A Random Dynamical Systems Perspective on Isochronicity for Stochastic Oscillations

Abstract

For an attracting periodic orbit (limit cycle) of a deterministic dynamical system, one defines the isochron for each point of the orbit as the cross-section with fixed return time under the flow. Equivalently, isochrons can be characterized as stable manifolds foliating neighborhoods of the limit cycle or as level sets of an isochron map. In recent years, there has been a lively discussion in the mathematical physics community on how to define isochrons for stochastic oscillations, i.e. limit cycles or heteroclinic cycles exposed to stochastic noise. The main discussion has concerned an approach finding stochastic isochrons as sections of equal expected return times versus the idea of considering eigenfunctions of the backward Kolmogorov operator. We discuss the problem in the framework of random dynamical systems and introduce a new rigorous definition of stochastic isochrons as random stable manifolds for random periodic solutions with noise-dependent period. This allows us to establish a random version of isochron maps whose level sets coincide with the random stable manifolds. Finally, we discuss links between the random dynamical systems interpretation and the equal expected return time approach via averaged quantities.

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

“…In Reference [35], Thomas and Lindner proposed a definition of the asymptotic phase for strongly stochastic oscillators without relying on the limit-cycle solution of the deterministic system, where they used the slowest decaying eigenfunction of the backward Fokker-Planck operator as the phase function based on the consideration of the mean first passage time. In this section, we show that their definition can be viewed as a natural extension of the deterministic definition in the sense that it is given by the argument of the Koopman eigenfunction associated with the fundamental frequency [38,39].…”

“…To cope with this problem, Schwabedal and Pikovsky [34] introduced a definition of the phase in terms of the mean first return time, and Thomas and Lindner [35] proposed a definition of the asymptotic phase in terms of the slowest decaying eigenfunction of the backward Fokker-Planck (Kolmogorov) operator describing the mean first passage time, both of which yield phase values that increase with a constant frequency on average for stochastic oscillations in a similar way to the ordinary asymptotic phase for deterministic oscillators. Recently, we pointed out that the definition of the stochastic asymptotic phase by Thomas and Lindner [35] can be seen as a natural extension of the deterministic definition from the viewpoint of the Koopman operator theory; namely, it is given by the argument of the Koopman eigenfunction associated with the fundamental frequency [38] (see also Reference [39]) and extended this idea to the definition of the asymptotic phase for quantum oscillatory systems.…”

Non-equilibrium thermodynamics and the free energy principle in biology

“…A different strategy is to rely on an analogous theorem called "Multiplicative Ergodic Theorem," which can be used in differentiable random dynamical systems to prove the existence of random dynamical attractors (Engel and Kuehn 2021). The problem here is that this notion of ergodicity implies that the base flow preserves the probability P, which means that the model of noise (environmental fluctuations) is fixed and the probability measure is invariant (Engel and Kuehn 2021). In other words, in ergodic random dynamical systems, there is a one-to-one correspondence between invariant random measures (noise) and stationary measures of the associated stochastic process (dynamics of the system).…”

The asymptotic frequency of stochastic oscillators

“…The authors of [7], [8], [13], [19], [21], [34], [35], [41], define phase maps by considering the unperturbed (deterministic) oscillator. Meanwhile, [11], [12], [17], [33], attempt to account for the non-deterministic behaviour of a stochastic oscillator when definining a phase map. As will be seen, the phase maps of [11], [12], [17], [33], do not strictly meet our definition of a phase map (see Remark 1.1).…”

“…Meanwhile, [11], [12], [17], [33], attempt to account for the non-deterministic behaviour of a stochastic oscillator when definining a phase map. As will be seen, the phase maps of [11], [12], [17], [33], do not strictly meet our definition of a phase map (see Remark 1.1).…”