The Symbolic Partition of Chaotic Flows Based on Ordinal Patterns

Haipeng Li; Yueheng Lan

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arXiv · 2607.14490

The Symbolic Partition of Chaotic Flows Based on Ordinal Patterns

Haipeng Li·Yueheng Lan

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Abstract

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.

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Haipeng Li, Yueheng Lan. 2026-07-16. The Symbolic Partition of Chaotic Flows Based on Ordinal Patterns. https://arxiv.org/abs/2607.14490

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