Free ebooks ==> www.Ebook777.com www.Ebook777.com Recent Advances in Applied Nonlinear Dynamics with Numerical Analysis Fractional Dynamics, Network Dynamics, Classical Dynamics and Fractal Dynamics with their Numerical Simulations 8637hc_9789814436458_tp.indd 28/11/12 4:18 PM Free ebooks ==> www.Ebook777.com INTERDISCIPLINARY MATHEMATICAL SCIENCES* Series Editor: Jinqiao Duan (University of California, Los Angeles, USA) Editorial Board: Ludwig Arnold, Roberto Camassa, Peter Constantin, Charles Doering, Paul Fischer, Andrei V Fursikov, Xiaofan Li, Sergey V Lototsky, Fred R McMorris, Daniel Schertzer, Bjorn Schmalfuss, Yuefei Wang, Xiangdong Ye, and Jerzy Zabczyk Published Vol 5: The Hilbert–Huang Transform and Its Applications eds Norden E Huang & Samuel S P Shen Vol 6: Meshfree Approximation Methods with MATLAB Gregory E Fasshauer Vol 7: Variational Methods for Strongly Indefinite Problems Yanheng Ding Vol 8: Recent Development in Stochastic Dynamics and Stochastic Analysis eds Jinqiao Duan, Shunlong Luo & Caishi Wang Vol 9: Perspectives in Mathematical Sciences eds Yisong Yang, Xinchu Fu & Jinqiao Duan Vol 10: Ordinal and Relational Clustering (with CD-ROM) Melvin F Janowitz Vol 11: Advances in Interdisciplinary Applied Discrete Mathematics eds Hemanshu Kaul & Henry Martyn Mulder Vol 12: New Trends in Stochastic Analysis and Related Topics: A Volume in Honour of Professor K D Elworthy eds Huaizhong Zhao & Aubrey Truman Vol 13: Stochastic Analysis and Applications to Finance: Essays in Honour of Jia-an Yan eds Tusheng Zhang & Xunyu Zhou Vol 14: Recent Developments in Computational Finance: Foundations, Algorithms and Applications eds Thomas Gerstner & Peter Kloeden Vol 15: Recent Advances in Applied Nonlinear Dynamics with Numerical Analysis: Fractional Dynamics, Network Dynamics, Classical Dynamics and Fractal Dynamics with Their Numerical Simulations eds Changpin Li, Yujiang Wu & Ruisong Ye *For the complete list of titles in this series, please go to http://www.worldscientific.com/series/ims www.Ebook777.com RokTing - Recent Advs in Applied Nonlinear.pmd 11/21/2012, 4:43 PM Interdisciplinary Mathematical Sciences – Vol 15 Recent Advances in Applied Nonlinear Dynamics with Numerical Analysis Fractional Dynamics, Network Dynamics, Classical Dynamics and Fractal Dynamics with their Numerical Simulations Editors Changpin Li Shanghai University, China Yujiang Wu Lanzhou University, China Ruisong Ye Shantou University, China World Scientific NEW JERSEY 8637hc_9789814436458_tp.indd • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TA I P E I • CHENNAI 28/11/12 4:18 PM Free ebooks ==> www.Ebook777.com Published by World Scientific Publishing Co Pte Ltd Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Interdisciplinary Mathematical Sciences — Vol 15 RECENT ADVANCES IN APPLIED NONLINEAR DYNAMICS WITH NUMERICAL ANALYSIS Fractional Dynamics, Network Dynamics, Classical Dynamics and Fractal Dynamics with Their Numerical Simulations Copyright © 2013 by World Scientific Publishing Co Pte Ltd All rights reserved This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA In this case permission to photocopy is not required from the publisher ISBN 978-981-4436-45-8 Printed in Singapore www.Ebook777.com RokTing - Recent Advs in Applied Nonlinear.pmd 11/21/2012, 4:43 PM November 21, 2012 14:53 World Scientific Book - 9.75in x 6.5in Professor Zhong-hua Yang v ws-book975x65-rev Free ebooks ==> www.Ebook777.com This page intentionally left blank www.Ebook777.com October 5, 2012 15:23 World Scientific Book - 9.75in x 6.5in ws-book975x65-rev Foreword Almost no one doubts that Dynamics is always an exciting and serviceable topic in science and engineering Since the founder of dynamical systems, H Poincar´e, there have been great theoretical achievements and successful applications Meanwhile, complex and multifarious dynamical evolutions and new social requests produce new branches in the field of dynamical systems, such as fractional dynamics, network dynamics, and various genuine applications in industrial and agricultural production as well as national construction Although fractional calculus, in allowing integrals and