Correlation times in stochastic equations with delayed feedback and multiplicative noise

Mathieu Gaudreault, Juliana Militão Berbert, Jorge Viñals

Research output: Contribution to journalArticle

13 Citations (Scopus)

Abstract

We obtain the characteristic correlation time associated with a model stochastic differential equation that includes the normal form of a pitchfork bifurcation and delayed feedback. In particular, the validity of the common assumption of statistical independence between the state at time t and that at t-τ, where τ is the delay time, is examined. We find that the correlation time diverges at the model's bifurcation line, thus signaling a sharp bifurcation threshold, and the failure of statistical independence near threshold. We determine the correlation time both by numerical integration of the governing equation, and analytically in the limit of small τ. The correlation time T diverges as T~a⊃-1, where a is the control parameter so that a=0 is the bifurcation threshold. The small-τ expansion correctly predicts the location of the bifurcation threshold, but there are systematic deviations in the magnitude of the correlation time.

Original languageEnglish (US)
Article number011903
JournalPhysical Review E - Statistical, Nonlinear, and Soft Matter Physics
Volume83
Issue number1
DOIs
StatePublished - Jan 11 2011

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Delayed Feedback
Multiplicative Noise
Stochastic Equations
Bifurcation
Statistical Independence
thresholds
Diverge
Pitchfork Bifurcation
Delay Time
numerical integration
Control Parameter
Numerical integration
Normal Form
Governing equation
differential equations
time lag
Deviation
Differential equation
deviation
Predict

Cite this

Correlation times in stochastic equations with delayed feedback and multiplicative noise. / Gaudreault, Mathieu; Berbert, Juliana Militão; Viñals, Jorge.

In: Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, Vol. 83, No. 1, 011903, 11.01.2011.

Research output: Contribution to journalArticle

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