Delay-Coordinate Maps and the Spectra of Koopman Operators

Suddhasattwa Das, Dimitrios Giannakis

Research output: Contribution to journalArticle

Abstract

The Koopman operator induced by a dynamical system is inherently linear and provides an alternate method of studying many properties of the system, including attractor reconstruction and forecasting. Koopman eigenfunctions represent the non-mixing component of the dynamics. They factor the dynamics, which can be chaotic, into quasiperiodic rotations on tori. Here, we describe a method through which these eigenfunctions can be obtained from a kernel integral operator, which also annihilates the continuous spectrum. We show that incorporating a large number of delay coordinates in constructing the kernel of that operator results, in the limit of infinitely many delays, in the creation of a map into the point spectrum subspace of the Koopman operator. This enables efficient approximation of Koopman eigenfunctions in systems with pure point or mixed spectra. We illustrate our results with applications to product dynamical systems with mixed spectra.

Original languageEnglish (US)
JournalJournal of Statistical Physics
DOIs
StatePublished - Jan 1 2019

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Eigenfunctions
operators
eigenvectors
Operator
Dynamical system
dynamical systems
Kernel Operator
Point Spectrum
Continuous Spectrum
Integral Operator
Alternate
Forecasting
Attractor
continuous spectra
Torus
Subspace
forecasting
kernel
Approximation
products

Keywords

  • Delay-coordinate maps
  • Galerkin approximation
  • Kernel methods
  • Koopman eigenfunctions
  • Koopman operators
  • Point spectrum

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Delay-Coordinate Maps and the Spectra of Koopman Operators. / Das, Suddhasattwa; Giannakis, Dimitrios.

In: Journal of Statistical Physics, 01.01.2019.

Research output: Contribution to journalArticle

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