### 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 language | English (US) |
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Article number | 011903 |

Journal | Physical Review E - Statistical, Nonlinear, and Soft Matter Physics |

Volume | 83 |

Issue number | 1 |

DOIs | |

State | Published - Jan 11 2011 |

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### Cite this

*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*,

*83*(1), [011903]. https://doi.org/10.1103/PhysRevE.83.011903

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

Research output: Contribution to journal › Article

*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*, vol. 83, no. 1, 011903. https://doi.org/10.1103/PhysRevE.83.011903

}

TY - JOUR

T1 - Correlation times in stochastic equations with delayed feedback and multiplicative noise

AU - Gaudreault, Mathieu

AU - Berbert, Juliana Militão

AU - Viñals, Jorge

PY - 2011/1/11

Y1 - 2011/1/11

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=78751515887&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78751515887&partnerID=8YFLogxK

U2 - 10.1103/PhysRevE.83.011903

DO - 10.1103/PhysRevE.83.011903

M3 - Article

AN - SCOPUS:78751515887

VL - 83

JO - Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

JF - Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

SN - 1539-3755

IS - 1

M1 - 011903

ER -