含有二阶幂零鞍点的双同宿环附近的极限环分支

摘要

Determining the number of limit cycles of a planar differential system is related to the second part of Hilbert's 16th problem.In this paper,a near-Hamiltonian system is studied,where the unperturbed system has a double homoclinic loop with second order nilpotent saddle and has three families of periodic orbits around the loop.By investigating the expansions of the first order Melnikov functions near the double homoclinic loop as well as their coefficients,the numbers of limit cycles that may appear around the loop are obtained.To be specific,it is shown that there may exist 1 1,13,14 or 16 limit cycles under some conditions.Moreover,an example is given to illustrate the theoretical result.%研究平面微分系统的极限环个数问题与Hilbert第十六问题的第二部分.考虑一类near-Hamiltonian系统,其未扰系统有一个含有二阶幂零鞍点的双同宿环且在双同宿环附近有三族周期轨.研究了首阶Melnikov函数在双同宿环附近的展开式和展开式的各项系数,得出了此类系统在双同宿环附近可以出现的极限环个数.具体来说,证得此类系统在某些条件下可在双同宿环附近出现11,13,14和16个极限环,并给出了应用实例.