Existence and uniqueness of global Koopman eigenfunctions for stable fixed points and periodic orbits

A quantitative analysis of Koopman operator methods for system identification and predictions

“…It was notably instrumental in Von Neumann's proof of the mean ergodic theorem [18], and a first connection of chaotic behavior to the existence of continuous parts in the spectrum of the Koopman semigroup was noted in [19]. More recently, spectral properties of the Koopman semigroup have been shown to be related to a variety of aspects of the underlying dynamical system such as ergodicity, stability, existence of different time scales, mixing and non-mixing properties (see for example [20][21][22][23][24][25]). An important feature in these works is that the spectral analysis of the Koopman operator is done in a rather geometrical spirit in combination with linearization results and conjugation techniques, focusing on local behaviour near equilibria or attractors.…”

Estimating Phase from Observed Trajectories Using the Temporal 1-Form

“…We note that, while there has been work to define generalized notions of asymptotic phase for stochastic oscillators [44][45][46][47] (see [61,62] for a spirited discussion), in this paper we restrict ourselves to estimation of the classical asymptotic phase of a deterministic oscillator using data from a perturbed version of the underlying deterministic system. We also note that there are operator-theoretic methods of recent interest revealing phase as the generator of a family of eigenfunctions of the Koopman operator with purely imaginary eigenvalues [63][64][65]. These eigenfunctions and others can be estimated using Fourier/Laplace averages [63][64][65][66][67] when dynamical equations are known, and from data using Dynamic Mode Decomposition [68], at least in certain cases.…”

“…We also note that there are operator-theoretic methods of recent interest revealing phase as the generator of a family of eigenfunctions of the Koopman operator with purely imaginary eigenvalues [63][64][65]. These eigenfunctions and others can be estimated using Fourier/Laplace averages [63][64][65][66][67] when dynamical equations are known, and from data using Dynamic Mode Decomposition [68], at least in certain cases. However, the results of these emerging spectral methods seem to be sensitive to the choice of observables and their nonlinearities [69].…”

Asymptotic Phase and Amplitude for Classical and Semiclassical Stochastic Oscillators via Koopman Operator Theory

“…Recently, the asymptotic phase and isochrons (level sets of the asymptotic phase), classical notions in the theory of nonlinear oscillations since Winfree [8] and Guckenheimer [9], have been studied from a viewpoint of the Koopman operator theory by Mauroy, Mezić, and Moehlis [10], and their relationship with the Koopman eigenfunction associated with the fundamental frequency of the oscillator has been clarified [10][11][12][13][14]. Moreover, they have shown that the (asymptotic) amplitude and isostables, which characterize deviation of the system state from the limit cycle and extend the Floquet coordinates [13,15,16] to the nonlinear regime, can be introduced naturally in terms of the Koopman eigenfunctions associated with the Floquet exponents with non-zero real parts [10][11][12][13][14]17]. By using the asymptotic phase and amplitude functions, we can obtain a reduced description of limitcycle oscillators, which is useful for the analysis and control of synchronization dynamics of limit-cycle oscillators [18][19][20][21][22][23][24].…”