Bifurcation Of Limit Cycles For Two Kinds Of Planar Differential Systems

In this paper,we mainly study limit cycles bifurcations of two kinds of plane differ-ential systems.For the unperturbed Liénard system x=y,y=-g(x)in the case that deg g(x)=7 and the system has 2,3,4 or 5 singular points,we give all the different topological phase portraits.In all the phase portraits given in this paper,we find the expansion of Melnikov function near a double heteroclinic loop with two nilpotent cusps and a hyperbolic saddle has not been studied,which is one of the main result of this paper.We further give the first several coefficients in the expansion.By using these coefficients,we give conditions to obtain limit cycles near the two heteroclinic loops and the double heteroclinic loop,which are applied to study the number of limit cycles for a non-centrally symmetric Liénard and a centrally symmetric Liénard system.When a Hamiltonian system has a heteroclinic loop passing through two nilpotent saddles of smooth type,we give the expansion of the Melnikov function near the hetero-clinic loop for a general near-Hamiltonian system.We also give the conditions to obtain limit cycles near the heteroclinic loop.By using the result given in this paper and the existing results about the number of limit cycles near a nilpotent center,we study the number of limit cycles for a near-Hamiltonian system.