弱噪声扰动下非线性动力系统非平衡定态的准稳态及逃逸研究

Nonlinear dynamical systems can present various complex phenomena when suffering weak noise disturbances. Research of dynamical behaviors of steady-states, especially non-equilibrium steady states under disturbances helps to understand the internal mechanisms of these phenomena and obtain quantitative characterizations of significance in the practical problems. Based on Freidlin-Wentzell theory, this project focuses on the quasi-stationary state distributions and the most probable escape path of non-equilibrium steady states under weak Gauss white noise perturbations: (1)High-order WKB approximation of quasi-stationary state distributions of periodic attractors is proposed; high-order approximation of quasi-stationary state distributions of quasiperiodic attractors is developed; quantitative method of quasi-stationary distribution of chaotic attractors is established through analyzing their internal structures. (2) Based on the knowledge of singularities of escape paths, method for solving the most probable escape path is proposed; based on understanding the global properties and structures of chaotic attractors, characters of escape paths of typical chaotic systems are revealed; efficient Monte-Carlo simulation method of escape problem is developed. The research of this project can help to reveal intrinsic mechanism of complex phenomena widely existing in the real systems, while the research is basic and frontier, possesses important academic values, and has important significance of potential applications for a wide range of scientific and engineering disciplines.

非线性动力系统在弱噪声扰动下会产生各种复杂现象，研究其定态，特别是非平衡定态受扰后的动力学行为有助于理解这些现象的内在机制，并得到对实际问题有重要意义的定量表征。本项目基于Freidlin-Wentzell理论，重点研究非线性动力系统非平衡定态受到弱Gauss白噪声扰动时的准稳态分布和最可能逃逸路径：（1）提出周期吸引子准稳态分布的高阶WKB近似；发展准周期吸引子准稳态分布的高阶近似计算方法；通过对混沌吸引子内部结构的分析建立混沌吸引子准稳态分布的量化方法。（2）基于对逃逸路径奇异性的认识，提出全局最可能逃逸路径的求解方法；基于对全局特性以及混沌吸引子结构的认识，揭示典型混沌系统逃逸路径的特性；发展逃逸问题高效的Monte-Carlo模拟方法。本项目的研究有助于揭示广泛存在于真实系统中复杂现象的内在机制，是基础性的前沿研究，具有重要学术价值，并且对广泛科学与工程学科有着重要潜在应用意义。

随机性在自然现象和工程问题中是不可避免的。研究随机扰动下结构的振动、随机参数系统的动力学性质，以及随机扰动下非线性动力系统响应的扩散性和保持性，都具有重要的学术价值和工程意义。本项目执行过程中的主要研究内容为：(1)弱随机扰动下非线性动力系统非平衡定态(周期吸引子、准周期吸引子)的准稳态分布计算。(2)转子/定子碰摩系统同频全周碰摩响应在随机扰动下的扩散性和保持性。(3)夹心板结构在随机分布载荷作用下的振动控制策略。(4)随机功能梯度材料的参数识别和振动控制。.本项目取得的主要成果有：(1) 基于弱噪声激励下闭轨吸引子概率密度可分解为吸引子横向和纵向两个独立部分的假设，提出了一种计算闭轨随机吸引子概率密度分布的方法：概率密度分解法。分别解决了连续时间系统极限环、离散时间系统准周期环以及非自治系统准周期环的准稳态概率密度计算方法。(2)计算了随机扰动下转子/定子同频全周碰摩的扩散特性， 提出了置信区域的可视化方法，对轴心轨迹图准稳态分布随参数的变化进行了研究和机理分析。利用响应的置信椭球研究了噪声诱导的状态切换现象，通过全局不变集分布揭示了这种噪声诱导逃逸的机理。(3)发展了一种分散化的振动控制方法：独立模态压电分流振动控制法，提出了模态可控性的概念，结合模态范数，建立了受分布随机载荷激励的夹心板的振动控制策略。(4)提出了利用随机有限元模型修正，对功能梯度材料随机组分材料性能参数及体积指数进行统计识别的方法，并利用线性矩阵不等式方法对随机功能梯度材料进行了鲁棒控制器设计。