derivatives of any positive real order (the term “fractional” is kept only for the historical reasons) even complex number order, has almost the same history as the classical calculus, fractional dynamics is still in the budding stage As far as we know, the beginning era of fractional dynamics very possibly originates from a paper on the Lyapunov exponents of the fractional differential systems published in Chaos in 2010 On the other hand, there have existed a huge number of publications in network dynamics albeit it appeared in 1990’s Besides, network dynamics has penetrated into various sources and more and more theories and applications will be prominently emerged With the rapid developments of the nonlinear dynamics, this volume timely collects contributions of recent advances in fractional dynamics, network dynamics, fractal dynamics and the classical dynamics The contents cover applied theories, numerical algorithms and computations, and applications in this regard First chapter contributes to surveys on Gronwall inequalities where the singular case has been emphasized which are often used in the fractional differential systems In the second chapter, recent results of existence and uniqueness of the solutions to the fractional differential equations are presented In the next chapter, the finite element method and calculation for the fractional differential equations are summarized and introduced In following three chapters, the numerical method and calculations for fractional differential equations are proposed and numerically realized, where the fractional step method, the spectral method, and the discontinuous finite element method, are used to solve the fractional differential equations, respectively In the seventh chapter, recent results on the asymptotic expansion of a singularly perturbed problem under curvilinear coordinates are shown with the aid of classical vii October 5, 2012 15:23 World Scientific Book - 9.75in x 6.5in ws-book975x65-rev Free ebooks ==> www.Ebook777.com viii Recent Advances in Applied Nonlinear Dynamics with Numerical Analysis Laplace transformation Chapter contributes to investigating the typically dynamical numerical solver-incremental unknowns methods under the background of alternating directional implicit (ADI) scheme for a heat conduction equation Chapter generalizes the sharp estimates of the two-dimensional problems to the stability analysis of three-dimensional incompressible Navier-Stokes equations solved numerically by a colocated finite volume scheme In the tenth chapter, numerical algorithms for the computation of certain symmetric positive solutions and the detection of symmetry-breaking bifurcation points on these or other symmetric positive solutions for p-Henon equation are studied In the following chapter, recent results of block incremental unknowns for solving reaction-diffusion equations are presented Chapters 12, 13 and 19 contribute to network dynamics, where the models and synchronization dynamics are introduced and analyzed in details Chapter 14 focuses on chaotic dynamical systems on fractals and their applications to image encryption Chapter 15 makes contribution to the generation of the planar crystallographic symmetric patterns by discrete systems invariant with respect to planar crystallographic groups from a dynamical system point of view Chapter 16 investigates the complicated dynamics of a simple two-dimensional discrete dynamical system Chapter 17 discusses the bifurcations in the delayed ordinary differential equation and the next chapter introduces the numerical methods for the option pricing problems We are very grateful to all the authors for their contributions to this volume We specially thank Ms Tan Rok Ting for her sparing no pains to inform us, replying to us and explaining various details regarding this edited volume The mostly mentionable question is that the published year of this book happens to be the year of Professor Zhong-hua Yang’s 70th birthday We are privileged and honored to dedicate this edited book to Professor Zhong-hua Yang, our teacher and life-long friend CL acknowledges the financial support of the National Natural Science