Conditions for Impulsive Synchronization of Chaotic and Hyperchaotic Systems

Abstract

Experimental results show that chaotic and hyperchaotic systems can be synchronized by impulses sampled from one or two state variables. In this paper, we study the conditions under which chaotic and hyperchaotic systems can be synchronized by impulses sampled from a part of their state variables. By calculating the Lyapunov exponents of variational synchronization error systems, we show that this kind of impulsive synchronization can be applied to almost all hyperchaotic systems. We also study the selective synchronization of chaotic systems. In a selective synchronization scheme, the synchronizing signal is chosen in the time periods when the Lyapunov exponents of variational synchronization error systems are negative. Since only driving signals during the time periods when synchronization error can be reduced are applied to reduce the synchronization error, and no signal is applied during the time periods when synchronization error can be increased, selective synchronization scheme can be used to achieve synchronization even in the case when continuous synchronization schemes fail to work.

Impulsive control and synchronization of spatiotemporal chaos

“…We shall state now two lemmas ( [19], Theorem 3.1, p. 45 and Corollary 2.2, p 33, respectively) and prove two other ones in order to establish several results needed in obtaining certain criteria for system (13) to be equi-attractive in the large. Lemma 2.…”

“…Due to the fact that impulsive systems have a unique character represented by jump discontinuities at the moments of impulse, the classical approach of finding Lyapunov exponents of the error dynamics is not applicable in this case. We shall, therefore, apply a different technique, proposed in [13], to study the dynamics of the synchronization error systems described by ODEÕs. However, in this section, we shall extend this technique to analyze the Lyapunov exponents of error dynamics between two identical spatiotemporal chaotic systems.…”

“…In addition, we shall confirm the theoretical results by developing a technique which analyzes the Lyapunov exponents of the error dynamics generated from the impulsive synchronization of these spatiotemporal chaotic systems. This technique is an extension of the method proposed by [13] which deals with systems of ODEÕs only. We shall generalize this technique to PDEÕs by incorporating it with the numerical method of lines [33], and then we shall generate a numerical result representing a sufficient condition for impulsive synchronization.…”

Some new less conservative criteria for impulsive synchronization of a hyperchaotic Lorenz system based on small impulsive signals

“…The other point to study is that in the synchronization problem of the slave system, most of the methods rely on receiving the master system signal continuously which is not generally the case in communications. Impulsive synchronization is one of the methods proposed to overcome this problem [9][10][11]. In [9] the conditions under which chaotic and hyperchaotic systems can be synchronized by impulses determined from samples of their state variables were studied.…”

“…Impulsive synchronization is one of the methods proposed to overcome this problem [9][10][11]. In [9] the conditions under which chaotic and hyperchaotic systems can be synchronized by impulses determined from samples of their state variables were studied. In [10], a detailed mathematical analysis was provided to explain how the asymptotic stability of the sporadically driven system depends on the driving period in linear systems.…”