Center Problems And Bifurcations Of Limit Cycles For Nilpotent Systems

This paper mainly investigates the weakened Hilbert's 16th problem, which is the problem about the bifurcation of limit cycles after a center or focus of planar system is perturbed by polynomials. We consider nilpotent center conditions of quartic Hamiltonian systems, and the bifurcation of limit cycles for a class of special cubic Hamiltonian system with a nilpotent center after being perturbed by polynomials. Using the geometrical method, we obtain the bifurcation of limit cycles of a near-Hamiltonian system.In Chapter 1, we introduce Hilbert's 16th problem, its weakened problem and the development and the methods concerned in the paper. Then we list our main work.In Chapter 2, we discuss nilpotent center conditions of quartic Hamiltonian systems, and give two sufficient conditions. Through analyzing zeros of Abelian integral and using Descartes Theorem, we consider limit cycles bifurcation of a class of special cubic Hamiltonian systems.In Chapter 3, we study in detail a class of quinitic symmetrical Hamiltonian systems, by founding a Picard-Fuchs equation. By the geometrical technique, we investigate the limit cycle bifurcated from a third order nilpotent center.