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...

No abstract

The Birkhoff Ergodic Theorem concludes that time averages, i.e., Birkhoff averages, B N (f ) ∶= Σ N −1 n=0 f (x n ) N of a function f along a length N ergodic trajectory (x n ) of a function T converge to the space average ∫ f dµ, where µ is the unique invariant probability measure. Convergence of the time average to the space average is slow. We use a modified average of f (x n ) by giving very small weights to the "end" terms when n is near 0 or N − 1. When (x n ) is a trajectory on a quasiperiodic torus and f and T are C ∞ , our Weighted Birkhoff average (denoted WB N (f )) converges "super" fast to ∫ f dµ with respect to the number of iterates N , i.e. with error decaying faster than N −m for every integer m. Our goal is to show that our Weighted Birkhoff average is a powerful computational tool, and this paper illustrates its use for several examples where the quasiperiodic set is one or two dimensional. In particular, we compute rotation numbers and conjugacies (i.e. changes of variables) and their Fourier series, often with 30-digit accuracy. d j=1 a j ρ j = 0, then every a j = 0. We then say such a ρ is irrational.Let T be a C ∞ quasiperiodic map. The quasiperiodicity persists for most small perturbations by the Kolmogorov-Arnold-Moser theory. We believe that quasiperiodicity is one of only three types of invariant sets with a dense trajectory that can occur in typical smooth maps. The other two types are periodic sets and chaotic sets. See [1] for the statement of our formal conjecture of this triumvirate. For example, quasiperiodicity occurs in a system of weakly coupled oscillators, in which there is an invariant smooth *