The Number And Distributions Of Limit Cycles For Polynomial System

In this paper, the distrubutions and number of the limit cycles bifurcated from some kinds of polynomial systems are investigated using the methods of bifurcation theory and qualitative analysis.The first section is concerned with the number of limit cycles for a cubic Hamiltonian system under quartic perturbations. It gives rise to the conclusion: the Hilbert number H(4) for the second part of Hilbert's 16th problem satisfies H(4) 13.In the second and third sections, we deal with Lienard equations of the form x = y,y = P(x) + yQ(x,y), For the former, attention goes to perturbations of the Hamiltonian vector fields with an elliptic Hamiltonian of degree six, exhibiting a double figure eight-loop. The number of limit cycles and their distributions are given. For the latter, it is proved that the Hopf cyclicity is two, and the new configurations of the limit cycles bifurcated from the homoclinic loop or heteroclinic loop for quintic system with quintic perturbations are given.In the fourth section, we consider the perturbations of two non-Hamiltonian integrable systems. For the former, it is proved that the system under the polynomial perturbations has at most [n/2] limit cycles in the finite plane and the upper bound is sharp. The proof relies on a careful analysis of a related Abelian integral. For the latter,we obtain the linear estimation of the number of isolated zeros of the corresponding Abelian integral.In the fifth section, we discuss the number of the limit cycles of a king of Hamiltonian system by the method of Liapunov's, and obtain that 6 is the upper bound .