Bifurcations Of Homoclinic Orbit Accompanied With Nilpotent Critical Point

In this paper, we study the bifurcation problems of homoclinic orbit accompa-nied with nilpotent critical point for higher dimensional systems. Firstly, we give the normal form of the systems in a sufficiently small neighborhood of nilpotent critical point and reduce dimension of the center manifold by the polar coordinates transformation, then we study the n — 1 dimensional systems instead of the origi-nal n dimensional systems. Secondly, by constructing local coordinate systems we get Poincare maps in a sufficiently small neighborhood of nilpotent critical point and in a sufficiently small tube neighborhood of homoclinic orbit, then by combin-ing the two maps we can obtain the successor functions and bifurcation equations.Hence the study of the bifurcation problems are converted into the study of the ex-istence of nonnegative solutions of the bifurcation equations. Finally, we study the existence, coexistence, number of homoclinic orbits, periodic orbits and heterolinic orbits bifurcated from the unperturbed systems, and get some conditions of homo-clinic orbits, periodic orbits and heterolinic orbits bifurcated from the unperturbed systems, moreover draw the corresponding bifurcation graphs.This paper consists of three chapters. In chapter one, the research background and current situation of bifurcation theory are briefly introduced, and the main works of this paper are given. In chapter two, the bifurcation problems of homo-clinic orbit accompanied with nilpotent critical point are discussed. Which consists of four sections: the basic assumptions, the construction of the local coordinate systems, the Poincare maps in a sufficiently small neighborhood of nilpotent critical point and in a sufficiently small tube neighborhood of homoclinic orbit, the main results of bifurcations. In which, the main results of bifurcations have three different situations: ? = 0;? ? 0, d > 0;? ? 0, d <0. In chapter three, some suggestions of bifurcations of homoclinic orbit accompanied with nilpotent critical point for higher dimensional systems are given to the further investigation.