Foundation of China (grant no 10872119), the Key Disciplines of Shanghai Municipality (grant no S30104), and the Key Program of Shanghai Municipal Education Commission (grant no 12ZZ084) Changpin Li, Shanghai University Yu-jiang Wu, Lanzhou University Ruisong Ye, Shantou University May 28, 2012 www.Ebook777.com October 5, 2012 15:23 World Scientific Book - 9.75in x 6.5in ws-book975x65-rev Preface This festschrift volume is dedicated to Professor Zhong-hua Yang on the occasion of his 70th birthday Zhong-hua Yang was born on October 5, 1942, in Shanghai, China He graduated in 1964 from Fudan University, a prestigious university in China After graduation, he was recruited to Shanghai University of Science and Technology (now is called Shanghai University), as a faculty member at the Department of Mathematics In 1982, Yang published his first research paper and in the same year he went to California Institute of Technology as a senior visiting scholar to work with the famous mathematician, Professor H.B Keller, for advanced studies on theory and computation of bifurcation Two years later, he returned to Shanghai University of Science and Technology, where he spent twenty years as a faculty member He has published 70 articles ranging in computational and applied mathematics, especially in computation of bifurcation In 1989, he was promoted to full professor and appointed as associate director of Department of Mathematics at the university In 1995, he was appointed as an advisor of the graduated students for Ph.D degree In 1988, 1992 and 1998, he was granted the Science and Technology Progress Award for three times by Ministry of Education, China In the 1990’s, Yang worked on bifurcation computation and applications for nonlinear problems, one of the projects in National “Climbing” Program He has received special government allowance from the State Council of China since 1992 He was then awarded Shanghai splendid educator in 1995 In 1996, he moved to Shanghai Normal University, and acted as vice dean of the School of Math Science (1997-2002) His academic positions and responsibilities also include: Editor of the journal Numerical Mathematics: A Journal of Chinese Universities (English Series), Council member of Shanghai Mathematics Society and reviewer for Mathematical Reviews In 2007, his book Nonlinear Bifurcation: Theory and Computation was published, which was the first monograph on bifurcation computation in China ix October 5, 2012 15:23 World Scientific Book - 9.75in x 6.5in Synchronization and its control between two coupled networks ws-book975x65-rev 377 where p˜˙ij = −eTi Γyj , q˜˙ij = −eTi Γ(xj + yi ), H˙ i = µi ei , µi > 0, then antisynchronization between networks (19.6) and (19.7) can be realized under the controllers Ui , i = 1, 2, · · · , N 19.2.3 Numerical examples Choosing the Chua’s systems as the network node dynamics The Chua’ system is described by [Chua et al (1993)], x˙ i1 = α(xi2 − xi1 − φ(xi1 )), x˙ i2 = xi1 − xi2 + xi3 , x˙ i3 = −βxi2 , where φ(xi1 ) = bxi1 + (a − b)[|xi1 + 1| − |xi1 − 1|] and a, b, α, β are parameters For simplicity, we choose f (·), g(·) as Chua’s systems with different parameters Network X: a = −1.27, b = −0.68, α = 10.00, β = 14.87; while network Y: a = −1.39, b = −0.75, α = 10.00, β = 18.60 The innercoupling matrix Γ is an identity matrix In the numerical simulation, the initial values of state vectors X0 , Y0 and control Ei (0), Hi (0) are randomly chosen in (0, 1) Let N [(yij (t) + xij (t))2 ] e(t) = i=1 j=1 be the 2-norm of the total anti-synchronization errors at time t, for t ∈ (0, +∞) Example 18.1 Consider the network (19.3)-(19.4) The network size N is taken as 10 and configuration matrices of networks X and Y are given as follows, −3 A= 1 0 −4 0 0 −3 0 1 −4 1 0 −3 1 −5 0 1 −4 1 1 0 1 0 1 1 0 0 , 1 0 −4 −3 0 −4 (19.9) October 5, 2012 15:23 World Scientific Book - 9.75in x 6.5in ws-book975x65-rev Free ebooks ==> www.Ebook777.com 378 Recent Advances in Applied Nonlinear Dynamics with Numerical Analysis −2 B= 0 0 −4 1 −4 1 1 −3 0 1 −4 0 0 −5 0 0 −3 1 0 1 0 0 0 0 0 0 1 1 0 −5 −4 −3 (19.10) If the controllers Ui = 0, i = 1, 2, · · · , 10, through the numerical simulation, we