Limit Cycles Of A Lienard System

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.In chapter 2, we mainly research a class of near-Hamilton system whose unperturbed system has two center points, one saddle point, a simple homoclinic loop and a double homoclinic loop. By using three lemmas, we respectively derive the Melnikov function expand expressions of the system near two center, simple homoclinic loop and double homoclinic loop. Through researching about the relevant Melnikov function expanding expressions, we obtain a lower bound of the number and the distribution of the limit cycles of the system. The homoclinic and Hopf bifurcations are considered together, adjoining to the four cases of the coefficients of the melnikov function expand expression near the double homoclinic loop, we derive four theorems. We give a detailed proof about the two theorems of four, the others can be proved analogously.In chapter 3, the number and distribution of the limit cycles for a class concrete quartic Lienard systems are researched. First, by using the lemmas obtained in chapter 2 we derive the relevant Melnikov function expanding expression in general case. Then using the theorems in chapter two, bifurcations of the limit cycles for the Lienard system are discussed. Finally, we prove this Lienard system can respectively bifurcate 4, 7, 7, 8, 10, 11,11 limit cycles under the perturbation of degrees 3, 4, 5, 6, 7, 8 and 9.