一类非对称五次退化系统的分支分析

摘要

研究了一类非对称五次退化系统的稳定性和分支问题.由局部稳定性分析和Hopf分支定理,讨论了Hopf分支及其稳定性;利用一阶Melnikov函数,分别得到了围绕一个平衡点的小同宿分支和围绕三个平衡点的大同宿分支的分支曲线;推导了对应的Picard-Fuchs方程,由此证明了在退化Hopf和退化同宿分支点之间存在重极限环分支,并得到分支曲线计算公式.各分支曲线将参数平面分割为不同区域,给出了完整的分支图和各区域上的相轨线结构.%The dynamics of a class of non-symmetric quintic degenerate systems are investigated.According to the local stability analysis and Hopf bifurcation theory,the existence and stability of Hopf bifurcation are discussed.By analysing Melnikov function,we study the bifurcation curves for the little homoclinic orbit encircling one equilibrium and the big homoclinic orbit encircling all three equilibria points.Furthermore,by using Picard-Fuchs equations, it is proved that double limit cycle bifurcations occur between the degenerate Hopf and homoclinic bifurcation points.Moreover,a formula for calculating the bifurcation value of double limit cycle is derived.Finally,the complete bifurcation diagrams and associate phase portraits are obtained.