| In the development of dynamic system theory,the theory of homoclinic and heteroclinic bifurcation has always been associated with the complexity of dynamics.The study of homoclinic and heteroclinic bifurcation has become increasingly important in understanding dynamics beyond hyperbolic theory,where homoclinic and heteroclinic connecting orbits are basic dynamic phenomena,and also the main mechanism of non-hyperbolic.Therefore,the research on them is not only of great theoretical significance,but also of great guiding significance for practical applications.In this thesis,we mainly apply exponential dichotomy and LyapunovSchmidt reduction to study the persistence of heteroclinic orbits and heteroclinic loops under periodic perturbation.This article is divided into five parts.In Chapter 1,we mainly introduce the research background and status as well as the main content of this paper.In Chapter 2,we illustrates some basic knowledge.The first section introduces exponential dichotomy.The second section introduces LyapunovSchmidt reduction method.The third section introduces the existence of bounded solutions.In Chapter 3,we introduce the persistence of heteroclinic orbits under periodic perturbation.In this chapter,we considered the persistence of degenerate heteroclinic orbits under periodic perturbations of m dimensional parameters.Assume that the variation equation of the unperturbed heteroclinic orbit has d linearly independent bounded solutions,and we assume the splitting index s of the unperturbed heteroclinic orbit.We use exponential dichotomy and Lyapunov-Schmidt reduction method to derive a bifurcation function,which is the mapping from Rd+m to Rd-s.In both cases of splitting index s≥ 0 and s<0,we obtain the existence condition of heteroclinic orbits under periodic perturbation.In Chapter 4,we introduce the persistence of heteroclinic loop under periodic perturbation.We consider an autonomous differential equation with a heteroclinic loop,whose heteroclinic loop consists of two heteroclinic orbits γ1 and γ2,and two hyperbolic equilibrium points p+ and p-.We assume that the variational equation along the heteroclinic orbit γl has di linearly independent bounded solution,and we give the splitting index s and-s of the heteroclinic orbit,respectively,and we use exponential dichotomy and Lyapunov-Schmidt reduction method to derive a bifurcation function,which is the mapping from Rd1+d2+2 to Rd1+d2.Under certain conditions,the existence of a zero point in a bifurcation function means that the perturbed system has a heteroclinic loop Γεnear the unperturbed heteroclinic loop.In Chapter 5,we provides a summary and outlook for the entire article. |
