Impulsive stabilization for control and synchronization of chaotic systems: theory and application to secure communication

Impulsive control of a chaotic system is ideal for designing digital control schemes where the control laws are generated by digital devices which are discrete in time. In this paper, several theorems on the stability of impulsive control systems are presented. These theorems are then used to find the conditions under which the chaotic systems can be asymptotically controlled to the origin by using impulsive control. Given the parameters of the chaotic system and the impulsive control law, an estimation of the upper bound of the impulse interval is given. We also present a theory of impulsive synchronization of two chaotic systems. A promising application of impulsive synchronization of chaotic systems to a secure communication scheme is presented. In this secure communication scheme, the transmitted signals are divided into small time frames. In each time frame, the synchronization impulses and the scrambled message signal are embedded. Conventional cryptographic methods are used to scramble the message signal. Simulation results based on a typical chaotic system; namely, Chua's oscillator, are provided.

[1]  Edward Ott,et al.  Controlling chaos , 2006, Scholarpedia.

[2]  Tao Yang,et al.  Recovery of digital signals from chaotic switching , 1995, Int. J. Circuit Theory Appl..

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

[4]  E.R. Hunt,et al.  Keeping chaos at bay , 1993, IEEE Spectrum.

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

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

[7]  A. Samoilenko,et al.  Impulsive differential equations , 1995 .

[8]  Leon O. Chua,et al.  Cryptography based on chaotic systems , 1997 .

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

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

[11]  Kevin M. Short,et al.  Steps Toward Unmasking Secure Communications , 1994 .

[12]  Guanrong Chen,et al.  From Chaos to Order - Perspectives and Methodologies in Controlling Chaotic Nonlinear Dynamical Systems , 1993 .

[13]  Kennedy,et al.  Predictive Poincaré control: A control theory for chaotic systems. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[14]  Leon O. Chua,et al.  ADAPTIVE SYNCHRONIZATION OF CHUA'S OSCILLATORS , 1996 .

[15]  Ying-Cheng Lai,et al.  Controlling chaos , 1994 .

[16]  Leon O. Chua,et al.  Global unfolding of Chua's circuit , 1993 .

[17]  Leon O. Chua,et al.  ON ADAPTIVE SYNCHRONIZATION AND CONTROL OF NONLINEAR DYNAMICAL SYSTEMS , 1996 .

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

[19]  Kestutis Pyragas.  Continuous control of chaos by self-controlling feedback , 1992 .

[20]  R. E. Amritkar,et al.  Synchronization of chaotic orbits: The effect of a finite time step. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.