Limit Cycle Bifurcation Problems By Perturbing A Class Of Piecewise Linear Hamiltonian Systems

The research of piecewise differential system is one of the hot topics in recent years.In this paper,we pay attention to the number of limit cycles for a class of piecewise smooth near-Hamiltonian systems.By using the expression of the first order Melnikov function,we study upper bound of the number of limit cycles.This paper is organized as follows.The first chapter mainly introduces the development of ordinary differential and dynamical systems,the research status of piecewise smooth Hamiltonian systems and the main conclusions.In the second chapter,after reviewing some basic knowledge,we introduce several lemmas which are helpful to prove our result.We mainly give the first order Melnikov function expression of the system and introduce the definition of ECT-system and related propositions.In Chapter three,according to the Melnikov function expression given in Chapter 2,we calculate the specific expansion of the first order Melnikov function at h = 0 under the small perturbation.By analyzing the relative properties of the first order Melnikov function's coefficients,we obtain the maximum number of zeros of Melnikov function.In the fourth chapter,through the definition of ECT-system and related propositions,we give the maximum number of zeros of Melnikov function on the open interval and the number of limit cycles bifurcating from the period annulus.