Small noise and long time phase diffusion in stochastic limit cycle oscillators

Giacomin, Giambattista; Poquet, Christophe; Shapira, Assaf

doi:10.1016/j.jde.2017.09.029

Cited by 21 publications

(33 citation statements)

References 29 publications

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“…The following theorem gives the scaling of the first hitting time for leaving the manifold of rotating solutions: it parallels analogous bounds for stochastic oscillators [24,10]. Recall that v t = u t − T βt u * .…”

Section: Stability Results -Variational Phasementioning

confidence: 86%

Stochastic Rotating Waves

Kuehn¹,

MacLaurin²,

Zucal³

2021

Preprint

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Stochastic dynamics has emerged as one of the key themes ranging from models in applications to theoretical foundations in mathematics. One class of stochastic dynamics problems that has received considerable attention recently are travelling wave patterns occurring in stochastic partial differential equations (SPDEs), i.e., how deterministic travelling waves behave under stochastic perturbations. In this paper, we start the mathematical study of related class of problems: stochastic rotating waves generated by SPDEs. We combine deterministic dynamics PDE techniques with methods from stochastic analysis. We establish two different approaches, the variational phase and the approximated variational phase, for defining stochastic phase variables along the rotating wave, which track the effect of noise on neutral spectral modes associated to the special Euclidean symmetry group of rotating waves. Furthermore, we prove transverse stability results for rotating waves showing that over certain time scales and for small noise, the stochastic rotating wave stays close to its deterministic counterpart.

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Section: Stability Results -Variational Phasementioning

confidence: 86%

Stochastic Rotating Waves

Kuehn¹,

MacLaurin²,

Zucal³

2021

Preprint

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“…Theorem 1 (Theorem A in [27], Theorem 2.1 in [26]). Consider the flow : R m ×R → R m for the ODE (2.1) with hyperbolic stable limit cycle γ = {γ (t)} t∈[0,τ γ ] .…”

Section: The Deterministic Casementioning

confidence: 99%

“…Examples for stochastic oscillators/oscillations can be found in a wide variety of applications such as neuroscience [4,12,31,43], ecology [37,39], bio-mechanics [25,35], geoscience [6,33], among many others. In addition, stochastic oscillations have become a recently very active research topic in the rigorous theory of stochastic dynamical systems with small noise [3,7,8,26].…”

Section: Introductionmentioning

confidence: 99%

A Random Dynamical Systems Perspective on Isochronicity for Stochastic Oscillations

Engel

Kuehn

2021

Commun. Math. Phys.

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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.

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“…The second issue is that there are no O( √ ǫ) corrections to the deterministic part of the phase equation so that one has to go to O(ǫ) in order to determine the leading order corrections to the drift term. There are two sources of O(ǫ) terms: one arises from the coupling between the phase and amplitude fluctuations transverse to the limit cycle, and the second arises from changing between Stratonovich and Ito versions of the SDE based on Ito's formula 14,[46][47][48] . The precise form of these terms will also depend on the particular choice of phase reduction method.…”

Section: A Phase Reductionmentioning

confidence: 99%

Synchronization of stochastic hybrid oscillators driven by a common switching environment

Bressloff

MacLaurin

2018

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Many systems in biology, physics and chemistry can be modeled through ordinary differential equations, which are piecewise smooth, but switch between different states according to a Markov jump process. In the fast switching limit, the dynamics converges to a deterministic ODE. In this paper we suppose that this limit ODE supports a stable limit cycle. We demonstrate that a set of such oscillators can synchronize when they are uncoupled, but they share the same switching Markov jump process. The latter is taken to represent the effect of a common randomly switching environment. We determine the leading order of the Lyapunov coefficient governing the rate of decay of the phase difference in the fast switching limit. The analysis bears some similarities to the classical analysis of synchronization of stochastic oscillators subject to common white noise. However the discrete nature of the Markov jump process raises some difficulties: in fact we find that the Lyapunov coefficient from the quasi-steady-state approximation differs from the Lyapunov coefficient one obtains from a second order perturbation expansion in the waiting time between jumps. Finally, we demonstrate synchronization numerically in the radial isochron clock model and show that the latter Lyapinov exponent is more accurate.

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Small noise and long time phase diffusion in stochastic limit cycle oscillators

Cited by 21 publications

References 29 publications

Stochastic Rotating Waves

Stochastic Rotating Waves

A Random Dynamical Systems Perspective on Isochronicity for Stochastic Oscillations

Synchronization of stochastic hybrid oscillators driven by a common switching environment

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