周期扰动下分离指标对异宿轨道分支的影响  

The Influence of Splitting Index on Heteroclinic Orbit Bifurcation Under Periodic Perturbation

作　　者：龙斌 徐珊珊 曹慧 李建全 Long Bin;Xu Shanshan;Cao Hui;Li Jianquan(Department of Mathematics,Shaanxi University of Science and Technology,Xi'an 710021)

机构地区：[1]陕西科技大学文理学院数学系,西安710021

出　　处：《数学物理学报（A辑）》2021年第5期1516-1528,共13页Acta Mathematica Scientia

基　　金：国家自然科学基金(11801343,12071268,11971281,11801342);陕西省自然科学基础研究计划(2018JQ1031);陕西科技大学博士科研启动基金(2017BJ-45)。

摘　　要：应用Lyapunov-Schmidt约化方法与指数二分性,该文研究了退化异宿轨道在具有m维参数周期扰动下的分支问题.假设沿着未扰动异宿轨道的变分方程具有d个线性无关的有界解.给出了未扰动异宿轨道的分离指标s.分支函数是从R^(d+m)到R^(d-s)的一个映射.分支函数零点的存在性就对应着扰动系统异宿轨道的存在性.如果分离指标s<0,则至少需要1-s维的周期扰动才能扰开未扰动的异宿轨.如果分离指标s≥0,则存在一个一维的周期小扰动即可扰开未扰动的异宿轨.By using the method of Lyapunov-Schmidt reduction and exponential dichotomies,we consider the degenerate heteroclinic orbit bifurcation with m dimensional periodic perturbations.The variational equation along the heteroclinic orbit has d(d≥1) bounded solutions.The splitting index of the unperturbed heteroclinic orbit is s.The bifurcation equation has d+m variables and d-s equations.The zeros of bifurcation function correspond to the existence of heteroclinic orbits for perturbed equation.If the splitting index s <0,it needs at least 1-s dimensional periodic perturbation can break the unperturbed heteroclinic orbit.If the splitting index s≥ 0,there is a small perturbation can break the unperturbed heteroclinic orbit.

关 键 词：退化的异宿轨分支 Lyapunov-Schmidt约化 指数二分性 

分 类 号：O175.1[理学—数学]