Numerical Detection and Continuation of Homoclinic Points and Their Bifurcations for Maps and Periodically Forced Systems

Abstract

This paper describes a driver HomMap to the standard local bifurcation software AUTO for numerical analysis of homoclinic and heteroclinic bifurcations in maps and periodically forced systems. The driver can detect loci of homoclinic points if the unstable or stable manifolds are of dimension one, and treat problems of two or more dimensions. The algorithms used and their implementations in AUTO with HomMap are explained. Three examples are given, namely the Hénon map and the single and coupled Duffing oscillators, in order to demonstrate the usefulness and capabilities of the driver.