Validity of Koopman mode decomposition under finite-dimensional approximation for continuous-spectrum systems

Ascertain the validity of the Koopman mode decomposition formula x_{s+Δs} = ∑_k e^{λ_k Δs} φ_k(x_s) v_k (Eq. \ref{eq:oscillation_x}) for the virtual dynamics d x_s = ν_t^{hk}(x_s) ds when the Koopman generator has a continuous spectrum (e.g., in chaotic systems) and the system is forcibly approximated by a finite-dimensional linear operator. Specifically, determine whether a finite set of oscillatory modes can accurately represent the virtual dynamics under these conditions.

Background

The paper introduces a thermodynamic framework that decomposes the housekeeping entropy production rate in overdamped nonlinear Langevin dynamics into contributions from oscillatory modes identified via Koopman mode decomposition. The central representation (Eq. \ref{eq:oscillation_x}) expresses the virtual dynamics driven by the housekeeping local mean velocity as a sum over modes determined by the Koopman generator’s eigenvalues and eigenfunctions.

This framework assumes that, under a finite-dimensional numerical approximation, the Koopman generator is skew-adjoint and hence diagonalizable, enabling a finite modal expansion. However, for systems with infinitely many modes or continuous spectra—such as chaotic systems—the Koopman generator may not be fully captured by a finite set of eigenvalues, raising doubts about the adequacy of the finite-dimensional modal representation.

The authors explicitly note uncertainty regarding the validity of Eq. \ref{eq:oscillation_x} in such cases, suggesting that while generalized dynamic mode decomposition methods exist for continuous spectra, it remains unresolved whether a finite set of oscillatory modes suffices to accurately represent the virtual dynamics in systems with continuous Koopman spectra.