The Number Of Limit Cycles For A Class Of Hamiltonian Systems

In recent years,piecewise smooth dynamical systems have received extensive attention,involving singularity analysis,number of limit cycles and so on.In this paper,we consider a class of near Hamiltonian systems and apply the first order Melnikov function method to study the number of limit cycles.This paper is divided into five parts,the specific research work is as follows:In the preface of Chapter 1,the bifurcation of limit cycles for a class of Hamiltonian systems with perturbations and some results of experts and scholars are briefly introduced.In Chapter 2,we introduce the related preparatory lemma and give the expression of Melnikov function and the definition of extended complete Chebyshev system(ECT System).In the third chapter,as one of the main contents of this paper,the first-order Melnikov function expansion of n= 2 is introduced and sorted in detail.The rank of the first-order Melnikov function is obtained by determinant transformation of relevant parameters and coefficients.The number of zeros of the first-order Melnikov function about h> 0 is determined by ECT system,and the number of limit cycles of n= 2 is verified.In Chapter 4,we obtain the expansion of the first-order Melnikov function of n= 3,and further analyze the parameters of the expression of the first-order Melnikov function under limited conditions.Finally,we obtain the upper bound of the number of zeros by judging the ECT system,and explore its limit cycle branches,and prove the number of limit cycles.In Chapter 5,we discuss the case of n= 4 and obtain the expansion of the first order Melnikov function under the disturbance of n= 4 by using the undetermined coefficient method and Taylor formula.Furthermore,we obtain the number of zeros under the disturbance of n= 4 by combining the expansion of the first order Melnikov function and the relationship between the coefficients.