非均匀采样线性系统的稳定Takens嵌入定理
Stable Takens' Embedding Theorem for Non-Uniformly-Sampled Linear Systems
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中文总结 AI 辅助
本文将稳定线性Takens嵌入定理扩展至非均匀采样线性系统场景，提出广义范德蒙德矩阵秩的猜想，证明若猜想成立，嵌入质量可收敛至相同渐近界，结果获数值模拟支持。
中文摘要 AI 辅助
Takens的时延嵌入定理给出了条件，在该条件下，由在动力系统吸引子上演化的轨迹的均匀采样时间序列构成的时延坐标映射，可忠实地表示原系统的动力学特性。非线性系统可能具有高度敏感性，而Takens定理未提供时延嵌入稳定性的保证。在线性场景中，时延嵌入稳定性的相关结论更易处理，且已针对具有均匀间隔时延的时延坐标映射得到证明。然而，在许多实验应用中，时间序列数据可能为非均匀采样，尤其存在于具有多时间尺度的系统中，或使用基于事件而非基于时间的采样技术时。本文中，我们将稳定线性Takens嵌入的定理扩展至时延坐标映射包含非均匀间隔时延的场景。我们对捕获时延嵌入时间结构的广义范德蒙德矩阵的秩提出一个猜想。我们证明，若该猜想成立，现有的稳定线性Takens嵌入定理可直接扩展至非均匀采样场景；且当时延坐标映射中使用大量时延，嵌入质量收敛至相同的渐近界，该结果得到数值模拟的支持。
英文摘要
Takens' time-delay embedding theorem provides conditions under which delay-coordinate maps, formed using uniformly-sampled time series of trajectories evolving on attractors of dynamical systems, can faithfully represent the dynamics of the original system. Nonlinear systems can be highly sensitive, and Takens' theorem does not provide guarantees about the stability of time-delay embeddings. In the linear setting, statements about the stability of time-delay embeddings are more tractable and have been proven for delay-coordinate maps with evenly-spaced delays. In many experimental applications, however, time series data may be non-uniformly-sampled, especially in systems with multiple timescales or when using event-based rather than time-based sampling techniques. In this paper, we extend the theorems for the stable linear Takens' embeddings to the setting where the delay-coordinate maps involve unevenly-spaced delays. We pose a conjecture about the rank of generalized Vandermonde matrices that capture the temporal structure of time-delay embeddings. We prove that, provided the conjecture holds, existing theorems about stable linear Takens' embeddings readily extend to unevenly-sampled settings, and the quality of the embedding converges to the same asymptotic bounds when using a large number of delays in the delay-coordinate map, a result which is supported by numerical simulations.