find that anti-synchronization doesn’t happen for arbitrary value of θ in [0, 1] Now we take the value of θ is 0.5, and use the adaptive controllers proposed in Theorem 19.1 to make these two networks anti-synchronize Fig 19.1 plots the anti-synchronization error for εi = 2, i = 1, · · · , 10 18 16 14 ||e(t)|| 12 10 0 0.5 1.5 t 2.5 3.5 Fig 19.1 Anti-synchronization error between networks (19.3) and (19.4) with εi = 2, i = 1, · · · , 10 Example 18.2 Consider the network model (19.6)-(19.7) Let configuration matrices A and B be (19.9) and (19.10) respectively Here, C and D are 10 × 10 matrices with random entries, chosen from a uniform distribution on the interval (0, 0.4) Anti-synchronization also doesn’t exist between networks (19.6)–(19.7) for Ui = 0, and the adaptive controllers in Theorem 19.2 are applied The antisynchronization between networks (19.6) and (19.7) is shown in Fig 19.2 with µi = 2, i = 1, · · · , 10 www.Ebook777.com October 5, 2012 15:23 World Scientific Book - 9.75in x 6.5in ws-book975x65-rev Synchronization and its control between two coupled networks 379 18 16 14 ||e(t)|| 12 10 0 10 t Fig 19.2 Anti-synchronization error between networks (19.6) and (19.7) with µi = 2, i = 1, · · · , 10 19.3 Pinning anti-synchronization between two general complex dynamical networks Consider the following drive-response networks N x˙ i (t) = f (xi (t), t) + c1 N aij Γxj (t) + c2 j=1 N y˙ i (t) = f (yi (t), t) + c1 bij Γxj (t − τ (t)), N aij Γyj (t) + c2 j=1 (19.11) j=1 bij Γyj (t − τ (t)) + ui , (19.12) j=1 for i = 1, 2, , N Here xi , yi ∈ Rn are respectively the state vector of the ith node in drive network X and response network Y, f : Rn × R+ → Rn is a continuously differentiable nonlinear vector-valued function, c1 , c2 > are the coupling strength, Γ is a inner-coupling matrix between nodes, τ (t) is the time-varying delay, and A = (aij )N ×N , B = (bij )N ×N are respectively the coupling matrices representing the topological structure of the networks X and Y for non-delayed configuration and delayed one The entries aij (bij ) are defined as follows: aij (bij ) > if there is a connection between node i and node j (i = j); otherwise aij (bij ) = (i = j), and N N the diagonal entries aii = − j=1,j=i aij , bii = − j=1,j=i bij , i = 1, 2, , N ui is a controller to be designed To investigate the anti-synchronization between networks X and Y, the pinning strategy will be introduced Without loss of generality, assuming that the first l (1 ≤ l ≤ N ) nodes of network Y are selected The pinning adaptive feedback October 5, 2012 15:23 World Scientific Book - 9.75in x 6.5in ws-book975x65-rev Free ebooks ==> www.Ebook777.com 380 Recent Advances in Applied Nonlinear Dynamics with Numerical Analysis controllers ui can be described by ui = −c1 di (t)Γei (t), d˙i (t) = qi eTi (t)Γei (t), i = 1, 2, , l, ui = 0, i = l + 1, , N, (19.13) where qi are positive constants A1 A3 AT3 A2 where A1 ∈ Rl×l , A2 ∈ R(N −l)×(N −l) , A3 ∈ Rl×(N −l) According to (19.11), (19.12) and (19.13), we obtain the following error dynamical network N e˙ i (t) = f (yi (t), t) + f (xi (t), t) + c1 aij Γej (t) j=1 N + c bij Γej (t − τ (t)) − c1 di (t)Γei (t), i = 1, 2, , l, j=1 (19.14) d˙i (t) = qi eTi (t)Γei (t), i = 1, 2, , l, N e ˙ (t) = f (y (t), t) + f (x (t), t) + c aij Γej (t) i i i j=1 N + c bij Γej (t − τ (t)), i = l + 1, , N Suppose that the matrix A is symmetric and irreducible Let A = j=1 19.3.1 Pinning anti-synchronization criterion In order to derive the anti-synchronization criterion, some useful assumptions and lemmas are presented as follows Assumption 19.3 For any x = (x1 , x2 , , xn )T ∈ Rn , y = (y1 , y2 , , yn )T ∈ Rn , there exists a positive constant L1 such that (y − x)T (f (y, t) − f (x, t)) ≤ L1 (y − x)T Γ(y − x), where Γ is a positive definite matrix Here x and y are time-varying vectors Assumption 19.4 f (x, t) is an odd function of x, i.e., f (−x, t) = −f (x, t) for arbitrary x ∈ Rn Assumption 19.5 τ (t) is a differential function with ≤ τ˙ (t) ≤ ε < Clearly, constant time delay is a special case of this assumption Lemma 19.2 (Schur complement) The following linear matrix inequality (LMI) A(x) B(x) (B(x))T C(x) > 