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Symbolic extended dynamic mode decomposition.

Connor Kennedy1, John Kaushagen1, Hong-Kun Zhang1

  • 1Department of Mathematics and Statistics, University of Massachusetts Amherst, Amherst, Massachusetts 01003, USA.

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Summary
This summary is machine-generated.

This study introduces a novel Extended Dynamic Mode Decomposition (EDMD) method using symbolic representations for chaotic systems. The new approach improves Koopman operator estimation and forecasting accuracy.

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Area of Science:

  • Dynamical systems theory
  • Data-driven scientific discovery
  • Numerical analysis

Background:

  • Extended Dynamic Mode Decomposition (EDMD) is a powerful tool for analyzing complex dynamical systems by estimating the Koopman operator.
  • A key challenge in EDMD is selecting an effective dictionary of observables.
  • Symbolic representations offer a structured way to define observables for certain dynamical systems.

Purpose of the Study:

  • To develop a new EDMD method tailored for systems with symbolic representations.
  • To construct an effective dictionary for EDMD using symbolic dynamics.
  • To analyze the theoretical convergence and practical estimation bounds of the proposed method.

Main Methods:

  • Constructing a dictionary from indicators of "cylinder sets" derived from a generating partition of the system's state space.
  • Applying EDMD to chaotic dynamical systems with known or estimable generating partitions.
  • Proving strong operator topology convergence for the projection and the EDMD estimate (Km).

Main Results:

  • Demonstrated strong operator topology convergence for the projection and the EDMD estimate (Km).
  • Established practical finite-step estimation bounds for the projection and Km.
  • Successfully applied the method to the dyadic and logistic maps for eigenspectrum estimation and forecasting.

Conclusions:

  • The proposed EDMD method effectively utilizes symbolic representations to construct an optimal dictionary.
  • The method provides theoretical guarantees on convergence and practical bounds for estimation accuracy.
  • This approach enhances the capabilities of EDMD for analyzing and predicting chaotic dynamical systems.