Bifurcations of heteroclinic loops with nonresonant eigenvalues

Abstract

In this paper, we use the way of local coordinates instead of the Floquet method to study the problems of homoclinic and periodic orbits bifurcated from heteroclinic loop for high-dimensional system. Under some transversal conditions and the non-twisted or twisted conditions, we discuss the existence, uniqueness, coexistence, and non-coexistence of 1-periodic orbit, 1-homoclinic orbit, and 1-heteroclinic orbit near the heteroclinic loop. We get some general conclusions only under the basic hypotheses, and the other conclusions under the two hyperbolic ratios of the heteroclinic loop are greater than 1. Meanwhile, the bifurcation surfaces and existence regions are given.