0, www.Ebook777.com October 5, 2012 15:23 World Scientific Book - 9.75in x 6.5in ws-book975x65-rev Synchronization and its control between two coupled networks 381 where A(x) = (A(x))T , C(x) = (C(x))T , is equivalent to one of the following conditions: (a) A(x) > and C(x) − (B(x))T (A(x))−1 B(x) > 0; (b) C(x) > and A(x) − B(x)(C(x))−1 (B(x))T > Here, the pinning adaptive anti-synchronization between the drive network X and the response network Y is investigated, and the main result is summarized in the following theorem Theorem 19.3 Suppose that Assumptions 19.3, 19.4 and 19.5 hold, Γ is a positive definite matrix and A is symmetric and irreducible If λmax (A2 ) < −L/c1 , (19.15) where P = (B T B)⊗ΓT , k = λmax (P )/λmin (IN ⊗Γ), L = L1 +(c2 k)/(2(1−ε))+c2 /2, then the networks X and Y can realize anti-synchronization 19.3.2 Numerical simulations In the numerical simulations throughout this subsection, the coupling matrices A and B obey the scale-free distribution of the BA network model [Barab´asi and Albert (1999)] with m0 = m = 8, N = 100 and the small-world model [Watts and Strogatz (1998)] with the link probability p = 0.1, m = 4, N = 100, respectively, the initial values are randomly chosen in the interval (0, 1) and the quantity E(t) = max{ xi (t) + yi (t) : i = 1, 2, , N }, for t ∈ [0, +∞) is used to measure the quality of the anti-synchronization process The dynamics at every node of both networks X and Y are described by Chua’s system [Chua et al (1993)], α(x2 (t) − x1 (t) − φ(x1 (t))), (19.16) x(t) ˙ = f (x(t), t) = x1 (t) − x2 (t) + x3 (t), − βx2 (t), where x(t) = (x1 (t), x2 (t), x3 (t))T , φ(x1 (t)) = bx1 (t)+ 21 (a−b)(|x1 (t)+1|−|x1 (t)−1|) and a = −1.27, b = −0.68, α = 10.00, β = 14.87 For simplicity, setting Γ = diag(2, 2, 2), c1 = 30, c2 = 1, τ (t) = 1, then one gets k = 72.5671, ε = It is easy to see that f (x(t), t) is an odd function of x(t) Thus, one has (y(t) − x(t))T (f (y(t), t) − f (x(t), t)) = −10(y1 (t) − x1 (t))2 + 11(y1 (t) − x1 (t))(y2 (t) − x2 (t)) − (y2 (t) − x2 (t))2 − 13.87(y2 (t) − x2 (t))(y3 (t) − x3 (t)) − 10(y1 (t) − x1 (t))(φ(y1 (t)) − φ(x1 (t))) ≤ 2.7(y1 (t) − x1 (t))2 − (y2 (t) − x2 (t))2 + 11(y1 (t) − x1 (t))(y2 (t) − x2 (t)) − 13.87(y2 (t) − x2 (t))(y3 (t) − x3 (t)) October 5, 2012 15:23 World Scientific Book - 9.75in x 6.5in ws-book975x65-rev Free ebooks ==> www.Ebook777.com 382 Recent Advances in Applied Nonlinear Dynamics with Numerical Analysis 11ρ 11 13.87η )(y1 (t) − x1 (t))2 + (−1 + + )(y2 (t) − x2 (t))2 2ρ 13.87 (y3 (t) − x3 (t))2 + 2η ≤ L1 (y(t) − x(t))T Γ(y(t) − x(t)), ≤ (2.7 + where L1 > is determined by choosing appropriate parameters ρ, η > Choosing ρ = 1.4823, η = 0.6390, one has L1 = 5.4263 Then, L = L1 + (c2 k)/(2(1 − ε)) + c2 /2 = 42.2098, −L/c1 = −1.4070 In [Yu et al (2009)], it was pointed out that it is better to use the high-degree pinning scheme when a small part of nodes is controlled Here, from numerical calculation, orbits of λmax (A2 ) as functions of the number of pinned nodes by highdegree (for matrix A), low-degree (for matrix A) and random pinning schemes are shown in Fig 19.3, and we observe that one only needs 31, 24 and nodes of network Y to realize the anti-synchronization between networks X and Y by using low-degree (for matrix A), random and high-degree (for matrix A) pinning schemes, respectively Hence, it is better to use the high-degree pinning scheme in this case Furthermore, it is found that we should use the low-degree pinning scheme when −L/c1 < −4 Now, we apply adaptive feedback control to the first most highly connected nodes of the non-delayed configuration matrix A based on (19.11), (19.12) and (19.13) with qik = 2, k = 1, , The evolution of anti-synchronization error E(t) is illustrated in Fig 19.4, which shows that the anti-synchronization between networks X and Y is achieved −2 −8 high degree low degree random max (A ) −6 λ −4 −10 −12 −14 10 20 30 40 50 60 Number of pinned nodes (l) 70 80 90 Fig 19.3 Orbits of λmax (A2 ) as functions of the number of pinned nodes by high-degree, lowdegree and random pinning schemes www.Ebook777.com October 5, 2012 15:23 World Scientific Book - 9.75in x 6.5in ws-book975x65-rev Synchronization