Bifurcations Of Limit Cycles For Some Polynomial Systems

In this thesis, the limit cycles bifurcated from some kinds of polynomial systems are investigated using the methods of bifurcation theory and qualitative analysis. There are five parts in this thesis. In the first one, we introduce the history of the qualitative and bifurcation theory of differential equations, some present results and our main work.The second section is concerned with the number of zeros of the Abelian integrals for a cubic Hamiltonian system with double homoclinic orbit under 2n + 1 -order perturbations. We estimate the upper bound of the number using Picard-Fuchs equation.In the third section, we deal with the limit cycles bifurcated from a class of one-parameter Hamiltonian system. The Abelian integral I(h) is investigated, from which we obtain the bifurcation of limit cycles.In the fourth section, we consider the higher order Melnikov function of a Hamiltonian system, which is perturbed under a class of quintic polynomial.In the fifth section, we obtain some sufficient conditions under which a class of a planar quintic system has at most two limit cycles by transforming it into Abel equation.