The Number Of The Limit Cycles For Some Non-smooth Liènard Systems

As an introduction, in the first chapter we introduce the background of our research and main topics that we will study in the following chapters. We also give a description of our methods and results detained in this thesis in the first chapter .And we give the conclusions which have been obtained in some papers.In the second chapter, our main purpose is to study the number of the limit cycles for some non-smooth Lienard systems. Based on the theorems which are proved in [1],we study further for the problems.the authors in [1] proved that for n=1,2 or 3 system x = y- Fn(x,a), y =-g(x), respectively has Hopf cyclicity 1, 3, 5 at the origin. In this paper we first give a formula for computing Bi, i = 7,8,9,10, and then apply Theorems2.1.2-2.1.4 to study the Hopf bifurcation of some non-smooth Lienard systems.In particular, we found the result of Hopf cyclicity for system x = y-Fn(x, a), y=-g(x), in the case n = 3 is false. The correct result is Hopf cyclicity 4.We discuss system x = y- Fn(x, a), y =-g(x), has Hopf cyclicity 2n-2 at the origin if Mn≠0 for n = 3, 4 and 5 respectively.This is our important result in the paper.