and its control between two coupled networks 383 2.5 E(t) 1.5 0.5 0 0.5 1.5 t 2.5 3.5 Fig 19.4 Anti-synchronization error between networks X and Y by using high-degree pinning scheme with l = and qik = 2, k = 1, , 19.4 Generalized synchronization between two networks Consider the following driving-response (or master-slave) configurations [Wu et al (2012)]: N1 x˙ i = f (xi (t)) + µ aij Γ1 xj , i = 1, , N1 , (19.17) j=1 N2 y˙ i = g(yi (t)) + D(yi , xj ) + ε bik Γ2 yk , k=1 i = 1, , N2 , j ∈ {1, , N1 }, (19.18) where D(yi , xj ) = x˙ j − g(xj ) + (H − ∂g(xj )/∂xj )(yi − xj ) Here xi , yi ∈ Rn ; f, g : Rn → Rn are continuously differentiable µ, ε > are coupling strengths, Γ1 , Γ2 are matrices (with order n) linking coupled variables A = (aij )N1 ×N1 , B = (bij )N2 ×N2 are respectively the inner connection matrices of the driving (or master) network X and response (or slave) network Y The matrix H is an arbitrary constant Hurwitz one (whose eigenvalues lie in open left semi-plane) The chosen interaction (19.17)– (19.18) is based on the open-plus-closed-loop (OPCL) method Definition 19.2 If there exist a transformation Φ : Rn×N1 → Rn×N2 , a manifold M = {(x, y) : y = Φ(x)}, and a subset N = Nx ×Ny ⊂ Rn×N1 ×Rn×N2 with M ⊂ N such that all trajectories of (19.17)–(19.18) with initial conditions in the attractive basin N approach the manifold M as time goes to +∞, then we say networks X October 5, 2012 15:23 World Scientific Book - 9.75in x 6.5in ws-book975x65-rev Free ebooks ==> www.Ebook777.com 384 Recent Advances in Applied Nonlinear Dynamics with Numerical Analysis and Y possess the property of generalized synchronization between x ∈ Rn×N1 and y ∈ Rn×N2 19.4.1 Generalized synchronization criterion Let φtx : Rn×N1 → Rn×N1 be the flow of network X , φt = (φtx , φty ) be the flow of system (19.17)–(19.18) with φty : Rn×(N1 +N2 ) → Rn×N2 We find that the map Φ relates to the flow φty To more accurately characterize the conditions of the occurrence of generalized synchronization for (19.17)–(19.18), the following criterion is presented Lemma 19.3 Generalized synchronization occurs in (19.17)–(19.18) if and only if for all (x0 , y0 ) ∈ N the response network Y is uniformly asymptotically stable, i.e., for arbitrarily given x0 ∈ Nx , and ∀y10 , y20 ∈ Ny , lim y(t, x0 , y10 ) − t→+∞ y(t, x0 , y20 ) = In the following, in order to study the asymptotic stability of the network Y, we construct an auxiliary network Z as follows, N2 z˙i = g(zi (t)) + D(zi , xj ) + ε bik Γ2 zk , i = 1, , N2 , j ∈ {1, , N1 },(19.19) k=1 where D(zi , xj ) = x˙ j − g(xj ) + (H − ∂g(xj )/∂xj )(zi − xj ) Letting ei = yi − zi , and linearizing the error system around xj , one has N2 e˙ i = Hei + ε bik Γ2 ek , i = 1, , N2 (19.20) k=1 If we set e = (e1 , , eN2 ) ∈ Rn×N2 , then (19.20) can be rewritten in the compact form, e˙ = He + εΓ2 eB T (19.21) Then, the following lemma can be obtained Lemma 19.4 Assume that B has mk multiple eigenvalues λk ∈ R where k = 1, , , λk = αk + jβk ∈ C (αk , βk ∈ R, βk = 0) where k = + 1, , , and k=1 mk = N2 If the real parts of all eigenvalues of H + ελk Γ2 (for k = 1, , ) are negative, and (H T + H) + εαk (ΓT2 + Γ2 ) < for k = + 1, , , then the zero solution to the matrix equation (19.21) is asymptotically stable Combining Lemma 19.3 and Lemma 19.4, one has the following main result: Theorem 19.4 Assume that B has mk multiple eigenvalues λk ∈ R where k = 1, , , λk = αk + jβk ∈ C (αk , βk ∈ R, βk = 0) where k = + 1, , , and k=1 mk = N2 If the real parts of all eigenvalues of H + ελk Γ2 (for k = 1, , ) are negative, and (H T +H)+εαk (ΓT2 +Γ2 ) < for k = +1, , , then generalized synchronization occurs in (19.17)–(19.18) www.Ebook777.com October 5, 2012 15:23 World Scientific Book - 9.75in x 6.5in Synchronization and its control between two coupled networks 19.4.2 ws-book975x65-rev 385 Numerical examples In the network X, the dynamics at every node follows the Răossler system x i1 = + xi1 (xi2 − 4), x˙ i2 = −xi1 − xi3 , x˙ i3 = xi2 + 0.45xi3 , i = 1, , 10 The inner-coupling matrix is as follows, −3 0 −4 0 −3 0 −4 1 0 −3 A= 1 −5 1 −4 1 1 0 1 0 1 1 0 0 1 0 −4 −3 0 −4 (19.22) (19.23) We may simply set Γ1 = diag(1, 0, 0), µ = In