Bifurcations Of Compound Loop And Heteroclinic Loop For Two Kinds Of Planar Polynomial Systems

In this paper we mainly discuss the bifurcations of compound loop and heteroclinic loop of two types of planar polynomial systems.In Chapter one,we introduce the research background,research status,research topic and main results of this paper.In Chapter two,we study the expansions of Melnikov function near a heteroclinic loop with two nilpotent cusps of order m,and a heteroclinic loop with two nilpotent cusps of order two and three.Further,we discuss the number of limit cycles near the heteroclinic loop.In Chapter three,we mainly study the limitycle bifurcations of a kind of unper?turbed Li(?)nard system x=y?y=-g(x),where deg g(x)=7.When the unperturbed Li(?)nard system has six singular points or seven singular points,we give the different topological types of phase portraits for this system.In particular,for a near-Hamiltonian system,we give the expansion of the Melnikov function near a compound loop with a nilpotent cusp and two hyperbolic saddles.And the first few coefficients in the expansion are given.By the expansions of the Melnikov function near a cusp loop,a heteroclinic loop and a homoclinic loop,we give the conditions to obtain limit cycles for this kind of near-Hamiltonian system.Further,we apply these conditions to study the number of limit cycles for a concrete Li(?)nard system.