发表机构
Gdańsk University of Technology; Institute of Mathematics, Polish Academy of Sciences; Warsaw University（格但斯克理工大学; 波兰科学院数学研究所; 华沙大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究提出基于BEPE、DoD和欧拉特征摘要的向量场拓扑描述符框架，用于稳健比较动力系统状态，在多个基准系统及高维湍流中验证了其有效性、稳健性和计算效率。
AI 中文摘要
非线性动力系统的定性分析依赖于识别控制相空间组织的几何和拓扑结构。我们开发了一个基于向量场的新型拓扑和几何描述符的计算框架，能够对来自解析模型和采样数据的动力状态进行稳健比较。该框架包括起点-终点嵌入（BEPE）、方向密度（DoD）以及基于欧拉特征的摘要（欧拉特征曲线（ECC）和欧拉特征剖面（ECP））。BEPE捕捉向量场的局部变化，而DoD表征向量方向的全局组织。ECC和ECP提供基于过滤的向量场结构拓扑摘要，其中ECP将这些摘要扩展到多参数设置。这些描述符共同编码了关于流组织的互补几何和拓扑信息。我们制定了描述符的连续和采样版本，并建立了它们对扰动、有限采样和选定变换的稳健性。所提出方法的有效性在基准系统集合（Hopf分岔、FitzHugh-Nagumo模型、Lorenz系统）上得到了证明。在离散Hénon映射上的实验证明了该框架对离散动力系统生成的位移场的适用性，而高维湍流实验则评估了其超出低维相空间的可扩展性。与经典方法（包括基于Conley指数的方法和$L_p$型度量）的比较表明，所提出的描述符提供了互补的结构信息，同时具有有利的稳健性、可解释性和计算效率，在动力系统理论与拓扑数据分析之间建立了实际联系。
英文摘要
The qualitative analysis of nonlinear dynamical systems relies on identifying geometric and topological structures that govern phase-space organization. We develop a computational framework based on novel topological and geometric descriptors of vector fields that enables robust comparison of dynamical regimes from both analytical models and sampled data. The framework comprises the Begin-End Point Embedding (BEPE), Density of Directions (DOD) and Euler characteristic-based summaries (Euler Characteristic Curve (ECC) and Euler Characteristic Profile (ECP)). BEPE captures local variations in vector field, while DoD characterizes the global organization of vector directions. ECCs and ECPs provide filtration-based topological summaries of vector-field structure, with ECPs extending these summaries to multiparameter settings. Together, these descriptors encode complementary geometric and topological information about flow organization. We formulate both continuous and sampled versions of the descriptors and establish their robustness with respect to perturbations, finite sampling and selected transformations. The effectiveness of the proposed approach is demonstrated on a collection of benchmark systems (the Hopf bifurcation, the FitzHugh-Nagumo model, the Lorenz system). Experiments on the discrete Hénon map demonstrate the applicability of the framework to displacement fields generated by discrete dynamical systems, while experiments on high-dimensional turbulent flows assess its scalability beyond low-dimensional phase spaces. Comparisons with classical approaches, including Conley-index-based methods and $L_p$-type metrics, show that the proposed descriptors provide complementary structural information while offering favorable robustness, interpretability and computational efficiency, establishing a practical connection between dynamical systems theory and topological data analysis.
Comments45 pages, 19 figures
