Delay-coordinate maps, coherence, and approximate spectra of evolution operators

TL;DR

This paper develops a data-driven method using delay-coordinate maps and kernel operators to identify approximately cyclical observables in ergodic dynamical systems, with applications to systems like Lorenz 63.

Contribution

It introduces a novel approach to construct approximate Koopman eigenfunctions from delay-embedded data, applicable to systems with continuous spectra and no non-trivial eigenfunctions.

Findings

Constructed observables exhibit near-cyclic behavior with autocorrelation above 0.5 over 10 Lyapunov times.

Approximate eigenfunctions become more accurate as delay-embedding window increases.

Method applies to measure-preserving, ergodic systems with arbitrary spectral properties.

Abstract

The problem of data-driven identification of coherent observables of measure-preserving, ergodic dynamical systems is studied using kernel integral operator techniques. An approach is proposed whereby complex-valued observables with approximately cyclical behavior are constructed from a pair eigenfunctions of integral operators built from delay-coordinate mapped data. It is shown that these observables are $\epsilon$-approximate eigenfunctions of the Koopman evolution operator of the system, with a bound $\epsilon$ controlled by the length of the delay-embedding window, the evolution time, and appropriate spectral gap parameters. In particular, $ \epsilon$ can be made arbitrarily small as the embedding window increases so long as the corresponding eigenvalues remain sufficiently isolated in the spectrum of the integral operator. It is also shown that the time-autocorrelation functions of such observables are $\epsilon$-approximate Koopman eigenvalue, exhibiting a well-defined characteristic oscillatory frequency (estimated using the Koopman generator) and a slowly-decaying modulating envelope. The results hold for measure-preserving, ergodic dynamical systems of arbitrary spectral character, including mixing systems with continuous spectrum and no non-constant Koopman eigenfunctions in $L^2$. Numerical examples reveal a coherent observable of the Lorenz 63 system whose autocorrelation function remains above 0.5 in modulus over approximately 10 Lyapunov timescales.