Cubical Hamilton Vector Fields Topological Classification

In this dissertation,we mainly discuss the globally topological structure of the cubical Hamiltonian vector fields in the plane.As in the paper [37],Llibre has studied the quadratic Hamitonian vector fields and has got 29 kinds of topological structure. Here,we use the tools and the methods of the paper [19].The main techniques used in this paper include the thought of algebraic classification of Llibre's,the idea to high-order critical point of professor Zhang Zhifen,Li Xuemin and Lu Yulin etc.The discussing process follows the below steps :At first,by means of the theroy of algebraic invariant,we make algebraric classification to binary quatic homogeneous polynomial and refer to the result of Yang Xin'an and Zhang Jianfeng. According to the thought of the direct sum in the plane,we made the topological classification to the cubical Hamiltonian vector fields and get the classification of / ?IX.in the first part of the paper,we mainly discuss degree one plus degree three Hamiltonian vector fields which has at least one finite critical point,that is,/' - IX'.For these systems /' - IX', we study their finite and infinite critical point respectively.In light of these critical pionts' classification and nature,we get every system's globally topological phase diagram and the coefficient condiction of these diagrams.But,some of these systems have not been obtdined perfect result.Second,in the second part of the paper,we study degree one plus degree two and plus degree three Hamiltonian system with one or two linear solution and we get their globally topological phase diagram respectively.In the last part of the paper,we find the limit cycle condiction of the cubical Hamiltonian vector fields under small perturbation -elliptic perturbation.