Noise-driven self-excited oscillators: Diffusion between limit cycles

Epele, LN; Fanchiotti, H.; Spina, Alejandro; Vucetich, H.

doi:10.1103/physreva.31.2631

Cited by 6 publications

(3 citation statements)

References 24 publications

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“…Limit cycles attractors perturbed by Gaussian noise are considered in the physics and other natural sciences literature since quite some time [10,15,16,22,23,27]. As an application of our main result we work out the example of the Van der Pol oscillator perturbed by multiplicative α-stable noise.…”

Section: Introductionmentioning

confidence: 92%

The Exit Problem from a Neighborhood of the Global Attractor for Dynamical Systems Perturbed by Heavy-Tailed Lévy Processes

Högele

Pavlyukevich

2013

Stochastic Analysis and Applications

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We consider a finite dimensional deterministic dynamical system with a global attractor A with a unique ergodic measure P concentrated on it, which is uniformly parametrized by the mean of the trajectories in a bounded set D containing A. We perturbe this dynamical system by a multiplicative heavy tailed Lévy noise of small intensity ε > 0 and solve the asymptotic first exit time and location problem from a bounded domain D around the attractor A in the limit of ε ց 0. In contrast to the case of Gaussian perturbations, the exit time has the asymptotically algebraic exit rate as a function of ε, just as in the case when A is a stable fixed point (see for instance [9,18,24]). In the small noise limit, we determine the joint law of the first time and the exit location from D c . As an example, we study the first exit problem from a neighbourhood of a stable limit cycle for the Van der Pol oscillator perturbed by multiplicative α-stable Lévy noise.

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Section: Introductionmentioning

confidence: 92%

The Exit Problem from a Neighborhood of the Global Attractor for Dynamical Systems Perturbed by Heavy-Tailed Lévy Processes

Högele

Pavlyukevich

2013

Stochastic Analysis and Applications

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“…In the physics and other natural sciences, Gaussian perturbations of dynamical systems with limit cycle attractors have been considered since quite some time, see e.g. Epele et al [13], Moran and Goldbeter [35], Hill et al [20], Kurrer and Schulten [32], Liu and Crawford [34], and Saet and Viviani [42]. As an application of our main result we present two examples in detail: the Duffing equation with two point attractors and a planar system from [35] with two stable limit cycles which lie in one another.…”

Section: The Unique Time Scale and Total Communication Of States In Tmentioning

confidence: 99%

“…For Q defined in (13) we construct the matrix Q 0 := Q 0 0 0 and denote by m 0 a continuous-time Markov chain on the state space {1, . .…”

Section: Proof Of Theorem 22mentioning

confidence: 99%

Metastability in a class of hyperbolic dynamical systems perturbed by heavy-tailed Lévy type noise

Högele

Pavlyukevich

2015

Stoch. Dyn.

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We consider a finite dimensional deterministic dynamical system with finitely many local attractors K ι , each of which supports a unique ergodic probability measure P ι , perturbed by a multiplicative non-Gaussian heavy-tailed Lévy noise of small intensity ε > 0. We show that the random system exhibits a metastable behavior: there exists a unique ε-dependent time scale on which the system reminds of a continuous time Markov chain on the set of the invariant measures P ι . In particular our approach covers the case of dynamical systems of Morse-Smale type, whose attractors consist of points and limit cycles, perturbed by multiplicative α-stable Lévy noise in the Itô, Stratonovich and Marcus sense. As examples we consider α-stable Lévy perturbations of the Duffing equation and Pareto perturbations of a biochemical birhythmic system with two nested limit cycles.

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Escape from a metastable state

1986

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Many important processes in science involve the escape of a particle over a barrier. In this review, we report, extend, and interpret various theories of noiseactivated escape. We discuss the connection between many-body transition state theory and Kramers' original diffusive Brownian motion approach (both in oneand multidimensional potential fields) and emphasize the physical situation inherent in Kramers' rate for weak friction. A rate theory accounting for memory friction is presented together with a set of criteria which test its validity. The complications and peculiarities of noise-activated escape in driven systems exhibiting multiple, locally stable stationary nonequilibrium states are identified and illustrated. At lower temperatures, quantum tunneling effects begin to play an increasingly important role. Early approaches and more recent developments of the quantum version of Kramers approach are discussed, thereby providing a description for dissipative escape at all temperatures.

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Noise-driven self-excited oscillators: Diffusion between limit cycles

Cited by 6 publications

References 24 publications

The Exit Problem from a Neighborhood of the Global Attractor for Dynamical Systems Perturbed by Heavy-Tailed Lévy Processes

The Exit Problem from a Neighborhood of the Global Attractor for Dynamical Systems Perturbed by Heavy-Tailed Lévy Processes

Metastability in a class of hyperbolic dynamical systems perturbed by heavy-tailed Lévy type noise

Escape from a metastable state

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