AI 中文总结
研究针对连续混沌流符号划分难题，将基于柯普曼分析的方法扩展至此。通过基于序数模式构建庞加莱截面并结合返回映射方法，实现连续混沌流有效符号划分，经测试证明方法鲁棒性，还应用于多个系统，成功将方法从映射拓展到连续流。
AI 中文摘要
符号动力学作为混沌系统研究领域的关键工具，促使人们对各种符号划分方法进行广泛研究。大多数现有方法在划分多元混沌状态空间时具有启发式和经验性的局限性。此前研究中，我们成功采用KA方法并通过GKA方法在混沌映射上获得了初步的粗符号边界并细化了符号边界，但该方法应用于连续混沌流时失败。为应对上述挑战，本研究将基于柯普曼分析的符号划分方法扩展到连续混沌流。对于一般混沌流，利用柯普曼分析识别合适的庞加莱截面。我们基于序数模式构建不同的候选庞加莱截面，并将基于序数模式的庞加莱截面构建方法与基于柯普曼分析的返回映射方法相结合，实现了连续混沌流的有效符号划分。还进行了噪声扰动测试，证明了所提方法的鲁棒性。该基于序数模式的分析应用于罗塞尔系统、洛伦兹系统、陆系统和陈系统。本研究借助序数模式成功地将有效的符号划分引入连续系统，实现了该方法从映射到连续流的可靠转移。
英文摘要
As a crucial tool in the field of chaotic systems study, Symbolic dynamics prompts extensive research into various methods for symbolic partitioning. The limitations of the majority of these methods are usually heuristic and empirical for partitioning the multivariate chaotic state space. Fortunately, we successfully take KA method and obtain primary coarse symbolic boundary and refine the symbolic boundary via GKA method on chaotic map in our previous studies. However, this method fails when applied to continuous chaotic flows. The trajectories of complex continuous chaotic flows are complex, and while their mechanisms are quite different from those of chaotic maps, they are by no means completely distinct -- after all, both are governed by the same fundamental laws of chaos. In response to the aforementioned challenges, a modified approach should be developed to overcome the failure of the existing method for continuous chaotic flows. In this study, we extend the Koopman-analysis-based symbolic partitioning approach to continuous chaotic flows. For general chaotic flows, Koopman analysis is employed to identify suitable Poincare sections. In this work, we construct different candidate Poincare sections based on ordinal patterns. By combining this ordinal-pattern-based Poincare section construction method with the Koopman-analysis-based return-map approach, we achieve effective symbolic partitioning of continuous chaotic flows. Noise perturbation tests are also conducted, demonstrating the robustness of the proposed method. This ordinal-pattern-based analysis is applied to Rossler system, Lorenz system, Lu system and Chen system. This study, with the aid of ordinal patterns, successfully introduces an effective symbolic partition into continuous systems, achieving a faithful transfer of the method from maps to continuous flows.
