Complexity in a Hybrid van der Pol System

Abstract

In this work, we study, from a numerical point of view, the dynamics of a specific hybrid map which is that of the kicked van der Pol system. The dynamics of this system is generated by a two-stage procedure: the first stage is the time-[Formula: see text] map of the vector field associated with the van der Pol equation, and the second stage is a translation. We propose a numerical method for computing local (un)stable manifolds of a given fixed point, which leads to high order polynomial parameterization of the embedding. Such a representation of the dynamics on the manifold is obtained in terms of a simple conjugacy relation and by constructing two contracting operators. We illustrate our techniques by plotting (for specific values of the parameters) both stable and unstable manifolds displaying homoclinic intersection. Furthermore, our numerical study reveals the presence of a strange attractor included in the closure of the unstable manifold of the fixed point.