Symbolic extended dynamic mode decomposition

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.