the network Y, the dynamics of the individual nodes is described by the Lorenz system y˙ i1 = σ(yi2 − yi1 ), y˙ i2 = γyi1 − yi1 yi3 − yi2 , y˙ i3 = yi1 yi2 − byi3 , i = 1, , 12, where σ = 10, γ = 28, b = 8/3 The 0 0 1 0 1 0 B= 1 0 0 0 0 0 0 1 0 0 (19.24) inner-coupling matrix is chosen as below, 0 0 1 0 0 0 0 0 0 0 0 0 1 0 (19.25) 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 We may set Γ2 = diag(1, 1, 1), j = in D(yi , xj ) The Hurwitz matrix H for the Lorenz system is −σ σ γ + p1 −1 p2 , p3 p4 −b October 5, 2012 15:23 World Scientific Book - 9.75in x 6.5in ws-book975x65-rev Free ebooks ==> www.Ebook777.com 386 Recent Advances in Applied Nonlinear Dynamics with Numerical Analysis where p1 , , p4 are parameters A suitable choice of pk is p1 < − γ, p2 = p3 = p4 = Here, let p1 = −29 < − γ for the current simulation If ε = 0.3, it is easy to find that the conditions of Theorem 19.4 hold In the numerical simulations throughout this subsection, the initial values are randomly chosen in the interval (0, 1) Figs 19.5 and 19.6 show the simulation results of the driving-response networks with OPCL configurations The evolution of state variables xi = (xi1 , xi2 , xi3 )T and yi = (yi1 , yi2 , yi3 )T are shown in Fig 19.5 (a)– (c) and Fig 19.6 (a)-(c), respectively Obviously, networks X and Y not reach complete synchronization Now we introduce an auxiliary network Z which is a replica of the response network Y Fig 19.7 displays ei = (ei1 , ei2 , ei3 )T = (yi1 − zi1 , yi2 − zi2 , yi3 − zi3 )T The appearance of complete synchronization between Y and Z implies that generalized synchronization between X and Y of system (19.17)–(19.18) occurs 15 10 xi2, i=1, ,10 xi1, i=1, ,10 10 −2 −4 (a) 0 10 12 14 t (b) −6 10 12 14 t xi3, i=1, ,10 −5 (c) Fig 19.5 19.5 −10 10 12 14 t Diagrams of the state variables in network X Conclusion In this chapter, anti-synchronization and generalized synchronization between two coupled networks have been presented On condition that two networks are not ultimately connected and synchronization is necessary, we should design the controllers to realize it Based on this principle, two adaptive controllers are given to realize anti-synchronization with nonlinear signal’s connection and the inter-network actions However, it is impossible to ensure synchronization by adding controllers to all nodes due to the complexity of the dynamical network So researchers try to control a complex network by pinning part of nodes And pinning controllers have www.Ebook777.com October 5, 2012 15:23 World Scientific Book - 9.75in x 6.5in ws-book975x65-rev Synchronization and its control between two coupled networks 14 12 12 10 387 10 yi2, i=1, ,12 yi1, i=1, ,12 4 −2 −2 (a) −4 10 12 14 t −6 (b) 10 12 14 t yi3, i=1, ,12 −5 −10 (c) 0.6 0.6 0.4 0.4 0.2 0.2 −0.2 14 −0.2 −0.4 −0.6 −0.8 12 Diagrams of the state variables in network Y −0.4 (a) 10 t ei2, i=1, ,12 ei1, i=1, ,12 Fig 19.6 −0.6 10 12 14 t (b) −0.8 10 12 14 t 0.6 0.4 ei3, i=1, ,12 0.2 −0.2 −0.4 −0.6 (c) −0.8 10 12 14 t Fig 19.7 Synchronization between networks Y and Z In (a)-(c), (ei1 , ei2 , ei3 )T = (yi1 −zi1 , yi2 − zi2 , yi3 − zi3 )T , i = 1, · · · , 12 been proposed to study anti-synchronization between two coupled complex networks with non-delayed and delayed coupling Furthermore, the criterion for the occurrence of generalized synchronization in master-slave networks is introduced The theoretical criterion is based on the uniform asymptotical stability of the response network, which can be verified by utilizing the Lyapunov stability theorem Free ebooks ==> www.Ebook777.com This page intentionally left blank www.Ebook777.com October 5, 2012 15:23 World Scientific Book - 9.75in x 6.5in ws-book975x65-rev Bibliography Albert, R and Barab´ asi, A L (2002) Statistic mechanics of complex networks, Rev Mod Phys 74, pp 47–91 Arenas, A., Diaz-Guilera, A., Kurths, J., Moreno, Y and Zhou, C S (2008) Synchronization in complex networks, Phys Rep 469, pp 93–153 Barab´ asi, A L and Albert, R (1999) Emergence of scaling in random networks, Science 286, pp 509–512 Boccaletti, S., Latora, V., Moreno, Y., Chavez, M., and Hwang, D.