Hopf bifurcation with additive noise

Trans. Amer. Math. Soc.

Lamb

Rasmussen

2019

Self Cite

We introduce the notion of Lyapunov exponents for random dynamical systems, conditioned to trajectories that stay within a bounded domain for asymptotically long times. This is motivated by the desire to characterize local dynamical properties in the presence of unbounded noise (when almost all trajectories are unbounded). We illustrate its use in the analysis of local bifurcations in this context.The theory of conditioned Lyapunov exponents of stochastic differential equations builds on the stochastic analysis of quasi-stationary distributions for killed processes and associated quasi-ergodic distributions. We show that conditioned Lyapunov exponents describe the local stability behaviour of trajectories that remain within a bounded domain and -in particular -that negative conditioned Lyapunov exponents imply local synchronisation. Furthermore, a conditioned dichotomy spectrum is introduced and its main characteristics are established. 1 We note that random dynamical systems with bounded noise suffer less from this problem, but such systems are generally not amenable to techniques from stochastic analysis and we will not consider these here. We refer to [2,13,14,19] for alternative approaches to bifurcations of random dynamical systems in the bounded noise context.

Section: )mentioning

confidence: 99%

Conditioned Lyapunov exponents for random dynamical systems

Trans. Amer. Math. Soc.

Lamb

Rasmussen

2019

Self Cite

“…A (deterministic) Hopf bifurcation occurs if, by the variation of a model parameter, an asymptotically stable equilibrium loses stability under the emission of a small attracting limit cycle. Numerical studies [29] suggest that the mechanism of shear-induced chaos is at play also in stochastic Hopf bifurcations, but while analytical proofs of parameter regimes with negative top Lyapunov exponents are within reach [10,11], until now, there are no rigorous results concerning the existence of parameter regimes with positive top Lyapunov exponents in this context. The results of this paper may well be relevant to shed more light on this problem.…”

Section: Introductionmentioning

confidence: 99%

Bifurcation Analysis of a Stochastically Driven Limit Cycle

Lamb

Rasmussen

2019

Commun. Math. Phys.

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We establish the existence of a bifurcation from an attractive random equilibrium to shear-induced chaos for a stochastically driven limit cycle, indicated by a change of sign of the first Lyapunov exponent.This addresses an open problem posed by Kevin Lin and Lai-Sang Young in [21,30], extending results by Qiudong Wang and Lai-Sang Young [28] on periodically kicked limit cycles to the stochastic context.

“…It has been shown in [20] that for b small enough the first Laypunov exponent λ 1 < 0 is negative such that the corresponding random attractor A is indeed a singleton. For b large, one can see numerically that the attractor becomes chaotic.…”

Section: Chaotic Random Attractors and Singletonsmentioning

confidence: 99%

A Random Dynamical Systems Perspective on Isochronicity for Stochastic Oscillations

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.