Study on the Geometrical Properties and Existence of Orbit Homoclinic to a Saddle Point in n-Dimensional Autonomous Vector Field with Polynomials

Abstract

According to the theory of stable and unstable manifolds of an equilibrium point, we firstly find out some geometrical properties of orbits on the stable and unstable manifolds of a saddle point under some brief conditions of nonlinear terms composed of polynomials for [Formula: see text]-dimensional time continuous system. These properties show that the orbits on stable and unstable manifolds of the saddle point will stay on the corresponding stable and unstable subspaces in the [Formula: see text]-neighborhood of the saddle point. Furthermore, the necessary conditions of existence for orbit homoclinic to a saddle point are exposed. Some examples including homoclinic bifurcation are given to indicate the application of the results. Finally, the conclusions are presented.