-U (2006) Complex networks: structure and dynamics, Phys Rep 424, pp 175–308 Chen, J R., Jiao, L C., Wu, J S and Wang, X H (2009) Adaptive synchronization between two different complex networks with time-varying delay coupling, Chin Phys Lett 26, pp 060505 Chen, T P., Liu, X W and Lu, W L (2007) Pinning complex networks by a single controller, IEEE Trans Circuits Syst I 54, pp 1317–1326 Chen, M and Zhou, D (2006) Synchronization in uncertain complex netwroks, Chaos 16, pp 013101 Chua, L O., Itoh, M., Kocarev, L and Eckert, K (1993) Chaos synchronization in Chua’s circuit, J Circ Syst Comput 3, pp 93–108 Duan, Z S., Chen, G R and Huang, L (2007) Complex network synchronizability: Analysis and control, Phys Rev E 76, pp 056103 Huang, D (2006) Adaptive-feedback control algorithm, Phys Rev E 73, pp 066204 Li, C P., Sun, W G and Kurths, J (2007) Synchronization between two coupled complex networks, Phys Rev E 76, pp 046204 Li, C P., Xu, C X., Sun, W G., Xu, J and Kurths, J (2009) Outer synchronization of coupled discrete-time networks, Chaos 19, pp 013106 Li, Y., Liu, Z R and Zhang, J B (2008) Synchronization between different networks, Chin Phys Lett 25, pp 874–877 Li, X., Wang, X F and Chen, G R (2004) Pinning a complex dynamical network to its equilibrium, IEEE Trans Circuits Syst I 51, pp 2074–2087 Newman, M E J (2003) The structure and function of complex networks, SIAM Review 45, pp 167–256 Shang, Y., Chen, M Y and Kurths, J (2009) Generalized synchronization of complex networks, Phys Rev E 80, pp 027201 Sorrentino, F and Ott, E (2008) Adaptive synchronization of dynamics on evolving complex networks, Phys Rev Lett 100, pp 114101 Sun, M., Zeng, C Y and Tian, L X (2009) Generalized projective synchronization between two complex networks with time-varying coupling delay, Chin Phys Lett 389 October 5, 2012 15:23 World Scientific Book - 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9.75in x 6.5in ws-book975x65-rev Index alternating directional scheme, 144, 146 asymptotic expansion, 123, 126, 137 average path length, 226, 228, 230, 234 invariant map, 307–310, 312–314, 317, 321 binomial tree method, 364 branch switching, 193, 197 Lyapunov-Schmidt reduction, 340, 341, 343 Caputo derivative, 51, 62, 86, 106 chaotic dynamical system, 283, 284 colocated, 160, 162 complex network, 252–254, 373, 374, 386 curvilinear coordinates, 122, 123 multilevel method, 206 Koch networks, 227, 236, 246 Navier-Stokes equations, 159, 160 p-Henon equation, 191, 193 periodic solution, 340–344, 347, 348, 352–355 permutation, 282, 283, 291–294 planar crystallographic group, 307, 308 delay differential equation, 340, 355 diffusion, 281–283, 291–294, 300, 301 discontinuous Galerkin method, 107–109 random walk, 226, 227, 236, 247 reaction-diffusion equation, 206 Riemann–Liouville derivative, 51, 86 existence, 26–28, 33, 39, 42 finite element method, 54, 56, 61 fractional Cauchy problem, 105, 107 fractional conservation law, 70 fractional differential equation, 54 fractional Fokker–Planck equation, 84, 85 fractional step method, 70, 71, 73 singularly perturbed problem, 121, 122 spatiotemporal chaos, 253, 254, 258, 264 spectral method, 84, 85, 89, 90 stability, 165, 169, 170, 181, 182, 186, 187 symmetry-breaking bifurcation, 193, 194, 196–199 synchronization, 251–254 generalized synchronization, 374, 383, 384, 386, 387 Gronwall inequality, 2, tiling pattern, 306, 307, 317, 319–321 two dimensional heat equation, 144 heat equation, 144 uniqueness, 27, 28, 33, 37, 42 image encryption, 281, 282, 287, 289–291, 293–295, 301 incremental unknowns, 143, 146, 205, 206, 213 weakly singular Gronwall inequalities, 13 391 .. .Recent Advances in Applied Nonlinear Dynamics with Numerical Analysis Fractional Dynamics, Network Dynamics, Classical Dynamics and Fractal Dynamics with their Numerical Simulations... Sciences – Vol 15 Recent Advances in Applied Nonlinear Dynamics with Numerical Analysis Fractional Dynamics, Network Dynamics, Classical Dynamics and Fractal Dynamics with their Numerical Simulations... 9.7 5in x 6. 5in ws-book975x65-rev Free ebooks ==> www.Ebook777.com Recent Advances in Applied Nonlinear Dynamics with Numerical Analysis the powerful tools in the theoretical analysis of the integral