.

(6)

c) When ω = 0 and |x|h(x) ∈ L1 (Rd ), d = 3, 4 problem (1) admits a unique solution in H 2 (Rd ) if and only if (h(x), 1)L2 (Rd ) = 0, d = 3, 4.

(7)

d) When ω = 0 and |x|h(x) ∈ L1 (Rd ), d ≥ 5 problem (1) possesses a unique solution in H 2 (Rd ). Lemma 7. Let ω = 0 and h(x) ∈ L1 (Rd ) ∩ L2 (Rd ) with d ≥ 5. Then problem (1) admits a unique solution in H 2 (Rd ).



Regular and Singular Pulse and Front Solutions and Possible Isochronous Behavior in the Extended-Reduced Ostrovsky Equation: Phase-Plane, Multi-Infinite Series and Variational Formulations U. Tanriver1†, G. Gambino2, S. Roy Choudhury3 1 Department

of Mathematics, Texas A&M University-Texarkana, USA of Mathematics, University of Palermo, Italy 3 Department of Mathematics, University of Central Florida, Orlando, USA 2 Department

Submission Info Communicated by Valentin Afraimovich Received 10 March 2015 Accepted 7 November 2015 Available online 1 April 2016 Keywords Extended-Reduced Ostrovsky equation Regular and singular pulse solutions Front solutions Variational solitary waves Isochronous behavior Homoclinic and heteroclinic orbits

Abstract In this paper we employ three recent analytical approaches to investigate several classes of traveling wave solutions of the so-called extendedreduced Ostrovsky Equation (exROE). A recent extension of phase-plane analysis is first employed to show the existence of breaking kink wave solutions and smooth periodic wave (compacton) solutions. Next, smooth traveling waves are derived using a recent technique to derive convergent multi-infinite series solutions for the homoclinic orbits of the travelingwave equations for the exROE equation. These correspond to pulse solutions respectively of the original PDEs. We perform many numerical tests in different parameter regime to pinpoint real saddle equilibrium points of the corresponding traveling-wave equations, as well as ensure simultaneous convergence and continuity of the multi-infinite series solutions for the homoclinic orbits anchored by these saddle points. Unlike the majority of unaccelerated convergent series, high accuracy is attained with relatively few terms. And finally, variational methods are employed to generate families of both regular and embedded solitary wave solutions for the exROE PDE. The technique for obtaining the embedded solitons incorporates several recent generalizations of the usual variational technique and it is thus topical in itself. One unusual feature of the solitary waves derived here is that we are able to obtain them in analytical form (within the assumed ansatz for the trial functions). Thus, a direct error analysis is performed, showing the accuracy of the resulting solitary waves. Given the importance of solitary wave solutions in wave dynamics and information propagation in nonlinear PDEs, as well as the fact that not much is known about solutions of the family of generalized exROE equations considered here, the results obtained are both new and timely. ©2016 L&H Scientific Publishing, LLC. All rights reserved.

† Corresponding

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


86

U. Tanriver, G. Gambino, S. Roy Choudhury/ Discontinuity, Nonlinearity, and Complexity 5(1) (2016) 85–100

1 Introduction In the papers [1], [2], extensive classifications have been made of the periodic, solitary (pulse), and compacton traveling wave solutions of the so-called extended-reduced Ostrovsky equation (exROE), the long-wave limit [3] of the so-called Ostrovsky equation [4] governing nonlinear internal waves in rotating flows. In this paper, we investigate solutions of this equation by three other techniques. Various analytical methods have been developed to construct solitary waves of physically important nonlinear partial differential equations (NLPDEs), including variational methods, diverse series solution techniques, the extended tanh −function method, Hirota’s method, truncated regular and invariant Pain-levé expansions, and various others. Three of these techniques are applied to the exROE equation in this paper. First, novel phase-plane methods are used to consider singular solutions of the exROE equation, in particular breaking kink or front solutions. We next employ one recently developed technique to construct convergent, multi-infinite, series solutions for regular solitary waves of the exROE equation (or equivalently, homoclinic orbits of its traveling-wave equation). In addition, in an alternative approach, the variational method is employed to construct regular solitary waves of the exROE NLPDE directly, and also attempt to construct embedded solitons of the PDE using several recent extensions of the variational approach. The remainder of the paper is organized as follows. In Section 2, the traveling wave ODE of the exROE equation is considered. A recently developed technique (see [5], [6]) is employed to construct convergent series solutions for its homoclinic and heteroclinic orbits, corresponding to solitary wave and front (pulse) solutions of the original exROE NLPDE. A Lagrangian for the exROE equation is developed in Section 3. Section 4 then considers the linear spectrum of the exROE equation to isolate the parameter regimes where regular solitary waves exist. A Gaussian ansatz or trial function for these solitary waves is then substituted into the Lagrangian, and its Euler-Lagrange equations are solved to derive the optimum soliton or ansatz parameters in the usual way (within the functional Gaussian form of the ansatz). 2 Singular Solutions of the Extended-Reduced Ostrovsky equation Let us consider the following traveling wave equation associated to the extended reduced Ostrovsky equation (exROE): 1 d dφ φ (φ ) + (p + q)φ 2 + (pc + β )φ = k, (1) dz dz 2 where φ = φ (z), p, q and β are constant coefficients and k is a constant of integration. Recall that the exROE has been derived in [1] from the Hirota-Satsuma-type shallow water wave equation, and has been also discussed in [2]. Equation (1) is equivalent to the following 2-dimensional system: ⎧ dφ ⎪ ⎪ = y, ⎪ ⎪ ⎨ dz (2) ⎪ 2 − 1 (p + q)φ 2 ⎪ k − (pc + β ) φ − φ y dy ⎪ 2 ⎪ = , ⎩ dz φ2 which is the traveling wave system for (1). The system (2) belongs to the following class of system: dφ = y, dz

dy Q(φ , y) = , dz f (φ , y)

