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

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.