Using Melnikov's method to solve Silnikov's problems

Abstract

SynopsisA function space approach is employed to obtain bifurcation functions for which the zeros correspond to the occurrence of periodic or aperiodic solutions near heteroclinic or homoclinic cycles. The bifurcation function for the existence of homoclinic solutions is the limiting case where the period is infinite. Examples include generalisations of Silnikov's main theorems and a retreatment of a singularly perturbed delay differential equation.