Applying Lin's method to constructing heteroclinic orbits near the heteroclinic chain

Abstract

In this paper, we apply Lin's method to study the existence of heteroclinic orbits near the degenerate heteroclinic chain under ‐dimensional periodic perturbations. The heteroclinic chain consists of two degenerate heteroclinic orbits and connected by three hyperbolic saddle points . Assume that the degeneracy of the unperturbed heteroclinic orbit is , the splitting index is . By applying Lin's method, we construct heteroclinic orbits connected and near the unperturbed heteroclinic chain. The existence of these orbits is equivalent to finding zeros of the corresponding bifurcation function. The lower order terms of the bifurcation function is the map from to . Using the contraction mapping principle, we provide a detailed analysis on how zeros can exist based on different cases of splitting indices , and then obtain the existence of the heteroclinic orbits which backward asymptotic to and forward asymptotic to .