Dynamics of the Van der Pol equation

This paper is a review of approaches to understanding "chaotic" dynamics in the forced Van der Pol equation. In addition, it discusses the phenomena of entrainment and phase locking from the point of view of dynamical systems theory. There are two principal regions of the parameter space where chaotic motion has been analyzed. The first occurs in a nearly linear system near resonance. Here one uses the method of averaging to initially reduce the problem to a two-dimensional one. This two-dimensional problem is analyzed by using bifurcation theory and topological methods. The appearance of homoclinic orbits in the averaged equations signals the presence of more complicated dynamics for the original problem. The second region of parameter space one examines is the one in which the Van der Pol equation describes a relaxation oscillation. In this situation one can approximate the dynamics by the iteration of a noninvertible one-dimensional mapping. This process is described together with the use of symbolic dynamics in parametrizing the resulting limit sets.