Synchronization of Impulsive Real and Complex Van der Pol Oscillators

Ali Al-Qahtani¹, Houari B. Khenous¹, Shaban Aly^2*

¹Department of Mathematics, Faculty of Science, King Khalid University, Abha, KSA.
²Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut, Egypt.

DOI: 10.4236/am.2015.66084                

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.

Keywords

Real and Complex Van der Pol, Chaotic Systems, Error Systems, Impulsive Synchronization

Conflicts of Interest

The authors declare no conflicts of interest.

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