(3)

called the second type of singular traveling wave system in [7], where f (φ , y) and Q(φ , y) are sufficiently regular functions satisfying the following condition: y

∂ f (φ , y) ∂ Q(φ , y) ≡ 0, + ∂φ ∂y

(4)


which implies there exists a first integral of Eq. (3). Notice that

dy dz

=

Q(φ ,y) f (φ ,y)

is not defined on the set of real planar

curves f (φ , y) = 0 and when the phase point (φ , y) passes through every branch of f (φ , y) = 0, the quantity changes sign [7]. In this case: f (φ , y) = φ 2

and

87

1 Q(φ , y) = k − (pc + β )φ − φ y2 − (p + q)φ 2 . 2

dy dz

(5)

The singular curve is φ 2 = 0. We make the coordinate transformation dz = φ 2 d ζ for φ 2 = 0 to obtain the following regular system associated to (2): ⎧ dφ ⎪ = yφ 2 , ⎪ ⎪ ⎨ dζ (6) ⎪ ⎪ 1 dy ⎪ ⎩ = k − (pc + β )φ − φ y2 − (p + q)φ 2 . dζ 2 The systems of equations in (2) and (6) have the same invariant curve solutions, the main difference between Eqs. (2) and (6) is the parametric representation of the orbit: near φ = 0, Eq. (6) uses the fast time variable ζ , while Eq. (2) uses the slow time variable z (see [6, 8] for details). Since the first integral of both Eqs. (2) and (6) are the same, thus both of them have the same phase orbits, except on the straight lines φ = 0 and we can study the associated regular system of Eq. (6) in order to get the phase portraits of Eq. (2). Via standard linear stability analysis we compute the following equilibria of the system (6) when p + q = 0: −(pc + β ) ± (pc + β )2 + 2k(p + q) , 0). (7) P± ≡ ( p+q The equilibria P± in (7) exist real when (pc + β )2 + 2k(p + q) ≥ 0. To test their stability, we compute the jacobian matrix associated to the system (6) evaluated in P± : φ¯±2 0 , (8) −(pc + β ) − (p + q)φ¯± 0 where φ¯± is the first coordinate of the point P± . It is straightforward to show that P+ , when it exists (i.e. for β )2 (pc+β )2 k ≥ − (pc+ 2(p+q) ), is a center and P− , when it exists (i.e. for k ≥ − 2(p+q) ), is a saddle, see the phase portrait in Fig.1 (the parameters correspond to Fig.14b) of [2]). The oval curve on the right of the singular line φ = 0 in Fig.1 gives rise to a smooth periodic wave solution (compacton) of the original equation. The family of open curves which approaches on the left of the singular straight line φ = 0 gives rise to uncountably infinitely many bounded breaking wave solutions. Moreover, the stable and unstable manifolds of the saddle point P− gives rise to a one-sided breaking kink wave solution and a one-sided breaking anti-kink wave solution. β )2 When p + q = 0 and k = − (pc+ 2(p+q) , the two equilibria P+ and P− coincide and they are equal to P ≡ β (− pc+ p+q , 0). In this case the linear stability analysis fails, as the eigenvalues are equal to zero and the phase portrait is presented in Fig.2 (for parameters corresponding to Fig. 14a) of [2]). The family of open curves approaching the singular line from the right give rise to uncountably infinitely many bounded breaking wave solutions. When p + q = 0 the regular system reduces to: ⎧ dφ ⎪ = yφ 2 , ⎪ ⎪ ⎨ dζ (9) ⎪ ⎪ dy ⎪ 2 ⎩ = k − (pc + β )φ − φ y , dζ

88


8 6 4 2 y

0 −2 −4 −6 −8 −8

−6

−4

−2 φ 0

2

4

6

2

β) Fig. 1 The phase portraits of system (6) when p + q = 0 and k ≥ − (pc+ 2(p+q) . The parameters are chosen as p = 1, q = 2, k = 1, c = 1, β = 1 and the equilibria are the center P+ ≡ (0.3874, 0) and the saddle P− ≡ (−1.7208, 0). The straight line φ = 0 is drawn in red.

2 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −3

−2

−1

0

1

2

φ

2

β) Fig. 2 The phase portraits of system (6) when p + q = 0 and k = − (pc+ 2(p+q) . The parameters are chosen as p = 0.5, q = 0.5, c = 1, β = 1, k = −1.125 and the equilibrium is P ≡ (−1.5, 0). The straight line φ = 0 is drawn in red.

k and, when pc + β = 0 the only equilibrium is P ≡ ( pc+ β , 0). The characteristic equation corresponding to this equilibrium is straightforwardly obtained as follows:

λ2 +

k2 = 0, pc + β

therefore P is a center when pc + β > 0 and it is a saddle when pc + β < 0, see the corresponding phase portraits in Fig. 3. Notice that in Fig.(3)(a) a limit cycle arises corresponding to the time-dependent first integral. Morover, the family of open curves which approaches on the left of the singular straight line φ = 0 gives rise to uncountably infinitely many bounded breaking wave solutions. The parameters in Fig.(3)(a) to those in Fig. 10 of [2]. In Fig.(3)(b), where the parameters correspond to Fig. 1b) of [2], the stable and unstable manifolds of the saddle point P gives rise to a one-sided breaking kink wave solution and a one-sided breaking anti-kink wave


