Testing Topological Conjugacy of Time Series

Justyna Signerska-Rynkowska; Paweł Dłotko; Michał Lipiński

doi:10.1137/23m1594728

Testing Topological Conjugacy of Time Series

Abstract

This paper considers a problem of testing, from a finite sample, a topological conjugacy of two trajectories coming from dynamical systems (X, f ) and (Y, g). More precisely, given x1, . . . , xn \subset X and y1, . . . , yn \subset Y such that xi+1 = f (xi) and yi+1 = g(yi) as well as h : X \rightarrow Y, we deliver a number of tests to check if f and g are topologically conjugated via h. The values of the tests are close to zero for systems conjugate by h and large for systems that are not. Convergence of the test values, in the case when the sample size goes to infinity, is established. We provide a number of numerical examples indicating scalability and robustness of the presented methods. In addition, we show howthe presented method gives rise to a test of sufficient embedding dimension, mentioned in Takens' embedding theorem. Our methods also apply to the situation when we are given two observables of deterministic processes, of a form of one or higher dimensional time series. In this case, their similarity can be assessed by comparing the dynamics of their Takens' reconstructions. Finally, we include a proof-of-concept study using the presented methods to search for an approximation of the homeomorphism conjugating given systems.

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DOI:: Digital Object Identifier (open in new tab) 10.1137/23M1594728
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Keywords

CONJUGACY, SEMICONJUGACY, EMBEDDING, NONLINEAR TIME-SERIES ANALYSIS, FALSE NEAREST NEIGHBORS, SIMILARITY MEASURES, K-NEAREST NEIGHBORS

Details

Category:

Articles

Type:

artykuły w czasopismach

Published in:

SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS no. 23, pages 2939 - 2982,
ISSN: 1536-0040

Language:

English

Publication year:

2024

Bibliographic description:

Signerska-Rynkowska J., Dłotko P., Lipiński M.: Testing Topological Conjugacy of Time Series// SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS -Vol. 23,iss. 4 (2024), s.2939-2982

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