Bifurcation Of Some Homoclinic Orbits And Heteroclinic Orbits

This thesis is devoted to investigating the bifurcation of some homoclinic orbits and heteroclinic orbits with special hypothesis. The work is divided into five chapters.In chapter 1, the background and research status of bifurcation theory are briefly given. Meanwhile, we introduce the main results achieved in this paper.In chpter 2, the bifurcations are stadied near a primary symmetric homoclinic orbit to a saddle-center in a 4-dimensional reversible system. By establishing a new kind of moving frame along the primary homoclinic orbit and using the Melnikov functions, we construct a Poicare map to investigate the existence of sufficiently small nonnegative so-lutions of the induced bifurcation equations. Moreover, sufficient conditions are given to guarantee the existence of 1-homoclinic orbits,1-periodic orbits,R-symmetric 1-periodic orbits, R-symmetric 2-periodic orbits and R-symmetric 2-homoclinic orbits. Also, this new kind of moving frame is first introduced for the homoclinic orbit to a saddle-center, which will greatly simplify the original system.In chapter 3, local moving frame is constructed to analyze the bifurcation of heterodi-mensional cycle with a weak inclination flip in four dimensional systems. Under other generic hypotheses, and based on bifurcation equations, the persistence of the heterodi-mensional cycle, the existence of 1-homoclinic orbit,1-periodic orbit and two-fold or three-fold 1-periodic orbit are proved. Meanwhile, the corresponding bifurcation sufaces are given. Finaly, a concrete example of vector field with all given hypotheses shows the existence of the systems which have a heterodimensional cycle with the weak inclination flip. The example given here provides a useful reference for solving the difficult problem of constructing a Non-Hamilton system with weinclination flips, and also for solving the dilemma of rigorously proving in theory that the unstable manifold and the stable mani-fold satisfy the strong inclination property, i.e., inclination flips don't occur on these two manifolds. In chapter 4, we study a codimension-three bifurcation of double homoclinic orbits with resonant eigenvalues in four dimensional vector fields. Using the moving frame and Poincare map, the existence codition of (12-) (or (21-)) homoclinic orbit, (12-) periodic orbit and two-fold (12-) periodic orbit are obtained. Also, two saddle-noddle bifurcation surfaces and corresponding bifurction diagrams are also given in case two orbits are all twisted.In chapter 5, a concrete nonlinear system with degree two and the least dimension, dimension 3, is given by a transformation for a planar vector field, and it is shown that the system has a heterodimensional cycle. Then, by using the Silnikov coordinate and the moving frame, the bifurcation of the cycle is studied under the perturbation of degree three. The method given here provides a useful reference for constructing homoclinic, heteroclinic and heterodimensional cycles with various other kinds of degeneracy.