89

3

4 3

2

2 1

0

y

y

1

−1

0 −1

−2

−2

−3 −3

−4 −2

−1

0

1 φ

2

3

−4

4

−3

(a)

−2

−1 φ

0

1

2

(b)

Fig. 3 The phase portraits of system (6) when p + q = 0. The straight line φ = 0 is drawn in red. (a) pc + β > 0. The parameters are chosen as p = 0.5, q = −0.5, k = 1, c = 1, β = 0.2 and the equilibrium is the center P ≡ (1.4286, 0). (b) pc + β < 0. The parameters are chosen as p = 0.5, q = −0.5, k = 1, c = 1, β = −0.8 and the equilibrium is the saddle P ≡ (−3.33, 0).

10

y

5

0

−5

−10 −5

0 φ

5

Fig. 4 The phase portraits of system (6) when p + q = 0 and pc + β = 0. The parameters are chosen as p = 0.5, q = −0.5, k = 1, c = 2, β = −1. The singular straight line φ = 0 is drawn in red.

solution. When p + q = 0 and pc + β = 0 the regular system (6) does not admit any equilibrium and the typical phase portrait is reported in Fig.4. The family of open curves in Fig. 4 approaching the singular line from the left gives rise to uncountably infinitely many bounded breaking wave solutions. The parameters in Fig.4 correspond to Fig. 1 of [2]. 3 Regular pulse and front solutions of the exROE: analytic solutions for homoclinic orbits In this section, we change gears and consider regular pulse and front solutions of the exROE in (1) by calculating convergent, multi-infinite series solutions for the possible homoclinic orbits. We employ a recently developed approach [5, 9], using the method of undetermined coefficients to derive convergent analytic series

90


3 2

y

1 0

−1 −2 −3 −3

−2

−1 φ

0

1

Fig. 5 The phase portraits of system (6) when p + q = 0. The parameters are chosen as p = 0.5, q = 0.5, k = −1, c = 1, β = 1. The equilibrium P− ≡ (−2, 0) is a saddle and the equilibrium P+ ≡ (−1, 0) is a center.

for homoclinic orbits of Eq. (1). When p + q = 0 the point P− is a saddle, and the phase portrait of the system (6) shows a homoclinic orbit to this point, see Fig.5 for typical parameters. We look for a solution of the following form: ⎧ + ⎨ φ (z) φ (z) = x0 ⎩ − φ (z) where:

∞

φ + (z) = x0 +

∑ ahehα z ,

z>0 z=0 z 0 are undetermined constants and ah , gh , with h ≥ 1, are, at the outset, arbitrary coefficients. Substituting the series (11) for φ + (z) we obtain the following expressions for each term of (1):

φ2 = φ φz2 =

∞ h−1

∞

h= 2 j= 1

h=1

∑

∑ ah− j a j ehα z + 2x0 ∑ ahehα z + x20,

(12)

∞ h−1 j−1

∑ ∑ ∑ al a j−l ah− j ( j − l)l α 2ekα z

(13)

h= 3 j= 2 l= 1

+x0

∞ h−1

∑ ∑ (h − j) jα 2 ah− j a j ekα z ,

h= 2 j= 1

φzz φ 2 =

∞ h−1 j−1

∑∑

h= 3 j= 2 l= 1

+2x0

∞

∑ al a j−l ah− j (h − j)2 α 2ekα z + x20 ∑ ah (hα 2 )ehα z

∞ h−1

∑ ∑ (h − j)2α 2 ah− j a j ekα z .

h= 2 j= 1

h=1

(14)


91

Using (12)—(14) into the Eq. (1) we obtain: ∞ 1 (p + q)x20 + (pc + β )x0 − k + ∑ (x20 (hα )2 + (p + q)x0 + pc + β )ah ehα z 2 h=1 ∞ h−1

+∑

1

∑ (x0 (h − j) jα 2 + 2x0 (h − j)2α 2 + 2 (p + q))ah− j a j ehα z

(15)

h= 2 j= 1

∞ h−1 j−1

+∑

∑ ∑ ((h − j)2 + ( j − l)l)al a j−l ah− j α 2 ehα z = 0.

h= 3 j= 2 l= 1

As x0 is an equilibrium of the equation (1), the first three terms in (15) are identical to zero. Comparing the coefficients of ehα z for each h, one has for h = 1: (α 2 x20 + (p + q)x0 + pc + β )a1 = 0. Assuming a1 = 0 (otherwise ah = 0 for all h > 1 by induction), results in the two possible values of α : (p + q)x0 + pc + β (p + q)x0 + pc + β α1 = − , α2 = − − . 2 x0 x20

(16)

(17)

We are dealing with the case when the equilibrium x0 is a saddle (see the first section for details), therefore it results that α1 > 0 and α2 < 0. In this case, as our series solution (10) needs to converge for z > 0, we pick the negative root α = α2 . For h = 2 we have: 1 F(2α2 )a2 = −(3α22 x0 + (p + q))a21 , 2

(18)

where F(hα2 ) = (hα2 )2 x20 + (p + q)x0 + pc + β . For h = 3 we obtain: a3 = −

(14α22 x0 + p + q)a1 a2 + 2α 2 a31 . F(3α2 )

(19)

And so on, for h > 3 one has: ah =

∞ h−1 1 1 [ ∑ (x0 (h − j) jα 2 + 2x0 (h − j)2 α 2 + (p + q))ah− j a j ∑ F(hα2 ) h= 4 j= 1 2 h−1 j−1

+∑

∑ ((h − j)2 + ( j − l)l)al a j−l ah− j α 2].

