Analysis On The Periodic Solutions In A Class Of Smooth And Non-smooth Systems

Firstly,we obtain some results on the number of zeros of the developed Melnikov function and the number of limit cycles of piecewise smooth systems.These results are regarded as the supplement of the well-known developed Melnikov function method.Sec-ondly,by using this method,two kinds of piecewise smooth systems are studied.Through complicated calculation,we find the number of zeros of its Melnikov function.And then some theorems on the maximum number of limit cycles bifurcated from the periodic orbits are acquired.Another research of this thesis is the number and stability problems on the periodic solutions of one--dimensional periodic equations-We give several methods to s-tudy these problems,including the Poincare map,normal form and the averaging method,which can be used to determine the number of their periodic solutions.As for the stability of the zeros,we obtain some results by studying their multiplicity.This thesis contains five chapters as follows.Chapter 1 is devoted to the introduction concerning the background and the method used in this paper.In Chapter 2,we introduce the Melnikov function method for piecewise smooth sys-tems.For the piecewise smooth near-Hamiltonian systems,we summarize all the known results on the developed Melnikov function,based on which some theorems on the periodic annular bifurcation and Hopf bifurcation are also given.In Chapter 3,we study the bifurcation problem of a class of non-smooth Lienard systems in two cases:the switch line on x-axis and y-axis,respectively.We first investigate two kinds of systems with multiple parameters.By calculating the first and second items of the developed Melnikov function,we derive the maximum number of limit cycles of these two systems bifurcated from the periodic orbits.Then,according to the relationship between the systems we discuss the original systems.A new lower bound of limit cycles is obtained.Some examples are also given to illustrate our results.In Chapter 4,a class of switched systems in Lienard form is considered,whose left and right subsystems are polynomial systems.By using the developed Melnikov function method,we find a formula of the Melnikov function M(h).Then,in order to find the number of zeros of M(h)and the maximum number of limit cycles generated from the periodic loops,the items of h in M(h)are divided into three parts,the numbers of which are related to the main results in this chapter.At last,several theorems about the maximum number of limit cycles are given in different cases which are classified according to the polynomials' degrees of the system.In Chapter 5,we discuss the existence,stability and the number of periodic solutions in one-dimensional periodic systems.Some theories and methods to study these problems are given.The main results of this chapter consists of four parts.In the first part,we summarize and improve some known results,which contains the relationship between the number of periodic solutions and the Poincare map.In the second part,some sufficient conditions are given,when the zero x=0 is odd multiplicity,even multiplicity or center,respectively.We also provide the relationship between the multiplicity of a zero and its stability of one-dimensional periodic equations.The third part contains a normal form theorem which is suitable for general equations.The main idea of the forth part is the averaging method in one-dimensional periodic equations,which can be used to study the number of periodic solutions.