Spectral Analysis of the Koopman Operator for Partial Differential Equations

Abstract

We provide an overview of the Koopman operator analysis for a class of partial differential equations describing relaxation of the field variable to a stable stationary state. We introduce Koopman eigenfunctionals of the system and use the notion of conjugacy to develop spectral expansion of the Koopman operator. For linear systems such as the diffusion equation, the Koopman eigenfunctionals can be expressed as linear functionals of the field variable. The notion of inertial manifolds is shown to correspond to joint zero level sets of Koopman eigenfunctionals, and the notion of isostables is defined as the level sets of the slowest decaying Koopman eigenfunctional. Linear diffusion equation, nonlinear Burgers equation, and nonlinear phase-diffusion equation are analyzed as examples.

15 pages, 4 figures

References in corpus (4)

• Applied Koopmanism
• Phase-amplitude reduction of transient dynamics far from attractors for limit-cycling systems
• Data Fusion via Intrinsic Dynamic Variables: An Application of Data-Driven Koopman Spectral Analysis
• Phase reduction approach to synchronization of spatiotemporal rhythms in reaction-diffusion systems

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• Koopman analysis of the periodic Korteweg-de Vries equation
• Improvement of system identification of stochastic systems via Koopman generator and locally weighted expectation
• Koopman spectral analysis of elementary cellular automata
• Dynamic mode decomposition for detecting oscillatory transient activity via sparsity and smoothness regularization