模糊非线性系统的瞬态响应和隶属分布函数演化动力学研究

Complicated phenomena and behaviors widely observed in real systems are mostly transient processes of dynamics, some of which can be characterized by the dynamics of nonlinear systems with fuzzy uncertainty. Analysis on transient evolutionary dynamics of the membership distribution functions (MDFs) of the master equations in fuzzy nonlinear systems is still an open problem. This project is devoted to develop related theories and numerical methods for transient responses and evolutionary membership distribution functions (MDFs) in fuzzy nonlinear systems; to reveal the mechanism and characteristics of transient dynamical behaviors of a fuzzy response as well as its dependence on the possibility distributions of system-initial states; to make clear on the role of spatially topological structures of fuzzy invariant sets and invariant manifolds on the evolutionary process of MDFs; to quantitatively determine the possibility distribution of first escape and average escape time; to specify the globally evolutionary characteristics of the MDFs in the process of fuzzy noise-induced chaotic transients and multi-metastable state switches and transitions in order to obtain the scaling law of the chaotic transients and the possibility measure of the average time of the state switches and transitions. This project is to understand basic evolutionary mechanism of transient dynamical behaviors for complex systems under the interaction of fuzzy uncertainty (noise) and nonlinearity. It belongs to a fundamental and frontier research in nonlinear science and is of great theoretical value and has application potential in a wide range of disciplines in science and engineering.

真实系统展示的复杂现象和行为往往是瞬态的动力学过程，模糊非线性动力学已用来刻画某些真实系统，但其对应的主方程的隶属分布函数的瞬态演化分析仍是有待解决的科学问题。本项目拟发展模糊非线性系统的瞬态响应和隶属分布函数演化理论以及相应的数值方法，揭示模糊瞬态响应动力学行为的机理和规律，及其对初始状态可能性分布的依赖关系；阐明模糊不变集和不变流形空间拓扑结构对隶属分布函数演化过程的作用机制，定量描述首次逃逸和平均逃逸时间的可能性分布；研究混沌瞬态和多亚稳态切换转移过程的隶属分布函数全局演化特性，给出混沌瞬态的标度律描述和切换转移平均时间的可能性测度。本项目的研究试图揭示模糊不确定性(噪声)和非线性相互作用下复杂系统所展示的瞬态演化动力学行为的基本规律，属于非线性科学的基础性前沿研究，具有重要的科学意义，对广泛科学与工程学科有重要的潜在应用价值。

真实系统展示的复杂现象和行为往往是瞬态的动力学过程，模糊非线性动力学已用来刻画某些真实系统，但其对应的主方程的隶属分布函数的瞬态演化分析仍是有待解决的科学问题。本项目发展了模糊非线性系统的瞬态响应和隶属分布函数演化理论以及相应的数值方法，揭示了模糊瞬态响应动力学行为的机理和规律，其主要结果如下：.基于模糊主方程推导出具有模糊非线性形式短时演化律的微分方程，证明了模糊主方程的解与微分包含形式的模糊微分方程的解的等价性，发展了模糊非线性系统的隶属分布函数瞬态演化近似解析计算方法。提出了隶属分布函数短时近似的自适应插值方法，基于演化可能性向量，通过将自适应差值方法引入集值模糊参数空间计算一步转移隶属分布矩阵，建立了高效计算瞬态和稳态模糊响应的隶属分布函数的胞映射法（FGCM with AI），计算效率提升了三十到五十倍。针对具有高斯或波松白噪声的非线性系统，发展了近似稳态概率密度函数的求解方法和随机敏感性分析方法。提出了求解非线性动力系统全局吸引子的高效细分插值胞映射方法。揭示了模糊响应隶属分布函数瞬态演化的动力学行为的机理和规律，发现了模糊噪声诱导的余维二分岔，揭示了分段光滑系统噪声诱导的迁转机制和Stick-Slip分岔以及发现了一类鞍鞍相碰Wada 边界分岔新机制。发表国际期刊论文13篇。