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

Abstract

The asymptotic phase is a fundamental quantity for the analysis of deterministic limit-cycle oscillators, and generalized definitions of the asymptotic phase for stochastic oscillators have also been proposed. In this article, we show that the asymptotic phase and also amplitude can be defined for classical and semiclassical stochastic oscillators in a natural and unified manner by using the eigenfunctions of the Koopman operator of the system. We show that the proposed definition gives appropriate values of the phase and amplitude for strongly stochastic limit-cycle oscillators, excitable systems undergoing noise-induced oscillations, and also for quantum limit-cycle oscillators in the semiclassical regime.

Quantitative comparison of the mean–return-time phase and the stochastic asymptotic phase for noisy oscillators

“…For the MRT phase, its convergence to the deterministic phase in the case of vanishing noise has been investigated by Cao et al (2020) §2.4, who established convergence under additional regularity assumptions. For the stochastic asymptotic phase, its convergence to the deterministic phase for vanishing noise was anticipated by Thomas and Lindner (2014), see also the discussion in terms of the Koopman operator by Kato et al (2021) and derived in "Appendix D" in this manuscript using the same additional regularity assumptions as in Cao et al (2020). In line with these observations, although the intrinsic differences between both phases depend on each particular system, we have found the differences between the MRT phase and the stochastic asymptotic phase to grow as the noise is increased.…”

“…Thomas and Lindner (2014) defined a notion of stochastic asymptotic phase by means of the eigenfunctions of the Kolmogorov backwards operator. Since the Kolmogorov backwards operator and the stochastic Koopman operator are equivalent ( Črnjarić-Žic et al 2019), the setup in Thomas and Lindner (2014), which we next review, generalises the Koopman approach to obtain the phase of deterministic oscillators to stochastic systems (Mauroy and Mezić 2018;Kato et al 2021).…”

Estimating asymptotic phase and amplitude functions of limit-cycle oscillators from time series data

“…First, we verify the validity of the proposed method using the Stuart-Landau (SL) oscillator, for which the analytical solutions of the phase and amplitude functions are obtained [74]. The SL oscillator is described as follows:…”

A definition of the asymptotic phase for quantum nonlinear oscillators from the Koopman operator viewpoint

“…It is the basis for phase reduction [3][4][5][6][7][8], which gives low-dimensional phase equations approximately describing the oscillators under weak perturbations. Recently, it has been clarified that the asymptotic phase, which was originally introduced from a geometrical viewpoint [3], has a natural relationship with the Koopman eigenfunction associated with the fundamental frequency of the oscillator [2,[42][43][44][45].…”

“…The asymptotic phase plays a fundamental role in the analysis of classical limit-cycle oscillators, but a fully quantum-mechanical definition for quantum limit-cycle oscillators has been lacking. In this study, we propose a definition of the asymptotic phase for quantum nonlinear oscillators, which naturally extends the definition of the asymptotic phase for classical stochastic oscillatory systems [1] from the Koopman-operator viewpoint [2] and provides us with appropriate phase values for characterizing quantum synchronization.…”