Random Perturbations of Dynamical Systems

Freidlin, Mark; Wentzell, Alexander D.

doi:10.1007/978-3-642-25847-3

Cited by 587 publications

(680 citation statements)

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“…Its origins can be traced back at least to the works of Eyring [10] and Kramers [13], who studied it in the context of chemical reaction rates. In the context of perturbed dynamical systems, Freidlin and Wentzell [11] developed a systematic approach based on large deviation theory. This approach was extended by Berglund and Gentz [3] to cover stochastic bifurcation and stochastic resonance, and by Olivieri and Scoppola [20,21] to study dynamics of Markov chains with exponentially small transition probabilities.…”

Section: Introductionmentioning

confidence: 99%

Multi-scale metastable dynamics and the asymptotic stationary distribution of perturbed Markov chains

Betz

Roux

2016

Stochastic Processes and their Applications

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Abstract. We consider a simple but important class of metastable discrete time Markov chains, which we call perturbed Markov chains. Basically, we assume that the transition matrices depend on a parameter ε, and converge as ε → 0. We further assume that the chain is irreducible for ε > 0 but may have several essential communicating classes when ε = 0. This leads to metastable behavior, possibly on multiple time scales. For each of the relevant time scales, we derive two effective chains. The first one describes the (possibly irreversible) metastable dynamics, while the second one is reversible and describes metastable escape probabilities. Closed probabilistic expressions are given for the asymptotic transition probabilities of these chains, but we also show how to compute them in a fast and numerically stable way. As a consequence, we obtain efficient algorithms for computing the committor function and the limiting stationary distribution.

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Section: Introductionmentioning

confidence: 99%

Multi-scale metastable dynamics and the asymptotic stationary distribution of perturbed Markov chains

Betz

Roux

2016

Stochastic Processes and their Applications

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“…From the theory of large deviations [17,18], the probability of the uncontrolled system, i.e. u ≡ 0, transitioning from the attractor at x 0 through x 1 to the adjoining gyre is given by…”

Section: A Mean Escape Times For the Uncontrolled Casementioning

confidence: 99%

“…with boundary conditions x(t 0 ) = x 0 , and x(t f ) = x 1 [19,17,18,20]. 1 The average time to escape the gyre through x 1 is inversely proportional to the probability of occurrence of this most likely transition path and its associated noise profile.…”

Section: A Mean Escape Times For the Uncontrolled Casementioning

confidence: 99%

Exploiting Stochasticity for the Control of Transitions in Gyre Flows

Kularatne

Forgoston

Hsieh

2018

Robotics: Science and Systems XIV

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Abstract-We present a control strategy to control the intergyre switching time of an agent operating in a gyre flow. The proposed control strategy exploits the stochasticity of the underlying environment to affect inter-gyre transitions. We show how control can be used to enhance or abate the mean escape time and present a strategy to achieve a desired mean escape time. We show that the proposed control strategy can achieve any desired escape time in an interval governed by the maximum available control. We demonstrate the effectiveness of the strategy in simulations.

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“…Even if Π E and Π F are sparse matrices, the complexity of a numerical computation of the invariant distribution seems prohibitive unless N is small, because the size of the state space is of order N 3 (King and Masel 2007). However, we are mostly interested in the behavior of the system when becomes small, so that we can take advantage of the separation of time scales; this is the essence of the perturbation approach of Freidlin and Wentzell (1998).…”

Section: The Modelmentioning

confidence: 99%

“…Our analysis rests on stochastic techniques from evolutionary game theory (Foster and Young 1990;Fudenberg and Harris 1992;Kandori et al 1993;Young 1993;, which in turn are inspired by the general study of random perturbations of dynamical systems (Freidlin and Wentzell 1998). These techniques provide more precise predictions in settings where the unperturbed system has multiple local attractors, so that its long-run behavior depends on the initial conditions.…”

Section: Introductionmentioning

confidence: 99%

Phenotype Switching and Mutations in Random Environments

Fudenberg

Imhof

2011

Bull Math Biol

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Cell populations can benefit from changing phenotype when the environment changes. One mechanism for generating these changes is stochastic phenotype switching, whereby cells switch stochastically from one phenotype to another according to genetically determined rates, irrespective of the current environment, with the matching of phenotype to environment then determined by selective pressure. This mechanism has been observed in numerous contexts, but identifying the precise connection between switching rates and environmental changes remains an open problem. Here, we introduce a simple model to study the evolution of phenotype switching in a finite population subject to random environmental shocks. We compare the successes of competing genotypes with different switching rates, and analyze how the optimal switching rates depend on the frequency of environmental changes. If environmental changes are as rare as mutations, then the optimal switching rates mimic the rates of environmental changes. If the environment changes more frequently, then the optimal genotype either maximally favors fitness in the more common environment or has the maximal switching rate to each phenotype. Our results also explain why the optimum is relatively insensitive to fitness in each environment.

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Random Perturbations of Dynamical Systems

Cited by 587 publications

References 0 publications

Multi-scale metastable dynamics and the asymptotic stationary distribution of perturbed Markov chains

Multi-scale metastable dynamics and the asymptotic stationary distribution of perturbed Markov chains

Exploiting Stochasticity for the Control of Transitions in Gyre Flows

Phenotype Switching and Mutations in Random Environments

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