Existence And Bifurcation Of Limit Cycles For A Coupled SD Oscillator

In this thesis,we consider a coupled smooth and discontinuous(SD)oscillator with a parameter (0 ≤α≤ 1).By using qualitaive and bifurcation methods,we obtain some new results on its equivalent planar system in the following three cases:(ⅰ)The existence and uniqueness of limit cycle for 0 <α< 1;(ⅱ)The number of limit cycles in Hopf bifurcation from a nolpotent center for α= 0;(ⅲ)The number of limit cycles bifurcated from a generalized heteroclinic loop for α= 1.This paper includes the following three chapters:In the first chapter,the theoretical research statuses of limit cycles and SD oscillators is briefly introduced,and the main work of this paper is introduced.In the second chapter,we recall the first order Melnikov function methods for smooth near-Hamiltonian systems,and give the formula of the first order Melnikov function for piecewise near-Hamiltonian systems with three regions.In the third chapter,the existence and bifurcation of limit cycles of a coupled SD oscillators are studied.According to the different values of the parameter ,some new results on the existence and uniqueness of limit cycles,the Hopf bifurcation from a nilpotent center and limit cycle bifurcations near a generalized heteroclinic are obtained,respectively.