基于大偏差理论的复杂随机动力学行为研究

This research aims at investigating the phenomena of large deviations occurring distinctively in random dynamical systems, facilitating to reveal the interactions between randomness and nonlinearity ubiquitous in the physical world and the engendered complexities. Pertinent knowledge on the mechanisms remains deficient to date and further studies are still required. Major results of this research are as follows: based on the Freidlin-Wentzell large deviation theory, the mechanisms of crossing the quasi-threshold manifolds in several typical fast-slow dynamical systems under weak white noise excitations are investigated, leading to a deeper comprehension of the underlying exiting essence of the concept of excitability. Patterns of white-noise driven escapes from various types of chaotic attractors are investigated, giving rise to a revelation of how homoclinic tangencies possessed by nonhyperbolic chaotic attractors, coexisting unstable invariant sets resulting from bifurcations of quasi--hyperbolic chaotic attractors and the other structures like chaotic saddles effect their escaping behaviors respectively. By constructing and developing the techniques and the applications of the large deviation theory in the case of colored noise, induced escapes of several nonlinear systems having disparate global structures are investigated. In order to precisely delineate the random switches between open and closed conformations of ion channels due to intrinsic thermal fluctuations, we derive the minimum action principle for stochastic hybrid systems and develop appropriate calculating methods. This research endeavors to provide reasonable explanations for the stochastic dynamical mechanisms of many physical phenomena from the perspective of exit problems.

本项目旨在研究随机动力系统所特有的大偏差现象，以揭示客观世界中随机性与非线性因素之间的相互作用机理和由此产生的复杂性。目前对于此类机理的认识还相当匮乏。本项目主要研究内容包括：基于大偏差理论，研究几类典型的快-慢动力系统在弱白噪声作用下穿越（拟）阈值流形现象的发生机制，深化对其可激性概念背后离出本质的理解；考察弱白噪声作用下具有不同类型的混沌吸引子系统的离出方式，揭示非双曲吸引子所具有的同宿相切结构、拟双曲吸引子分岔所产生的不稳定不变集以及混沌鞍结构等诸因素对各自离出行为的作用机理；构建和发展色噪声情形下大偏差理论的应用方法，考察几类具有不同全局结构的非线性动力系统在色噪声作用下的离出现象；为了准确地刻画神经元系统由于热涨落导致的离子通道随机开、闭的客观现象，建立和发展随机混合系统的作用量极值原理及其计算方法。本项目力图从离出问题的角度为诸多物理现象的随机动力学机制提供合理的解释。

本项目旨在研究随机动力系统所特有的大偏差现象，以揭示客观世界中随机性与非线性因素之间的相互作用机理和由此产生的复杂性。目前对于此类机理的认识还相当匮乏。.本项目主要研究内容包括：基于大偏差理论，研究几类典型的快-慢动力系统在弱白噪声作用下穿越（拟）阈值流形现象的发生机制，深化对其可激性概念背后离出本质的理解；考察弱白噪声作用下具有不同类型的混沌吸引子系统的离出方式，揭示非双曲吸引子所具有的同宿相切结构、拟双曲吸引子分岔所产生的不稳定不变集以及混沌鞍结构等诸因素对各自离出行为的作用机理；构建和发展色噪声情形下大偏差理论的应用方法，考察几类具有不同全局结构的非线性动力系统在色噪声作用下的离出现象；为了准确地刻画神经元系统由于热涨落导致的离子通道随机开、闭的客观现象，建立和发展随机混合系统的作用量.极值原理及其计算方法。本项目力图从离出问题的角度为诸多物理现象的随机动力学机制提供合理的解释。