Bifurcation of Homoclinic Orbits in Autonomous Systems and in Chaotic Blue Sky Catastrophe

Abstract

Homoclinic and heteroclinic orbits are of great interest from applied point of view. In reaction-diffusion problems they form the profiles of traveling wave solutions. Their existence can also cause complicated dynamics in the threedimensional systems, as in The Shilnikov example. The classical approach to predict the bifurcations of these homoclinic orbits, in driven or planar autonomous systems, deals with the Andronov-Melnikov method. This approach is mainly based on the distance between manifolds. Recently, another analytical method to predict the homoclinic bifurcations for autonomous systems was presented in [1]. This approach consists of attacking directly the period of periodic solution. More precisely, the condition we considered at such bifurcations is the limit of the period goes to infinity. The purpose of this paper is to present a new criterion to approximate such bifurcations. This criterion involves principally the periodic orbit and the hyperbolic point (saddle). The period is not needed in this approach. We will apply this new criterion to predict a chaotic blue sky catastrophe. Comparisons to numerical simulations and previous methods ([1], [2] and [3]) are reported.