(20)

j= 2 l= 1

Therefore for all h the series coefficients ah can be iteratively computed in terms of a1 : ah = ϕh ah1 ,

(21)

where ϕh , h > 1 are functions which can be obtained using Eq. (18)-(20). The first part of the homoclinic orbit corresponding to z > 0 has thus been determined in terms of a1 : ∞

φ + (z) = x0 + a1 eα2 z + ∑ ϕh ah1 ehα2 z .

(22)

h=2

Notice that the Eq. (1) is reversible under the standard reversibility of classical mechanical systems: z → −z,

(φ , φz , φzz ) → (φ , −φz , φzz ).

(23)

92


−0.5 −1

1.5

t=1 t=5 t=10

1 ah

φ

−1.5 −2

0.5

−2.5 −3 −3.5 −10

0 0

x

10

20

1

5

h 10

15

(a)

(b) Fig. 6 The parameter are chosen as in Fig.5. (a) The series solution φ (z) in (10) for the homoclinic orbit to the saddle point (−2, 0) plotted as a function of x for different values of t, showing traveling wave nature of the solution. Here a1 = 1.5595 is the only solution of the continuity equation (25) truncated at M = 20. (b) Plot of ah in (20) versus h shows the series coefficients are converging.

Mathematically, this property would translate to solutions having odd parity in z. Therefore the series solution for z < 0 can be easily obtained based on the intrinsic symmetry property of the equation, i.e.: ∞

φ − (z) = x0 − a1 eα2 z − ∑ ϕh ah1 ehα2 z .

(24)

h=2

We want to construct a solution continuous at z = 0, therefore we impose: ∞

x0 + a1 + ∑ ϕh ah1 = 0.

(25)

h=2

Hence we choose a1 as the nontrivial solutions of the above polynomial equation (25). In practice the Eq. (25) is numerically solved and the corresponding series solutions are not unique. Let us now choose the system parameter as in Fig.5. Following the above given computation of the series coefficients, we build the homoclinic orbit to the saddle point P− ≡ (−2, 0). Truncating the series solution up to h = 20, the corresponding homoclinic orbit solution is unique as the continuity condition (25) admits only the solution a1 = 1.5595 leading to a convergent series coefficients ak , see Fig.6(b), and the continuous series solution for the homoclinic orbit appears as in Fig.6(a), where its traveling nature is also shown. 4 Lagrangian via Jacobi’s Last Multiplier In this section, we derive a Lagrangian for the traveling wave equation (1) of the exROE equation. While this may be done by simply matching the terms in this equation to those in the Euler-Lagrange equation, we use an alternative approach here using the technique of Jacobi’s Last Multiplier. In Section 5, this Lagrangian will be employed to construct solitary wave solutions of the exROE equation, having amplitude and width parameters optimized to satisfy the corresponding Euler-Lagrange equations. Jacobi [10] first described his method for the “Last Multiplier” (which we shall refer to as the Jacobi Last Multiplier, or JLM for short) in Konigsberg over 1842 − 1843. It essentially yields an extra first integral for dynamical systems by locally reducing an n-dimensional system to a two-dimensional vector field on the intersection of the n − 2 level sets formed by the first integrals. After the work of Jacobi, the JLM received a fair


93

amount of attention, including in a classic paper by Sophus Lie [11] placing it within his general framework of infinitesimal transformations. In 1874 Lie [11] showed that one could use point symmetries to determine last multipliers. A clear formulation in terms of solutions or first integrals and symmetries is given by L.Bianchi [12]. Subsequently it was used for computing first integrals of some ordinary differential equations (ODEs). The relation between the Jacobi multiplier denoted by M, and the Lagrangian L for any second-order ODE was derived by Rao [13], following some investigations in the early twentieth century [14]. After Rao’s work, the JLM does not appear to have been extensively employed in work on dynamical systems till it was recently used by Leach and Nucci to derive Lagrangians for a variety of ODE systems [15–17]. Recently more geometric formulation of JLM has been studied in [18]. The study of isochronous behavior, i.e. periodic behavior with a single period, in dynamical systems has also been a subject of great interest over the past decade [19, 20]. One important reason for this has been the surprising fact that many, if not most, systems may be converted to nearby isochronous systems by a process of so-called ω -modification. There are also recent theoretical results [21] proving that, up to a translation or the addition of a constant, planar polynomial systems exhibiting isochronicity are described by either the linear simple harmonic oscillator potential or the isotonic potential. In this section, we first use the JLM to derive a Lagrangian for the traveling-wave equation of the exROE equation. And then we also investigate possible isochronous behavior in this traveling-wave equation (corresponding to singly-periodic wavetrains of the exROE PDE), we therefore also attempt to map the potential term to either the simple harmonic oscillator (SHO) or the isotonic potential for specific values of the coefficient parameters of the exROE equation. 4.1

