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

Dimitrios Giannakis

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Article number8
JournalResearch in Mathematical Sciences
Issue number1
StatePublished - Mar 2021


  • Delay-coordinate maps
  • Ergodic dynamical systems
  • Feature extraction
  • Kernel integral operators
  • Koopman operators

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Mathematics (miscellaneous)
  • Computational Mathematics
  • Applied Mathematics


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