Latent Modes of Nonlinear Flows: A Koopman Theory Analysis

The authors propose a framework for analyzing and modeling complex nonlinear local and nonlocal flows using Koopman theory and its discrete approximation, dynamic mode decomposition (DMD). They investigate the conditions for linearization, dimensionality reduction, and flow representation in a general setting, and discuss the limitations of DMD. A new mode decomposition technique is proposed based on the dynamics' typical time profile.

Format: Paperback / softback
Length: 75 pages
Publication date: 29 June 2023
Publisher: Cambridge University Press

The extraction of latent underlying structures from complex nonlinear local and nonlocal flows is a crucial step in their analysis and modeling. In this Element, the authors aim to establish a coherent framework through Koopman theory and its related popular discrete approximation, dynamic mode decomposition (DMD). They delve into the conditions for performing appropriate linearization, dimensionality reduction, and representation of flows in a highly general setting. The core components of this framework revolve around Koopman eigenfunctions (KEFs), for which existence conditions are formulated. This perspective involves viewing the dynamic as a curve in state-space, laying the groundwork for system reconstruction, global controllability, and observability for nonlinear dynamics.

The authors also examine the limitations of DMD by analyzing Koopman theory and proposing a novel mode decomposition technique based on the typical time profile of the dynamics. This approach aims to enhance the understanding and analysis of complex systems characterized by nonlinear dynamics.

By employing this framework, researchers can gain insights into the behavior and dynamics of complex systems, enabling the development of more accurate models and the implementation of effective control strategies. The study of nonlinear dynamics has significant applications in various fields, including physics, engineering, biology, and social sciences, and this Element contributes to the ongoing efforts to unravel the mysteries of these complex systems.

ISBN-13: 9781009323857