Synchronization of Impulsive Real and Complex Van der Pol Oscillators

Nonlinear systems involving impulse effects, appear as a natural description of observed evolu-tion phenomena of several real world problems, for example, many biological phenomena involv-ing thresholds, bursting rhythm models in medicine and biology, optimal control models in eco-nomics, 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 synchro-nization 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 evolu-tion is globally stable, by constructing appropriate Lyapunov functions. This means that, for a rel-atively 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