Impulsive control and synchronization of spatiotemporal chaos

The impulsive control of spatiotemporal chaos of a particular type of non-linear partial differential equations has been investigated. A criterion for the solutions of these partial differential equations to be equi-attractive in the large is determined and an estimate for the basin of attraction is given in terms of the impulse durations and the magnitude of the impulses. Extending these results to impulsively synchronize spatiotemporal chaos of the same type of partial differential equations is explored. A proof for the existence of a certain kind of impulses for synchronization such that the error dynamics is equi-attractive in the large, is established. A comparison of the developed theoretical model with other existent numerical models available in the literature has been studied. Several simulation results are given to confirm the theoretical results. Moreover, an investigation of the Lyapunov exponents of the error dynamics between impulsively synchronized spatiotemporal chaotic systems, is done to further confirm the theoretical results.

[1]  L. Chua,et al.  Impulsive control and synchronization of nonlinear dynamical systems and application to secure communication , 1997 .

[2]  V. Lakshmikantham,et al.  Stability Analysis in Terms of Two Measures , 1993 .

[3]  Ljupco Kocarev,et al.  Synchronization of Kuramoto-Sivashinsky equations using Spatially Local Coupling , 2000, Int. J. Bifurc. Chaos.

[4]  Yang Tao,et al.  Impulsive stabilization for control and synchronization of chaotic systems: theory and application to secure communication , 1997 .

[5]  Xinzhi Liu,et al.  Uniform Asymptotic Stability of Abstract Functional Differential Equations , 1997 .

[6]  Chun-Mei Yang,et al.  Control of Rössler system to periodic motions using impulsive control methods , 1997 .

[7]  Parlitz,et al.  Synchronization and control of coupled ginzburg-landau equations using local coupling , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[8]  Gang Hu,et al.  SYNCHRONIZATION OF SPATIOTEMPORAL CHAOS AND ITS APPLICATION TO MULTICHANNEL SPREAD-SPECTRUM COMMUNICATION , 1996 .

[9]  D. Bainov,et al.  The finite difference method for first order impulsive partial differential-functional equations , 1995, Computing.

[10]  Y. Soh,et al.  The stabilization and synchronization of Chua's oscillators via impulsive control , 2001 .

[11]  J García-Ojalvo,et al.  Spatiotemporal communication with synchronized optical chaos. , 2000, Physical review letters.

[12]  Chun-Mei Yang,et al.  Impulsive control of Lorenz system , 1997 .

[13]  Jean Bragard,et al.  Synchronization in Nonidentical Extended Systems , 1999 .

[14]  W. Schiesser.  The Numerical Method of Lines: Integration of Partial Differential Equations , 1991 .

[15]  Parlitz,et al.  Driving and synchronizing by chaotic impulses. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[16]  Tao Yang,et al.  Impulsive Systems and Control: Theory and Applications , 2001 .

[17]  Jianhong Wu,et al.  Comparison principles for impulsive parabolic equations with applications to models of single species growth , 1991, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[18]  Parlitz,et al.  Phase synchronization of coupled ginzburg-landau equations , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[19]  Chun-Mei Yang,et al.  Impulsive synchronization of Lorenz systems , 1997 .

[20]  Ljupco Kocarev,et al.  Synchronizing Spatiotemporal Chaos of Partial Differential Equations , 1997 .

[21]  Xinzhi Liu.  Stability results for impulsive differential systems with applications to population growth models , 1994 .

[22]  Ljupco Kocarev,et al.  SYNCHRONIZATION AND CONTROL OF SPATIALLY EXTENDED SYSTEMS USING SENSOR COUPLING , 1999 .

[23]  S Boccaletti,et al.  Controlling and synchronizing space time chaos. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[24]  E Schöll,et al.  Control of chaotic spatiotemporal spiking by time-delay autosynchronization. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[25]  M. Cross,et al.  Pattern formation outside of equilibrium , 1993 .

[26]  Xinzhi Liu,et al.  Impulsive stabilization and control of chaotic system , 2001 .

[27]  Allan R. Willms,et al.  Impulsive controllability of linear dynamical systems with applications to maneuvers of spacecraft , 1996 .

[28]  R. LeVeque.  Finite Volume Methods for Hyperbolic Problems: Characteristics and Riemann Problems for Linear Hyperbolic Equations , 2002 .

[29]  V. Lakshmikantham,et al.  Theory of Impulsive Differential Equations , 1989, Series in Modern Applied Mathematics.

[30]  R. LeVeque.  Numerical methods for conservation laws , 1990 .

[31]  Rajarshi Roy,et al.  Parallel communication with optical spatiotemporal chaos , 2001 .

[32]  Leon O. Chua,et al.  Impulsive control of nonautonomous chaotic systems using practical stabilization , 1998 .

[33]  Ljupco Kocarev,et al.  Synchronizing spatiotemporal chaos. , 1997, Chaos.

[34]  Parlitz,et al.  Synchronizing Spatiotemporal Chaos in Coupled Nonlinear Oscillators. , 1996, Physical review letters.

[35]  D. Bainov,et al.  Stability of solutions of first-order impulsive partial differential equations , 1994 .

[36]  Leon O. Chua,et al.  Conditions for impulsive Synchronization of Chaotic and hyperchaotic Systems , 2001, Int. J. Bifurc. Chaos.

[37]  J. E. Pearson.  Complex Patterns in a Simple System , 1993, Science.