AI 中文总结
本研究提出一种面向集合的方法，通过将状态空间网格化并转化为带权有向图，结合图算法与马尔可夫链，构建了评估映射是否具备混沌巡游性的算法，并在两类动力系统中验证了该算法的有效性。
AI 中文摘要
混沌巡游性（Chaotic Itinerancy，CI）由K.池田、I.津田和K.金子等学者在20世纪90年代初引起关注，是动力系统中轨迹在类吸引子附近经历有序运动、并在其间穿插混沌跃迁的现象。已发现存在混沌巡游性的映射包括耦合映射格（Coupled Map Lattices，CML）和全局耦合一维混沌映射（Globally Coupled one-dimensional chaotic Maps，GCM）。我们采用数值方法和图算法研究这类映射：具体而言，将状态空间划分为有限个紧子集构成的网格，并用网格元素的多值映射表示该映射，该映射可视为以网格元素为顶点、元素间映射为带权边的有向图。该设置提供了全局动力学的粗粒度视图，为使用马尔可夫链和高效图算法研究动力学特征创造了条件。特别是，可通过计算图中的强连通分量找到不变集；对图的转移矩阵进行分析，可得到其平稳分布并计算局部熵，以此作为系统膨胀或不稳定性的度量。利用这些工具，我们提出了一种评估特定映射是否具有混沌巡游性的算法，并将其应用于两类动力系统：全局耦合逻辑斯蒂映射系统，以及我们对大范围参数开展计算的耦合映射格（CML）变体系统。
英文摘要
Chaotic itinerancy (CI), brought to attention, among others, by K. Ikeda, I. Tsuda and K. Kaneko in the early 1990s, is a phenomenon in which trajectories in a dynamical system experience periods of ordered motion near quasi-attractors interspersed with chaotic transitions between them. Possible maps in which CI was found include coupled map lattices (CML) and globally coupled one-dimensional chaotic maps (GCM). We study such maps using numerical methods and graph algorithms. Specifically, we partition the state space into a finite grid of compact subsets, and we represent the map using a multivalued mapping of grid elements. This mapping can be perceived as a directed graph, with grid elements as vertices and individual mappings between them as weighted edges. This setup provides a coarse view of global dynamics and opens the opportunity for using Markov chains and efficient graph algorithms to study dynamical features. In particular, invariant sets can be found by computing strongly connected components in the graph. Analysis of the transition matrix of the graph makes it possible to find its stationary distribution and to compute local entropy as a measure of expansion or instability in the system. Using these tools, we propose an algorithm for assessing whether a certain map possesses the CI property and show its application to dynamical systems: a globally coupled system of logistic maps and a variant of a CML system for which we conduct computations for a large range of parameters.
Comments27 pages, 4 tables, 16 figures
