具有尖点环的非光滑微分系统的极限环分支

Limit Cycle Bifurcations of a Non-smooth Differential System with a Cuspidal Loop

研究具有尖点环的非光滑微分系统在n次多项式非光滑扰动下的极限环分支问题.首先把扰动微分系统的一阶Melnikov函数M(h)表示成几个具有多项式系数的生成积分的线性组合,并用数学归纳法证明这些多项式的系数是相互独立的常数.然后应用M(h)的渐近展式得到从原点和尖点环附近分支出极限环个数的下界.

This paper studies the limit cycle bifurcation problem of a non-smooth differential system with a cuspidal loop under non-smooth perturbation of polynomials of degree n.Firstly,the first order Melnikov function M(h) of the perturbed differential system is expressed as a linear combination of several generating integrals with polynomial coefficients,and the independence of coefficients of these polynomials is proved by mathematical induction.Then the lower bounds of the number of limit cycles bifurcating the origin and cuspidal loop are obtained by using the asymptotic expansions of M(h).