具幂零鞍点的Hamilton系统的周期环域的环性

The Cyclicity of Period Annuli of a Hamilton System with Nilpotent Saddle Points

研究具有幂零鞍点的三次Hamilton系统dx/dt=4x^2y+4y^3-y,dy/dt=4x^3-4xy^2+x的周期环域的环性.应用一阶Melnikov函数和Picard-Fuchs方程,得到该系统在n次实多项式扰动下从其周期环域中最多分支出4n+10个极限环(计重数).

In this paper,we study the cyclicity of period annuli of the following cubic Hamilton system with nilpotent saddle points dx/dt=4x^2y+4y^3-y,dy/dt=4x^3-4xy^2+x.By using the first order Melnikov function and Picard-Fuchs equation,we obtain that the above system under perturbations of real polynomials with degree n can bifurcate at most 4n+10 limit cycles(taking into account the multiplicity).