The spatiotemporal coupling in delay-coordinates dynamic mode decomposition

Area of Science:

• Dynamical Systems Analysis
• Data-Driven Science
• Fluid Dynamics

Background:

• Dynamic Mode Decomposition (DMD) is crucial for analyzing complex, high-dimensional dynamical systems without explicit equations.
• Standard DMD separates temporal and spatial information, providing a spectral representation via eigenvalues and modes.
• Delay-coordinates embedding augments observations with past time steps, expanding DMD's applicability.

Purpose of the Study:

• To investigate the spectral structure arising from applying DMD to delay-coordinates embeddings.
• To introduce and analyze the phenomenon of spatiotemporal coupling in delay-coordinates DMD.
• To develop an improved method for selecting DMD components in delay-coordinates analysis.

Main Methods:

• Augmenting observational data using delay-coordinates embedding.
• Applying Dynamic Mode Decomposition to the augmented data.
• Formulating and analyzing the spatiotemporal coupling in the resulting DMD components.
• Developing a novel component selection strategy based on the identified spatiotemporal coupling.

Main Results:

• Identified and analyzed a novel spectral structure termed 'spatiotemporal coupling' in delay-coordinates DMD.
• Proposed a new method for selecting essential DMD components from redundant sets.
• Demonstrated superior component selection performance compared to amplitude-based methods.
• Validated the method on noisy simulated signals and diverse dynamical systems.

Conclusions:

• Spatiotemporal coupling is an inherent characteristic of delay-coordinates DMD that impacts component interpretation.
• The proposed component selection method effectively extracts meaningful dynamics from delay-coordinates DMD.
• This approach facilitates more accurate and compact reduced-order modeling of complex systems.