随机扰动的非线性系统全局和局部动力学行为研究

This research aims to investigate the noise-induced global and local dynamical behaviors for the Hamiltonian systems which possess multiple fixed points, saddle points, homoclinic and heteroclinic cycles. Because of the multivalue property of the Hamiltonian and it's singularities within the vicinities of the saddle points and the boundary layers of the homoclinic and the heteroclinic cycles, the classical stochastic averaging method is not applicable, and so far, there is a deficient knowledge about the dynamical behaviors for such kind of systems. The content of this research includes: Based on the martingale theory and the singular perturbation method of Khasminskii, the stochastic averaging method on graph is introduced and developed. In view of the Hamiltonian diffusion process on graph, the boundary value problems of both Dynkin equation and FPK equation are investigated, and the results of which will lead to a deep-going understanding about the global dynamical behaviors which includes the exit problem and the global P-bifurcation behaviors. Based upon the theorem of L.Arnold, via Khasminskii transformation and the perturbation method of L.Arnold, the pth moment Lyapunov exponents of the multiple fixed points are investigated, and through which the local stochastic bifurcations are defined. By applying the analytical methods of the singular perturbation method of Khasminskii and Ludwig's ray method and the numerical method of cell mapping, for the original system, the distributions of the exit points on the homoclinic cycle and the heteroclinic one, the exit probabilities and the noise-induced variations of the structures of the attractors and the basins of attractions are investigated respectively. Through the research, we try to depict the integrated pictures about the global dynamical behaviors for the nonlinear stochastic systems.

本项目旨在研究噪声诱发具有多不动点、鞍点以及同（异）宿环的Hamilton系统的全局和局部动力学行为。此时，由于Hamilton函数的多值性以及在鞍点邻域、同（异）宿环附近的奇异性使得经典随机平均法不再适用。目前，对于此类系统动力学行为的认识还很匮乏。本研究主要内容包括：基于鞅理论和Khaminskii奇异摄动法建立和发展图（graph）上随机平均法的理论和方法。基于图上扩散过程，通过对其Dynkin方程和FPK方程边值问题的研究，促进对离出问题和P-分岔等全局动力学行为的认识。基于L.Arnold摄动法研究各不动点处的p阶矩Lyapunov指数以确定局部随机分岔条件。基于Khaminskii摄动法、射线方法、胞映射法等解析和数值方法研究原系统在同（异）宿环上的离出点分布、离出概率以及噪声诱发的吸引子和吸引域结构的变化。本研究力图给出此类非线性随机系统全局动力学行为描述一个完整的"拼图"。

本项目旨在研究噪声诱发具有多不动点、鞍点以及同（异）宿环的Hamilton系统的全局和局部动力学行为。此时，由于Hamilton 函数的多值性以及在鞍点邻域、同（异）宿环附近的奇异性使得经典随机平均法不再适用。目前，对于此类系统动力学行为的认识还很匮乏。本研究主要内容包括：基于鞅理论和Khaminskii 奇异摄动法建立和发展图（graph）上随机平均法的理论和方法。基于图上扩散过程，通过对其Dynkin 方程和FPK 方程边值问题的研究，促进对离出问题和P-分岔等全局动力学行为的认识。基于L.Arnold 摄动法研究各不动点处的p 阶矩Lyapunov 指数以确定局部随机分岔条件。基于Khaminskii 摄动法、射线方法、胞映射法等解析和数值方法研究原系统在同（异）宿环上的离出点分布、离出概率以及噪声诱发的吸引子和吸引域结构的变化。本研究力图给出此类非线性随机系统全局动力学行为描述一个完整的“拼图”。