On The Bifurcations Of Limit Cycles For Two Classes Of Two-dimensional Differential Systems

This paper mainly studies the bifurcation of limit cycles of two types of differential systems,one is a smooth system and the other is a piecewise smooth system.As we all know,Melnikov function and average theory are two important methods to study the limit cycles of differential systems.In this article,the bifurcation of limit cycles of a class of cubic polynomial system is studied by using Melnikov function method,and the limit cycle bifurcations of piecewise smooth system is studied by using first-order average theory,and several new results are obtained.This thesis is divided into four parts.Chapter 1 is the introduction,which mainly introduces the background sources,research status and research methods of this research topic,and puts forward the research work and innovations of this paper.Chapter 2,we study the bifurcation of limit cycles of a class of cubic polynomial (?)where b,b,Pi,j,qi,j are arbitrary constants,a,b?0,b2-4a>0,|pi,j|?K,|qi,j|?K,K is a positive constant.Using the Melnikov function method,when |?|>0 is sufficiently small,the maximum number of limit cycles bifurcating from the periodic the period annulus of origin is 5,which partly answers Sui,Zhao[International Journal of Bifurcation and Chaos,2018,28:1850063]proposed 5<H(3)?6.Chapter 3,we consider a class of piecewise smooth differential systems(?)where ?>0 is a small parameter,m is an arbitrary non-negative integer,?=arctan(y/x),Qk(x,y)=?i+j=0naijkxiyj is a real polynomial of degree n,S1={(x,y)|y?0},S2={(x,y)|y?0} and the characteristic function xs of a set S(?)R2 is defined by The main tool used for proving our results is the average function of first order for discon-tinuous piecewise vector fields.We obtain the maximum number of limit cycles.Applying the first-order average theory of the piecewise smooth differential system,it is obtained that when ?>0 is sufficiently small,the maximum number of limit cycles bifurcation from the period annulus of the unperturbed system is n and the upper bound n is reached.Chapter 4,we give the conclusions and discussion of this thesis.