Derivation of the Lagrangian via the JLM

Given a m-dimensional system of first order ODEs yi = fi (x, yi ), i = 1, . . . , m, the Jacobi last multiplier, denoted by M(x, yi ), is defined as an integrating factor of the system satisfing the following equation: d(log M) m ∂ fi (x, yi ) +∑ = 0, dx ∂ yi i=1

(26)

Since a second-order ODE y = f (x, y, y ) is equivalent to a 2-dimensional system of first order ODEs, the corresponding Jacobi multiplier M(x, y, y ) satisfies the following equation:

d(log M) ∂ f (x, y, y ) + = 0, dx ∂ y

(27)

see for details [17, 22–25]. Let us rewrite the Euler-Lagrange equation:

∂L d ∂L ( )= , dx ∂ y ∂y

(28)

2 2 ∂ 2L ∂ L ∂ L ∂L . + y + f (x, y, y ) 2 = ∂ x∂ y ∂ y∂ y ∂y ∂y

(29)

by inserting y = f (x, y, y ) as follows:

Assuming

∂ 2L 2 = 0 and differentiating equation (29) with respect to y , the following equation is obtained: ∂y ∂ 2L ∂f d log( 2 ) + = 0. dx ∂y ∂y

(30)

94


Comparing the equation (30) with (27), we find the equation which connects the JLM to the Lagrangian L [13, 14, 22]: ∂ 2L (31) M = 2 . ∂y Therefore, the appropriate Lagrangian for the system can be determined starting from the JLM. 4.2

Search for isochronous behavior via the JLM

Isochronous systems, whose motions are periodic with a single period in extended regions of phase-space (often the entire phase-space) have attracted significant interest in recent years, especially following the work of Calogero and his collaborators (see [19, 20] and references therein), which revealed the near-ubiquity of such dynamics “close” to numerous classes of dynamical systems. In addition, in [21] it is proved that, up to a possible translation and the addition of a constant, planar polynomial systems exhibiting isochronicity are described 2 2 2 2 2 by either the linear SHO potential V (x) = ω 2x , or the isotonic potential V (x) = ω 8x + cx2 . These are rational potential functions, and systems which may be mapped to them exhibit oscillatory solutions with the same period T = 2ωπ . Irrational potentials, such as some with discontinuous second derivatives, may also be isochronous. In [26, 27] Chouikha and Hill et al. studied conditions under which the so-called Cherkas system [28] with a center at the origin as well as a five-parameter of reversible cubic systems may exhibit isochronicity. However, the study of the isochronicity conditions is non-trivial, and the technique required considerable computational effort. The same problem was re-examined in [29, 30] using the JLM to derive the conditions for isochronous solution behavior much more directly and with far less computational effort (see [31] for complete review). Here we shall follow this latter approach to examine (1) for possible isochronous behavior. Once derived a Lagrangian via the use of the JLM, the next step is to attempt a transformation of variables which might map the Hamiltonian to that of the linear SHO or the isotonic potential. As discussed above, such a mapping would prove isochronous behavior of the original dynamical system [21]. 4.3

Lagrangian for the exROE traveling-wave equation

In this subsection, we consider (1) for the traveling waves of the exROE equation. To compute the JLM, we use the equation (27) which, for the exROE, becomes: d(log M) 2 − φ = 0. dz φ

(32)

The solution of the equation (32) is given by: M(φ ) = φ 2 .

(33)

Using the equation (31), we find the appropriate Lagrangian for (1) to be: 1 φ3 φ2 L(φ , φ ) = φ 2 φ 2 − (p + q) − (pc + β ) + kφ , 2 6 2

(34)

where the potential energy V (u) satisfies the following equation:

V (φ ) = (p + q)φ 2 /2 + (pc + β )φ − k.

(35)

Applying a Legendre transformation to the Lagrangian L in (34), one can find the corresponding Hamiltonian to be: 1 p (36) H = ( √ )2 +V (φ ), 2 M where the conjugate momentum p =

∂L ∂φ

= M(φ )φ .


95

Next, let us search for isochronous behavior via the use of the JLM. If such behavior were found, it would correspond to period traveling wavetrains in the exROE equation. Define the canonical variables: p and Q = Q(φ ) (37) P= √ M

that Q (φ ) = to be some function of φ such that the Poisson bracket [P, Q] = [p, q] is invariant. This implies 2 /2, implies that M(φ ). Assuming that there exists a linearizing transformation such that V ( φ ) → Q( φ ) V (φ ) = Q(φ )Q (φ ) = M(φ )Q(φ ), so that: V (φ ) = 2φ . Q(φ ) = M(φ )

Integrating Q (φ ) =

(38)

M(φ ), we obtain the following equation: k φ Q(φ ) = (p + q) + (pc + β ) − . 2 φ

(39)

Since Q cannot be determined uniquely, it cannot be a canonical variable. Thus, the potential cannot be directly mapped to a linear harmonic oscillator. Thus, at least within the famework of this method, we do not find any parameter sets for the paramter c for which the exROE traveling-wave equation (1) has isochronous solutions corresponding to singly-periodic traveling wavetrains of the exROE NLPDE. Hence, we turn next to the construction of solitary waves of the exROE equation using a variational approach. 5 Variational formulation 5.1

