Koopman Operator and its Approximations for Systems with Symmetries

Abstract

Nonlinear dynamical systems with symmetries exhibit a rich variety of behaviors, including complex attractor-basin portraits and enhanced and suppressed bifurcations. Symmetry arguments provide a way to study these collective behaviors and to simplify their analysis. The Koopman operator is an infinite dimensional linear operator that fully captures a system's nonlinear dynamics through the linear evolution of functions of the state space. Importantly, in contrast with local linearization, it preserves a system's global nonlinear features. We demonstrate how the presence of symmetries affects the Koopman operator structure and its spectral properties. In fact, we show that symmetry considerations can also simplify finding the Koopman operator approximations using the extended and kernel dynamic mode decomposition methods (EDMD and kernel DMD). Specifically, representation theory allows us to demonstrate that an isotypic component basis induces block diagonal structure in operator approximations, revealing hidden organization. Practically, if the data is symmetric, the EDMD and kernel DMD methods can be modified to give more efficient computation of the Koopman operator approximation and its eigenvalues, eigenfunctions, and eigenmodes. Rounding out the development, we discuss the effect of measurement noise.

References in corpus (20)

• Discovering governing equations from data: Sparse identification of nonlinear dynamical systems
• A Data-Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition
• Deep learning for universal linear embeddings of nonlinear dynamics
• Applied Koopmanism
• Symmetries, Cluster Synchronization, and Isolated Desynchronization in Complex Networks
• Extracting spatial-temporal coherent patterns in large-scale neural recordings using dynamic mode decomposition
• Extended dynamic mode decomposition with dictionary learning: a data-driven adaptive spectral decomposition of the Koopman operator
• On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operator
• Characterizing and correcting for the effect of sensor noise in the dynamic mode decomposition
• Complete Characterization of Stability of Cluster Synchronization in Complex Dynamical Networks
• On Automorphism Groups of Networks
• On the numerical approximation of the Perron-Frobenius and Koopman operator
• Observability and Controllability of Nonlinear Networks: The Role of Symmetry
• Stable Chimeras and Independently Synchronizable Clusters
• Applied Koopman Operator Theory for Power Systems Technology
• On the correspondence between Koopman mode decomposition, resolvent mode decomposition, and invariant solutions of the Navier-Stokes equations
• Topological Control of Synchronization Patterns: Trading Symmetry for Stability
• Crisis of the Chaotic Attractor of a Climate Model: A Transfer Operator Approach
• Patterns of patterns of synchronization: Noise induced attractor switching in rings of coupled nonlinear oscillators
• Sensing and Control in Symmetric Networks

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• Stochastic Adversarial Koopman Model for Dynamical Systems
• Sparse system identification by low-rank approximation
• Group-Convolutional Extended Dynamic Mode Decomposition
• Koopman Q-learning: Offline Reinforcement Learning via Symmetries of Dynamics