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.
Highlights
Spontaneous rhythmic oscillations and synchronization are widely observed in various fields of science and technology [1,2,3,4,5,6]
In this article, based on the Koopman operator theory for stochastic systems, we propose a definition of the asymptotic phase and amplitude for strongly stochastic oscillators
We proposed a definition of the asymptotic phase and amplitude functions for stochastic oscillatory systems by generalizing the definition for deterministic limit-cycle oscillators on the basis of the Koopman operator theory, motivated by the definition of the stochastic asymptotic phase introduced by Thomas and Lindner [35]
Summary
Spontaneous rhythmic oscillations and synchronization are widely observed in various fields of science and technology [1,2,3,4,5,6]. 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, Mezic, 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] 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]. The phase equation has been extensively used for the analysis of weakly coupled limit-cycle oscillators [1,2,3,4,5,6], and the amplitude equation has been used recently for the analysis and control of limit-cycle oscillators [18,19,20,21,24]
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