The variational approximation for regular solitons

The procedure for constructing regular solitary waves with exponentially decaying tails is well-known. It is widely employed in many areas of Applied Mathematics and goes by the name of the Rayleigh-Ritz method. In this section, we shall employ it to construct regular solitary waves of (1). The localized regular solitary wave solutions will be found by assuming a Gaussian trial function: u = A exp(−

z2 ) ρ2

(40)

and substituting (40) into the Lagrangian (34). Note that it is standard to use such Gaussian ansatzën for analytic tractability. This is true even for simpler nonlinear PDEs where exact solutions may be known, and have the usual sech or sech2 functional forms. The exponential trial function typically captures these more exact solitary wave forms extremely well in the core or central region of the soliton, with the two often being indistinguishable when plotted together. However, the accuracy is typically somewhat worse in the tails, sometimes with errors of upto a few percent there. Next, substituting the trial function into the Lagrangian and integrating over all space yields the ‘averaged Lagrangian’ or action (41): √ √ √ A π (9A3 − 4A2 3(p + q)ρ 2 − 18A 2ρ 2 (pc + β ) + 72kρ 2 ). 72ρ

(41)

The next step is to optimize the trial functions by varying the action with respect to the trial function parameters, viz. the core amplitude A, and the core width ρ . This determines the optimal parameters for the trial function or solitary wave solution, but within the particular functional form chosen for the trial function ansatz,

96


in this case a Gaussian. The resulting variational Euler-Lagrange equations, by varying A and ρ respectively, are the system of algebraic equations: √ √ 3A3 − 3(p + q)ρ 2 A2 − 3 2(pc + β )ρ 2 A + 6kρ 2 )) = 0, √ √ 9A3 + 4 3(p + q)ρ 2 A2 + 18 2(pc + β )ρ 2 A − 72kρ 2 = 0. is:

(42) (43)

Given their relative simplicity, and assuming a1 = 1/2, a3 = 1, a nontrivial solution to equations (42)-(43)

A=

ρ2 =

√ √ (−9 6(pc + β ) + 840 3(p + q)k + 486(pc + β )2 ) 14(p + q) 3A3 √ √ . (3A(pc + β ) 2 + A2 (p + q) 3 − 6k)

,

(44) (45)

The optimized variational soliton for the regular solitary waves of the traveling-wave equation (1) is given by the trial function (40) with the above A and ρ . Figure 7 shows the resulting regular solitary wave solution for various values of the parameters. Figure 8 shows a direct analysis of the accuracy of the variational regular solitary waves obtained above. In this instance, we are able to do a direct accuracy analysis since our variational solution for the regular solitary waves given by (40), (44) and (45) is, unlike for most variational solutions, an analytical one. Inserting this variational solution (40) (with (44) and (45)) into the traveling-wave ODE (1), the deviation of the left-hand side of (1) from zero gives a direct measure of the goodness of the variational solution. Figure 8 shows this left-hand side for p = 1, q = 1, β = 0.5 and k = 2. For positive values of the wave-speed c, the error is small. However, the error grows rapidly for c negative. 5.2

The variational approximation for embedded solitons

In the recent and novel variational approach to embedded solitary waves, the tail of a delocalized soliton is modeled by: (46) utail = α cos(κ (c)z). Our embedded solitary wave will be embedded in a sea of such delocalized solitons. The cosine functional form ensures an even solution, and the arbitrary function κ (c) will, as shown below, help to ensure the integrability of the action. Our ansatz for the embedded soliton uses a second order exponential core model plus the above tail model [32, 33]: u = A exp(−

z2 ) + utail . ρ2

(47)

Plugging this ansatz into the Lagrangian (34) and reducing the trigonometric powers to double and triple angles yields an equation with trigonometric functions of the double and triple angles, as well as terms linear in z. The former would make spatial integration or averaging of the Lagrangian divergent. However, it is possibly to rigorously establish, following a procedure analogous to proofs of Whitham’s averaged Lagrangian technique [34], that such terms may be averaged out, so we shall set them to zero a priori. The terms linear in z would also cause the Lagrangian to be non-integrable. To suppress these, we therefore set: 2 pc + β κ (c) = ± , (48) α


(a)

(b)

(c)

(d)

97

Fig. 7 (a) The regular soliton for p = 1, q = 1, β = 0.5 and k = 2, for different values of c. (b) The regular soliton for p = 1, q = 1, β = 0.5 and c = 2, for different values of k. (c) The regular soliton for p = 1, β = 0.5, c = 2 and k = 2, for different values of q. (d) The regular soliton for p = 1, q = 1, c = 2 and k = 2, plotted for different values of β .

which makes linear terms zero. Note that this step, and the preceding step of averaging out trigonometric functions of the higher angles are recent ones for the variational approximation of embedded solitary waves. They are not part of the traditional Rayleigh-Ritz method used for the construction of regular solitary waves. Next, the rest of the equation can be integrated to give the action: √ 1 2 2 4 3 2 2 α2 √ 1 2 A ρ (p + q) + (((−2((β − κ α + pc)ρ − ) 2A − (49) 8ρ 2 2 9 95 2 2 89 2 2 16 27 κ 2α 2 + pc)ρ 2 e 24 κ ρ (−2(p + q)α 2 + 8k)ρ 2 + A3 )e 24 κ ρ + α (− (β − 27 √ 2 2 2 ρ2 √ κ 71 83 2 2 27 27 2 1 − ρ 2 (p + q)e 6 (α e 4 + 2Ae23 ) + α (κ 2 ρ 2 + )Ae 24 κ ρ 8 8 2 √ 31 2 2 41 2 2 95 2 2 √ 27 + 3A2 (κ 2 ρ 2 + 3)e 8 κ ρ + α 2 ρ 2 κ 2 e 24 κ ρ ))e− 24 κ ρ π A). 4 As for the regular solitary waves, the action is now varied with respect to the core amplitude A, the core width

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Fig. 8 The error for p = 1, q = 1, β = 0.5 and k = 2: the error is small for positive c, but grows fast for c negative.

