Limit Cycle Bifurcation For A Kind Of Cubic Near Hamiltonian System

In this paper, we study limit cycles bifurcating from a center or a focus of Near-Hamiltonian systems. By means of concluding and computing in Mathematica program the expression of the Melnikov function near a center or a nilpotent saddle of Hamiltonian systems, we consider the bifurcation problem of the planar symmetric cubic Hamiltonian system having a nilpotent critical point (0,0). Perturbing the above system by cubic polyno-mials, we obtain the number of limit cycles bifurcating from any center and nilpotent saddle of the corresponding Hamiltonian system.In Chapter 1, we introduce some results of the second part of Hilbert’s 16th problem ,the bifurcation theory and methods of dynamical systems, and then list our main work.In Chapter 2, using the method of Melnikov function and Matmatica software,we ob-tain the classification of singular point of Hamiltonian systems and a cubic Hamiltonian system.In Chapter 3, we study the limit cycles bifurcating from a center of symmetric near-Hamiltonian systems by the computer algebra program-Mathmatica.In Chapter 4, we study the limit cycles bifurcating from a nilpotent saddle of symmetric Hamiltonian systems, perturbing the above system by cubic polynomials.