Limit Cycle Numbers For A Class Of Liénard-Rayleigh Oscillators

This paper is devoted to the study of limit cycle bifurcations for a class of cubic nearHamiltonian systems,which are in the form of Liénard-Rayleigh oscillators.By using geometric techniques to the first order Melnikov function,we provide a complete study on the maximum number of limit cycles in Hopf bifurcation near a center and Poincaré bifurcation from a period annulus.This paper is organized as follows.The first chapter,mainly introduces the research background,research content and main results of this topic.The second chapter,the preliminary knowledge of Melnikov function method and other relevant preliminary knowledge are given to prove the main results of this paper.In the third chapter,we mainly study the Hopf bifurcation of this system.According to the preliminary knowledge given in Chapter 2,the first order Melnikov function is Taylor expanded at origin.Then,by taking different values of the coefficients in the expansion,it obtains that the system has at most two limit cycles near the center.In the fourth chapter,we mainly study the Poincaré bifurcation of the system.On the basis of the previous work,the concrete expression of the first order Melnikov function of the system can be obtained.Then,by using geometric techniques of the expression,it obtains that the system has at most two limit cycles from a period annulus,and the sufficient conditions for the system to get different limit cycles can also be obtained.