ρ , and the small amplitude α of the oscillating tail. For strictly embedded solitary waves, which occur on isolated curves in the parameter space where a continuum or “sea” of delocalized solitary waves exist, the amplitude of the tail is strictly zero. Once again, this is an extra feature not encountered in the standard variational procedure. Hence, we also need to set α = 0 in these three variational equations to recover such embedded solitary waves. Implementing this, we have: √ √ (ρ 2 6k − 3 2(pc + β )A − 3(p + q)A2 ) + 3A3 = 0 √ √ (ρ 2 − 72k + 18 2(pc + β )A + 4 3(p + q)A2 ) + 9A3 = 0 √ 31ρ 2 κ 2 89ρ 2 κ 2 ) + 108(pc + β )ρ 2 exp( ) −8 3A2 (ρ 2 κ 2 + 3) exp( 8 24 √ 23ρ 2 κ 2 )=0 +27 2Aρ 2 (p + q) exp( 6

(50) (51) (52)

Subtracting the two equations (50), (51), one may obtain an expression for A in terms of ρ . Solving the two equations obtained by substituting this expression for A into the equation (50) and the equation (52) yields the solutions c = 0, ρ 2 = constant. Thus, no non-trivial embedded soliton solutions result for the exROE equation. One may also see this from a linearized or tail analysis of the traveling wave equation (1) which does not support oscillatory solutions. 6 Conclusions Three recent analytical approaches have been applied in this paper to treat the possible classes of traveling wave solutions of the so-called extended-reduced Ostrovsky (exROE) equations. A recent, novel application of phase-plane analysis is first employed to show the existence of breaking kink wave and smooth periodic (compacton) solutions in certain parameter regimes. Smooth traveling waves are next considered using a recent technique to derive convergent multi-infinite series solutions for the homoclinic orbits of the traveling-wave equations for the exROE equation, as well as for its generalized version with arbitrary coefficients. These correspond to pulse or solitary wave solutions respec-


99

tively of the original PDE. Unlike the majority of unaccelerated convergent series, high accuracy is attained with relatively few terms. We also show the traveling wave nature of these pulse and front solutions. Finally, variational methods are employed to treat families of both regular and embedded solitary wave solutions for the exROE PDE. The technique for obtaining the embedded solitons incorporates several recent generalizations of the usual variational technique and is thus topical in itself. One unusual feature of the solitary waves derived here is that we are able to obtain them in analytical form (within the assumed ansatz for the trial functions). Thus, a direct error analysis is performed, showing the accuracy of the resulting solitary waves. Given the importance of wave solutions in dynamics and information propagation, and the fact that quite little is known about solutions of the family of generalized exROE equations considered here, the results obtained are both new and topical. References [1] Parkes, E.J. (2008), Periodic and solitary travelling-wave solutions of an extended reduced Ostrovsky equation, SIGMA Symmetry Integrability Geom. Methods Appl., 4, Paper 053, 17. [2] Stepanyants, Y. A. (2008), Solutions classification to the extended reduced Ostrovsky equation, SIGMA Symmetry Integrability Geom. Methods Appl., 4, Paper 073, 19. [3] Stepanyants, Y.A. (2006), On stationary solutions of the reduced Ostrovsky equation: periodic waves, compactons and compound solitons, Chaos Solitons Fractals, 28(1), 193–204. [4] Ostrovskii, L.A. (1978), Nonlinear internal waves in the rotating ocean, Okeanologiia, 18, 181–191. [5] Choudhury, S.R. and Gambino, G. (2013), Convergent analytic solutions for homoclinic orbits in reversible and nonreversible systems, Nonlinear Dynam., 73(3), 1769–1782. [6] Rehman, T., Gambino, G., and Choudhury, S. R. (2014), Smooth and non-smooth traveling wave solutions of some generalized Camassa-Holm equations, Commun. Nonlinear Sci. Numer. Simulat., 19(6), 1746–1769. [7] Li, J. (2010), The dynamics of two classes of singular nonlinear travelling wave equations and loop solutions, In C. David and Z. Feng, editors, Solitary waves in fluid media, Bentham Science, Sharjah. [8] Li, J. and Dai, H. (2007), On the study of singular nonlinear traveling wave equations: dynamical approach, Science Press, Beijing. [9] Wang, X. (2009), Si’lnikov chaos and Hopf bifurcation analysis of Rucklidge system, Chaos Solitons Fractals, 42(4), 2208 – 2217. [10] Jacobi, C.G.J. (1844), Theoria novi multiplicatoris systemati aequationum differentalium vulgarium applicandi, J. fur ¨ Math., 27,199. [11] Lie, S. (1874), Veralgemeinerung und neue Verwerthung der Jacobischen Multiplicator- Theorie. Fordhandlinger i Videnokabs - Selshabet i Christiania, 255–274. [12] Whittaker, E. T. (1918), Lezione sulla teoria dei gruppi continui finiti di transformazioni, Enrico Spoerri Ed., Pisa. [13] Madhava Rao, B.S. (1940), On the reduction of dynamical equations to the Lagrangian form, Proc. Benares Math. Soc., n. Ser., 2, 53–59. [14] Whittaker, E. T. (1988), A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Cambridge University Press, Cambridge. [15] Nucci, M. C. (2005), Jacobi last multiplier and Lie symmetries: a novel application of an old relationship, J. Nonlinear Math. Phys., 12(2), 284–304. [16] Nucci, M. C. and Leach, P. G. L. (2008), Jacobi’s last multiplier and Lagrangians for multidimensional systems, J. Math. Phys., 49(7), 073517–8. [17] Nucci, M. C. and Tamizhmani, K. M. (2010), Lagrangians for dissipative nonlinear oscillators: the method of Jacobi last multiplier, J. Nonlinear Math. Phys., 17(2), 167–178. [18] Choudhury, A. G. Guha, P. and Khanra, B. (2009), On the Jacobi last multiplier, integrating factors and the Lagrangian formulation of differential equations of the Painlevé-Gambier classification, J. Math. Anal. Appl., 360(2), 651–664. [19] Calogero, F. (2001), Isochronous dynamical systems, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 369(1939), 1118–1136. [20] Calogero, F. (2012), Isochronous Systems, Oxford University Press, Oxford. [21] Chalykh, O.A. and Veselov, A.P. (2005), A remark on rational isochronous potentials, J. Nonlinear Math. Phys., 12(suppl. 1), 179–183. [22] Nucci, M. C. and Leach, P. G. L. (2002), Jacobi’s last multiplier and the complete symmetry group of the Euler-Poinsot system, J. Nonlinear Math. Phys., 9(suppl. 2), 110–121. Special issue in honour of P. G. L. Leach on the occasion of his 60th birthday. [23] Nucci, M. C. and Leach, P. G. L. (2004), Jacobi’s last multiplier and symmetries for the Kepler problem plus a lineal

