Noise-induced switching and extinction in systems with delay

Ira B. Schwartz, Lora Billings, Thomas W. Carr, M. I. Dykman

Research output: Contribution to journalArticle

8 Citations (Scopus)

Abstract

We consider the rates of noise-induced switching between the stable states of dissipative dynamical systems with delay and also the rates of noise-induced extinction, where such systems model population dynamics. We study a class of systems where the evolution depends on the dynamical variables at a preceding time with a fixed time delay, which we call hard delay. For weak noise, the rates of interattractor switching and extinction are exponentially small. Finding these rates to logarithmic accuracy is reduced to variational problems. The solutions of the variational problems give the most probable paths followed in switching or extinction. We show that the equations for the most probable paths are acausal and formulate the appropriate boundary conditions. Explicit results are obtained for small delay compared to the relaxation rate. We also develop a direct variational method to find the rates. We find that the analytical results agree well with the numerical simulations for both switching and extinction rates.

Original languageEnglish
Article number012139
JournalPhysical Review E - Statistical, Nonlinear, and Soft Matter Physics
Volume91
Issue number1
DOIs
StatePublished - 26 Jan 2015

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Extinction
extinction
Probable
Variational Problem
Dissipative Dynamical System
Path
Population Dynamics
Direct Method
Variational Methods
Time Delay
Logarithmic
Boundary conditions
Numerical Simulation
dynamical systems
time lag
boundary conditions
Model
simulation

Cite this

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abstract = "We consider the rates of noise-induced switching between the stable states of dissipative dynamical systems with delay and also the rates of noise-induced extinction, where such systems model population dynamics. We study a class of systems where the evolution depends on the dynamical variables at a preceding time with a fixed time delay, which we call hard delay. For weak noise, the rates of interattractor switching and extinction are exponentially small. Finding these rates to logarithmic accuracy is reduced to variational problems. The solutions of the variational problems give the most probable paths followed in switching or extinction. We show that the equations for the most probable paths are acausal and formulate the appropriate boundary conditions. Explicit results are obtained for small delay compared to the relaxation rate. We also develop a direct variational method to find the rates. We find that the analytical results agree well with the numerical simulations for both switching and extinction rates.",
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Noise-induced switching and extinction in systems with delay. / Schwartz, Ira B.; Billings, Lora; Carr, Thomas W.; Dykman, M. I.

In: Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, Vol. 91, No. 1, 012139, 26.01.2015.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Noise-induced switching and extinction in systems with delay

AU - Schwartz, Ira B.

AU - Billings, Lora

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AU - Dykman, M. I.

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AB - We consider the rates of noise-induced switching between the stable states of dissipative dynamical systems with delay and also the rates of noise-induced extinction, where such systems model population dynamics. We study a class of systems where the evolution depends on the dynamical variables at a preceding time with a fixed time delay, which we call hard delay. For weak noise, the rates of interattractor switching and extinction are exponentially small. Finding these rates to logarithmic accuracy is reduced to variational problems. The solutions of the variational problems give the most probable paths followed in switching or extinction. We show that the equations for the most probable paths are acausal and formulate the appropriate boundary conditions. Explicit results are obtained for small delay compared to the relaxation rate. We also develop a direct variational method to find the rates. We find that the analytical results agree well with the numerical simulations for both switching and extinction rates.

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