Persistent Homoclinic Or Heteroclinic Space For A Vector Field With Homoclinic Or Heteroclinic Orbits

In this dissertation we mainly study the new phenomenon on a vector field with homoclinic or heteroclinic orbits under autonomous perturbations.We first proposed the conceptions of persistent homoclinic or heteroclinic perturbation and persistent homoclin-ic or heteroclinic space.Under the persistent homoclinic or heteroclirnic perturbations,the homoclinic or heteroclinic orbit of a vector field persists.Both the theoretical deriva-tion and the numerical simulation verify this new phenomenon.This thesis is divided into three chapters:Chapter 1 is the research background.We first introduce the homoclinic tangles,symbolic dynamical systems and Smale horseshoe map.We list the related basic concepts,lemmas,theorems and so on.Then we introduce the classic Melnikov function.Finally,we present the high order Melnikov function in paper[24].Chapter 2 is the first part of our main results.We first deduce the high order Mel-nikov function of the vector field with homoclinic orbits under autonomous perturbations,and using the high order Melnikov function we construct the persistent homoclinic or het-eroclinic space.Secondly we provide the persistent homoclinic or heteroclinic spaces of specific examples.Chapter 3 is the second part of our main results.We first deduce the high order Melnikov function of the vector field with heteroclinic orbits under autonomous pertur-bations,and we reveal the differences and connections of them to the homoclinic cases.Then we construct the persistent homoclinic or heteroclinic space of the heteroclinic cases.Finally,we provide some persistent homoclinic or heteroclinic spaces of specific examples.