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story, J. Phys. A, 37(31), 7743–7753. [24] Nucci, M. C. (2007), Jacobi’s last multiplier, Lie symmetries, and hidden linearity: profusion of “goldfish”, Teoret. Mat. Fiz., 151(3), 495–509. [25] Tanriver, U., Choudhury, S. R. and G. Gambino (2015) Lagrangian dynamics and possible isochronous behavior in several classes of nonlinear second order oscillators via the use of Jacobi last multiplier, Int. J. Nonlinear Mech., 74, 100–107. [26] Conte, R. (2007), Partial integrability of the anharmonic oscillator, J. Nonlinear Math. Phys., 14(3), 454–465. [27] Hill, J. M., Lloyd, N. G. and Pearson, J. M. (2007) Algorithmic derivation of isochronicity conditions, Nonlinear Anal., 67(1), 52–69. [28] Cherkas, L. A. (1976), Conditions for a Lienard equation to have a center, Differensial ye Uravneniya, 12, 201–206. [29] Choudhury, A. G. and Guha, P. (2010), On isochronous cases of the Cherkas system and Jacobi’s last multiplier, J. Phys. A, 43(12), 125202. [30] Guha, P. and Choudhury, A. G. (2011), The role of the Jacobi last multiplier and isochronous systems, Pramana, 77(5), 917–927. [31] Guha, P. and Choudhury, A. G. (2013), The Jacobi last multiplier and isochronicity of Liénard type systems, Rev. Math. Phys., 25(6), 1330009–31. [32] Kaup, D.J. and Malomed, B. A. (2003), Embedded solitons in lagrangian and semi-lagrangian systems, Physica D, 184(14), 153 – 161. Complexity and Nonlinearity in Physical Systems – A Special Issue to Honor Alan Newell. [33] Kaup, D.J. and Vogel, T.K. (2007), Quantitative measurement of variational approximations, Phys. Lett. A, 362(4), 289–297. [34] Whitam, G.B. (1974), Linear and Nonlinear Waves, Wiley, New York.

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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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An Interdisciplinary Journal of Discontinuity, Nonlinearity, and Complexity Volume 5, Issue 1

March 2016

Contents Introducing Increasing Returns to Scale and Endogenous Technological Progress in the Structural Dynamic Economic Model SDEM-2 Dmitry V. Kovalevsky................................................................................................

1－8

The Existence of Optimal Control for Semilinear Distributed Degenerate Systems M. Plekhanova………...………………..…...…………………….………………..

9－18

Spin-transfer Torque and Topological Changes of Magnetic Textures Alberto Verga............................................................................................................

19－24

Equilibrium States Under Constraint in a Variational Problem on a Surface Panayotis Vyridis, M.K. Christophe Ndjatchi, Fernando Garcıa Flores, Julio Cesar Urbina……………………….…………………...……………..…………...

25－32

Stability of Hopfield Neural Networks with Delay and Piecewise Constant Argument M.U. Akhmet, M. Karacaoren………………………..…….....…...…...…………..

33－42

Vibrational Resonance in a System with a Signum Nonlinearity K. Abirami, S. Rajasekar, M.A.F. Sanjuan……………..…………………...…..….

43－58

Controllability of Nonlinear Fractional Delay Integrodifferential Systems R. Joice Nirmala, K. Balachandran....…...………………………..…………….....

59－73

Existence of Stationary Solutions for some Systems of Integro-Differential Equations Vitali Vougalter, Vitaly Volpert…….……………………………..…………….....

75－84

Regular and Singular Pulse and Front Solutions and Possible Isochronous Behavior in the Extended-ReducedOstrovskyEquation: Phase-Plane,Multi-Infinite Series and Variational Formulations U. Tanriver, G. Gambino, S. Roy Choudhury…….………...……..…………….....

85－100

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