Bifurcation of Nongeneric Homoclinic Orbit Accompanied by Pitchfork Bifurcation

Fushan Geng ,Song Li

doi:10.1155/2014/197914

Abstract

The bifurcation of a nongeneric homoclinic orbit (i.e., the orbit comes from the equilibrium along the unstable manifold instead of the center manifold) connecting a nonhyperbolic equilibrium is investigated, and the nonhyperbolic equilibrium undergoes a pitchfork bifurcation. The existence (resp., nonexistence) of a homoclinic orbit and an 1-periodic orbit are established when the pitchfork bifurcation does not happen, while as the nonhyperbolic equilibrium undergoes a pitchfork bifurcation, we obtain the sufficient conditions for the existence of homoclinic orbit and two or three heteroclinic orbits, and so forth. Moreover, we explore the difference between the bifurcation of the nongeneric homoclinic orbit and the generic one.

Highlights

It is well known that the nonhyperbolic equilibrium is unstable and always undergoes a saddle-node bifurcation
Zhu [1] gave the sufficient conditions for the existence of nongeneric heteroclinic orbits accompanied with saddle-node bifurcation by extending exponential trichotomy
Liu et al [3] considered the bifurcations of homoclinic orbit with a nonhyperbolic equilibrium for a high dimensional system; they achieved the persistence of homoclinic orbit and the bifurcation of periodic orbit for the system accompanied by a pitchfork bifurcation

Summary

Introduction

It is well known that the nonhyperbolic equilibrium is unstable and always undergoes a saddle-node (resp., transcritical or pitchfork) bifurcation. Zhu [1] gave the sufficient conditions for the existence of nongeneric heteroclinic orbits accompanied with saddle-node bifurcation by extending exponential trichotomy. Liu et al [3] considered the bifurcations of homoclinic orbit with a nonhyperbolic equilibrium for a high dimensional system; they achieved the persistence of homoclinic orbit and the bifurcation of periodic orbit for the system accompanied by a pitchfork bifurcation. Inspired by the above works, we deal with the nongeneric homoclinic bifurcation accompanied by a pitchfork bifurcation in a 4-dimensional system.

Hypotheses and Normal Form

Poincaré Map and Successor Function

The Main Results