Delay-Coordinate Maps and the Spectra of Koopman Operators

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 discrete spectrum subspace of the Koopman operator. This enables efficient approximation of Koopman eigenfunctions from high-dimensional data in systems with pure point or mixed spectra. arXiv:1706.08544v7 [math.DS] 5 Nov 2018Delay-coordinate maps and the spectra of Koopman operators 2 Ergodic theory, and in particular its operator-theoretic formulation [15,23], provides a natural framework to address these objectives. In this framework, the focus is on the action of the dynamical system on spaces of observables (functions of the state), as opposed to the dynamical flow itself. The advantage of this approach, first realized in the seminal work of Koopman [39], is that the action of a general dynamical system on spaces of observables is always linear. As a result, with appropriate regularity assumptions, the problem of identification and prediction of dynamically intrinsic coherent patterns can be formulated as an estimation problem for the spectrum of a linear evolution operator. In addition, for systems exhibiting ergodic behavior, spectral quantities such as eigenvalues and eigenfunctions can be statistically estimated from time-ordered data without prior knowledge of the state space geometry or the equations of motion. At the same time, spaces of observables are also infinite dimensional, so the issue of finitedimensional approximation of (potentially unbounded) operators becomes relevant.Starting from the techniques proposed in [22,50,52], the operator-theoretic approach to ergodic theory has stimulated the development of a broad range of techniques for data-driven modeling of dynamical systems. These methods employ either the Koopman [2,14,15,30,33,34,40,41,50,52,55,57,62,67] or the Perron-Frobenius (transfer) operators [21,22,[27][28][29], which are duals to one another in appropriate function spaces. The goal common to these techniques is to approximate spectral quantities for the operator in question, such as eigenvalues, eigenfunctions, and spectral projections, from measured values of observables along orbits of the dynamics. To that end, a diverse range of approaches has been employed, including state space partitions [21,22,[27][28][29], harmonic averaging [20,50,52], iterative methods [55,57], dictionary/basis representations [30,33,34,40,62,67], delay-coordinate embeddings [2,14,30,33], and spectral-mom...

Time-Delay Observables for Koopman: Theory and Applications

“…This convergence result is unusual for generic families of measurement functions, and motivates the application of Koopman spectral methods to delay coordinates for systems where incomplete measurement data is available. Das and Giannakis have also investigated the spectrum of the Koopman operator in delay coordinates [10]. HAVOK models are also closely related to the Prony approximation of the Koopman decomposition [42].…”

Operator-theoretic framework for forecasting nonlinear time series with kernel analog techniques

“…, }, (i) the eigenvalues λ i,n converge to λ i ; (ii) the RKHS functions ψ i,n converge, up to multiplication by a constant phase factor, to ψ i in C(U) norm; and (iii) each of the expansion coefficients α i,n (τ) converges to α i (τ). The first two of these claims are a consequence of the following lemma, which is based on [46,Theorem 15], [55,Corollary 2], and [56,Theorem 7].…”

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“…However, the HAVOK method builds models by time-delay embeddings, thus capturing a shadow of the latent variables [35]. Thus instead of advancing instantaneous linear or nonlinear measurements of the state of a system directly, as in DMD, it may be possible to obtain intrinsic measurement coordinates where the dynamics appear approximately linear, based on time-delayed measurements of the system [35,[89][90][91]. This perspective is data-driven, relying on the wealth of information from previous measurements to inform the future.…”