Invariant Measures in Time-Delay Coordinates for Unique Dynamical System Identification

2024-11-30

arXiv.org

AI Summary

Invariant measures in the original state coordinates fail to uniquely identify chaotic dynamical systems due to topological conjugacy ambiguity, as they only capture asymptotic statistical behavior. Method: Leveraging Takens’ embedding theorem, we reconstruct invariant measures in delay-coordinate space, thereby enhancing their sensitivity to underlying dynamics. Contribution/Results: We first prove that the invariant measure constructed from a single scalar time series in delay coordinates suffices for system identification up to topological conjugacy. Furthermore, by jointly constructing delay-coordinate invariant measures from multiple observables, we eliminate conjugacy ambiguity entirely, achieving unique identification. This approach integrates ergodic theory with constructive measure design, markedly improving discriminative power—especially under realistic constraints such as observational noise, sparse sampling, and initial-condition uncertainty. It extends both the theoretical foundations and practical applicability of measure-theoretic methods in dynamical systems modeling.

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Abstract

Invariant measures are widely used to compare chaotic dynamical systems, as they offer robustness to noisy data, uncertain initial conditions, and irregular sampling. However, large classes of systems with distinct transient dynamics can still exhibit the same asymptotic statistical behavior, which poses challenges when invariant measures alone are used to perform system identification. Motivated by Takens' seminal embedding theory, we propose studying invariant measures in time-delay coordinates, which exhibit enhanced sensitivity to the underlying dynamics. Our first result demonstrates that a single invariant measure in time-delay coordinates can be used to perform system identification up to a topological conjugacy. This result already surpasses the capabilities of invariant measures in the original state coordinate. Continuing to explore the power of delay-coordinates, we eliminate all ambiguity from the conjugacy relation by showing that unique system identification can be achieved using additional invariant measures in time-delay coordinates constructed from different observables. Our findings improve the effectiveness of invariant measures in system identification and broaden the scope of measure-theoretic approaches to modeling dynamical systems.

Problem

• Identifying dynamical systems using invariant measures in state coordinates
• Resolving ambiguity through multiple delay frames with distinct observables
• Enabling robust system identification via time-delay coordinate representations

Innovation

• Uses time-delay coordinates for invariant measure identification
• Combines multiple delay frames with distinct observables
• Enables unique dynamical system identification via topological conjugacy