TY - JOUR

T1 - Delay-coordinate maps, coherence, and approximate spectra of evolution operators

AU - Giannakis, Dimitrios

N1 - Funding Information:
The author is grateful to Andrew Majda for his guidance and mentorship during a postdoctoral position at the Courant Institute from 2009 to 2012. He is especially grateful for his friendship and collaboration over the years. This research was supported by NSF Grant 1842538, NSF Grant DMS 1854383, and ONR YIP Grant N00014-16-1-2649.
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG part of Springer Nature.
Copyright:
Copyright 2021 Elsevier B.V., All rights reserved.

PY - 2021/3

Y1 - 2021/3

N2 - 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 of eigenfunctions of integral operators built from delay-coordinate mapped data. It is shown that these observables are ε-approximate eigenfunctions of the Koopman evolution operator of the system, with a bound ε controlled by the length of the delay-embedding window, the evolution time, and appropriate spectral gap parameters. In particular, ε 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 ε-approximate Koopman eigenvalues, 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 L2. 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.

AB - 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 of eigenfunctions of integral operators built from delay-coordinate mapped data. It is shown that these observables are ε-approximate eigenfunctions of the Koopman evolution operator of the system, with a bound ε controlled by the length of the delay-embedding window, the evolution time, and appropriate spectral gap parameters. In particular, ε 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 ε-approximate Koopman eigenvalues, 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 L2. 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.

KW - Delay-coordinate maps

KW - Ergodic dynamical systems

KW - Feature extraction

KW - Kernel integral operators

KW - Koopman operators

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U2 - 10.1007/s40687-020-00239-y

DO - 10.1007/s40687-020-00239-y

M3 - Article

AN - SCOPUS:85100106743

VL - 8

JO - Research in Mathematical Sciences

JF - Research in Mathematical Sciences

SN - 2522-0144

IS - 1

M1 - 8

ER -