Abstract
Nonlinear systems involving impulse effects, appear as a natural description of observed evolution phenomena of several real world problems, for example, many biological phenomena involving thresholds, bursting rhythm models in medicine and biology, optimal control models in economics, population dynamics, etc., do exhibit impulsive effects. In a recent paper [1], both real and complex Van der Pol oscillators were introduced and shown to exhibit chaotic limit cycles and in [2] an active control and chaos synchronization was introduced. In this paper, impulsive synchronization for the real and complex Van der Pol oscillators is systematically investigated. We derive analytical expressions for impulsive control functions and show that the dynamics of error evolution is globally stable, by constructing appropriate Lyapunov functions. This means that, for a relatively large set of initial conditions, the differences between the master and slave systems vanish exponentially and synchronization is achieved. Numerical results are obtained to test the validity of the analytical expressions and illustrate the efficiency of these techniques for inducing chaos synchronization in our nonlinear oscillators.
Highlights
The stabilization and control of nonlinear systems is one of the most important properties of the systems and has been studied widely by many researchers in control theory
As the key technology of secure communication, chaotic synchronization has been widely developed since Pecora and Carroll [6] proposed the
The basic behavior and chaotic synchronization have been studied by several researchers
Summary
The stabilization and control of nonlinear systems is one of the most important properties of the systems and has been studied widely by many researchers in control theory (see Refs. [2]-[5]). Once the error system of the two coupled systems is asymptotically stable, they are said to be synchronized Speaking, these impulses are samples of the state variables of the master system at current discrete moments to drive the slave system. By using such a technique, we can increase the impulse distances and reduce the control cost In this work both real and complex Van der Pol oscillators were introduced and shown to exhibit chaotic limit cycles We use the impulsive control technique to achieve synchronization of both real and complex Van der Pol oscillators.
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