期刊
PHYSICA D-NONLINEAR PHENOMENA
卷 334, 期 -, 页码 49-59出版社
ELSEVIER
DOI: 10.1016/j.physd.2016.03.006
关键词
Topology; Delay-coordinate embedding; Nonlinear time-series analysis; Computational homology; Witness complex
资金
- National Science Foundation [CMMI-1245947, CNS-0720692, DMS-1211350]
- Directorate For Engineering
- Div Of Civil, Mechanical, & Manufact Inn [1537460] Funding Source: National Science Foundation
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [1211350] Funding Source: National Science Foundation
Computing the state-space topology of a dynamical system from scalar data requires accurate reconstruction of those dynamics and construction of an appropriate simplicial complex from the results. The reconstruction process involves a number of free parameters and the computation of homology for a large number of simplices can be expensive. This paper is a study of how to compute the homology efficiently and effectively without a full (diffeomorphic) reconstruction. Using trajectories from the classic Lorenz system, we reconstruct the dynamics using the method of delays, then build a simplicial complex whose vertices are a small subset of the data: the witness complex. Surprisingly, we find that the witness complex correctly resolves the homology of the underlying invariant set from noisy samples of that set even if the reconstruction dimension is well below the thresholds for assuring topological conjugacy between the true and reconstructed dynamics that are specified in the embedding theorems. We conjecture that this is because the requirements for reconstructing homology are less stringent: a homeomorphism is sufficient-as opposed to a diffeomorphism, as is necessary for the full dynamics. We provide preliminary evidence that a homeomorphism, in the form of a delay-coordinate reconstruction map, may exist at a lower dimension than that required to achieve an embedding. (C) 2016 Elsevier B.V. All rights reserved.
作者
我是这篇论文的作者
点击您的名字以认领此论文并将其添加到您的个人资料中。
推荐
暂无数据