Bifurcation Of Periodic Orbits Of Periodic Equations With Multiple Parameters By Averaging Method

In this paper,We investigate a class of n-dimensional periodic differential equations with multiple parameters by averaging method,and obtain sufficient conditions for the existence of multiple periodic orbits.As applications of the main results,a cubic system under cubic polynomial perturbation is studied.The paper is organized as follows.In section 1,we introduce the background and main topics of our research.Also,we will describe the methods and main results obtained in this thesis.In section 2,we introduce an analytic differential equation with multiple parameters ? and ? x=G(t,x,?)+?F(t,x,?,?),(t,x)?× ?,(0.2)where ?(?)Rm is an open subset,0<|?|<<|?|<<,1 G(t,x,?),F(t,x,?,?)are m-dimensional vector valued analytic functions with period T in the variable t.[10]devote to giving formulas of Mi(z),i= 0,1,…,n for system(0.2)when m? 1,the higher dimensional periodic differential system(0.2)is studied in section 3.In section 4,we study a cubic system as an application and prove that the system has at most four limit cycles.