学术报告：Cyclicity of periodic annulus and Hopf cyclicity in perturbing a hyper-elliptic Hamiltonian system with a degenerate heteroclinic loop
发布时间：2021-04-24 浏览次数：380
报告人：孙宪波（广西财经公司）
报告时间：2021年4月24日下午4：00-5:00
报告地点：育才数学楼2楼会议室
报告题目：Cyclicity of periodic annulus and Hopf cyclicity in perturbing a hyper-elliptic Hamiltonian system with a degenerate heteroclinic loop
报告人简介：孙宪波，广西财经公司教授，广西杰出青年基金获得者。主要从事微分方程定性理论及其应用研究。主持2项国家自然科学基金，在DCDS，JSC，BSM，JDE等业内期刊上发表学术论文20余篇。
报告摘要：In this talk, we discuss the cyclicity of periodic annulus and Hopf cyclicity in perturbing a quintic Hamiltonian system.The undamped system is hyper-elliptic, non-symmetric with a degenerate heteroclinic loop, which connects a hyperbolic saddle to a nilpotent saddle. We rigorously prove that the cyclicity is $3$ for periodic annulus when the weak damping term has the same degree as that of the associated Hamiltonian system.When the smooth polynomial damping term has degree $n$, first, a transformation based on the involution of the Hamiltonian is introduced, and then we analyze the coefficients involved in the bifurcation function to show that the Hopf cyclicity is $\big[\frac{2n+1}{3}\big]$. Further, for piecewise smooth polynomial damping with a switching manifold at the $y$-axis, we consider the damping terms to have degrees $l$ and $n$, respectively, and prove that the Hopf cyclicity of the origin is $\big[\frac{3l+2n+4}{3}\big]$ ($\big[\frac{3n+2l+4}{3}\big]$) when $l\geq n$ ($n\geq l$).