Bifurcation Of Some Degenerate Homoclinic And Heteroclinic Orbit

Homoclinic and heteroclinic orbit as one of the most interesting invariant set in dynamical system which has attracted the attention of many researchers.It is known that the Smale horeshoe described the chaos dynamics behavior for us,but what can induce chaos? By the Birkhoff-Smale theorem we know,if a map has a transverse homoclinic point,then it can appear Smale horseshoe,chaotic motion.So the research on homoclinic and heteroclinic bifurcation can help us to better understand the complex dynamics behavior.In this paper we will use exponential dichotomy,Fredholm alternative principle,Lyapunove-Schmidt method to study several degenerate homoclinic and heteroclinic bifurcation problems.We divide the dissertation into six chapters:In Chapter 1,we mainly introduce the background and development of the problems in this thesis,and then briefly state our main results and the symbols that are used in this article.In Chapter 2,we mainly introduce the essential research tools such as the method of Melnikov,the method of Lyapunov-Schmidt reduction,exponential dichotomy.In the first section we deal with the problem of a homoclinic orbit persist under the periodic perturbation for a planar Hamiltonian system by the method of Melnikov.In the second section we introduce how to solve a bounded linear operator equations by the method of Lyapunov-Schmidt reduction.This can help us reduce the dimension.In the third section we introduce the definition of exponential dichotomy in finite-dimensional and infinite-dimensional space,and how to use it in the study of the bifurcation of homoclinic and heteroclinic orbit.In Chapter 3,we consider an autonomous differential equation which has a heteroclinic orbit asymptotic to two hyperbolic equilibriums.By assuming that the variational equation along this heteroclinic orbit has three bounded solutions,and its dual equation has two bounded solutions,we study the bifurcation of the degenerate heteroclinic orbit under the periodic perturbation.Then we derive a bifurcation function from2(49)?(49)to2(49)by exponential dichotomy and Lyapunov-Schmidt reduction.The zeros of the bifurcation function correspond to the existence of the heteroclinic orbits for the perturbed equation.The low order terms of the Taylor expansion of the bifurcation functions are two real quadratic forms equations.The two real symmetric matrices which correspond to the two quadratic forms are formed by some Melnikov integral.Based on the eigenvalues of the two real symmetric matrices,we divide the quadratic forms equations into linear,hyperbolic,elliptic type.By the special circular rotation and hyperbolic rotation,the two real symmetric matrices can simultaneous diagonalization.So the two quadratic forms can change into standard form.We can easily find some condition which ensures the quadratic forms equation having two or four simple zeros in the standard form.Using the implicit function theorem then the bifurcation function has two or four zeros.So,two or four heteroclinic orbits bifurcated from the degenerate heteroclinic orbit under the periodic perturbation.The bounded solution of the variational equations along the two or four heteroclinic orbits is zero solution.It implies that the periodic map which corresponds to the perturbed equation has two or four transverse heteroclinic points.Hence the perturbed system has two or four Smale horseshoes.In Chapter 4,we consider a parabolic equation which has a degenerate homoclinic orbit asymptotic to a hyperbolic equilibrium.By assumming that the variational equation along with this homoclinic orbit has any finite bounded solutions,we study the bifurcation of the periodic solutions from the degenerate homoclinic orbit for a parabolic equation under periodic perturbation.First,we construct the solution of the perturbed equation by the exponential dichotomy and the variation of constant formula.Then we find some condition which ensures that this solution is a periodic solution by fredholm alternative theorem and Lyapunov-Schmidt.This condition is our bifurcation function.The zeros of the bifurcation function correspond to the existence of the periodic orbits for the perturbed equation.Under some conditions,the perturbed equation has periodic solutions which bifurcated from the degenerate homoclinic orbit for the unperturbed parabolic equation.In Chapter 5,we consider a special form of fast and slow systems.Suppose that the fast and slow variables are respectively to x and y.This special form can be obtained by using average to the slow variable.Assume that the unperturbed fast system has a degenerate homoclinic orbit ? in xoy plane which asymptotics to a hyperbolic equilibrium,and that the origin is a hyperbolic equilibrium for slow system.We study the periodic solutions bifurcated from the degenerate homoclinic orbit for fast system.Since the origin is a hyperbolic equilibrium,we obtain a bounded solution of slow system,and turn it into the fast system.By the exponential dichotomies and the Lyapunov-Schmidt reduction,the bifurcation function defined between two finite dimensional subspaces can be obtained.Under some solvability condition of the bifurcation,we obtain the periodic solutions near ?.Moreover we give an example to verify our conclusions.In Chapter 6,we simply summarize the thesis,as well hope that some results can be optimal and some assumptions can be weakened.