Stabilizing embedology: Geometry-preserving delay-coordinate maps
Stabilizing embedology: Geometry-preserving delay-coordinate maps
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DOI：
10.1103/physreve.97.022222
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发表时间：
2018-02-26
影响因子：
2.4
通讯作者：
Rozell, Christopher J.
中科院分区：
文献类型：
--
作者：
Eftekhari, Armin;Yap, Han Lun;Rozell, Christopher J.
延迟坐标映射是一种有效的和广泛使用的技术，用于重建和分析基于时间序列输出的非线性系统的动力学。延迟坐标映射的有效性长期以来一直得到Takens嵌入定理的支持，该定理保证延迟坐标映射使用时间序列输出来提供隐藏状态空间的重构，即系统吸引子的一对一嵌入。虽然这种拓扑保证确保重建中的不同点对应于原始状态空间中的不同点，但它并不表征这种嵌入的质量或说明特定参数如何影响重建。在本文中，我们扩展Takens的结果，建立条件下，延迟坐标映射是保证提供一个稳定的嵌入系统的吸引子。除了仅保留吸引子拓扑结构之外，稳定嵌入通过确保状态空间中的点之间的距离近似地保留来保留吸引子几何结构。特别是，我们发现，延迟坐标映射稳定嵌入一个动力系统的吸引子，如果系统的稳定秩是足够大的吸引子的维数成正比。稳定秩反映了延迟坐标映射中采样间隔与延迟数之间的关系。我们的理论研究结果提供指导，选择系统参数，呼应无关性和冗余之间的权衡，已在文献中进行了实证研究。我们的初步结果是说明吸引子是光滑的子流形的欧氏空间，与扩展的情况下提供的奇怪的吸引子。
Delay-coordinate mapping is an effective and widely used technique for reconstructing and analyzing the dynamics of a nonlinear system based on time-series outputs. The efficacy of delay-coordinate mapping has long been supported by Takens' embedding theorem, which guarantees that delay-coordinate maps use the time-series output to provide a reconstruction of the hidden state space that is a one-to-one embedding of the system's attractor. While this topological guarantee ensures that distinct points in the reconstruction correspond to distinct points in the original state space, it does not characterize the quality of this embedding or illuminate how the specific parameters affect the reconstruction. In this paper, we extend Takens' result by establishing conditions under which delay-coordinate mapping is guaranteed to provide a stable embedding of a system's attractor. Beyond only preserving the attractor topology, a stable embedding preserves the attractor geometry by ensuring that distances between points in the state space are approximately preserved. In particular, we find that delay-coordinate mapping stably embeds an attractor of a dynamical system if the stable rank of the system is large enough to be proportional to the dimension of the attractor. The stable rank reflects the relation between the sampling interval and the number of delays in delay-coordinate mapping. Our theoretical findings give guidance to choosing system parameters, echoing the tradeoff between irrelevancy and redundancy that has been heuristically investigated in the literature. Our initial result is stated for attractors that are smooth submanifolds of Euclidean space, with extensions provided for the case of strange attractors.