Analytical determination of the bifurcation thresholds in stochastic differential equations with delayed feedback

Gaudreault, Mathieu; Drolet, François; Viñals, Jorge

doi:10.1103/physreve.82.051124

Cited by 13 publications

(21 citation statements)

References 20 publications

(26 reference statements)

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“…Such an effect has been observed previously in scalar non-delayed systems in the presence of multiplicative noise only [26,27,[45][46][47][48]. However, an additive noise-induced transition had been found numerically in a scalar DDE near a Hopf bifurcation [10,49].…”

Section: Insights About Stochastic Systems Stabilitysupporting

confidence: 61%

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Reduced dynamics for delayed systems with harmonic or stochastic forcing

Lefebvre

Hutt

LeBlanc

et al. 2012

Chaos

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The analysis of nonlinear delay-differential systems (DDE's) subjected to external forcing is difficult due to the infinite dimensionality of the space in which they evolve. To simplify the analysis of such systems, the present work develops a non-homogeneous center manifold (CM) reduction scheme, which allows the derivation of a time-dependent order parameter equation in finite dimensions. This differential equation captures the major dynamical features of the delayed system. The forcing is assumed to be small compared to the amplitude of the autonomous system, in order to cause only small variations of the fixed points and of the autonomous CM. The time-dependent CM is shown to satisfy a non-homogeneous partial differential equation. We first briefly review CM theory for DDE's. Then we show, for the general scalar case, how an ansatz that separates the CM into one for the autonomous problem plus an additional time-dependent order-two correction leads to satisfying results. The paper then details the application to a transcritical bifurcation subjected to single or multiple periodic forcings. The validity limits of the reduction scheme are also highlighted. Finally, we characterize the specific case of additive stochastic driving of the transcritical bifurcation, where additive white noise shifts the mode of the probability density function of the state variable to larger amplitudes. PACS numbers:Delayed feedback systems have a widespread use accross the engineering, physical and biological sciences. In their applications, these are often subjected to temporally varying forcing to implement the active role played by external influences on the dynamics. However, due to the complexity of infinite dimensional systems, the way additive temporal fluctuations interact with retarded dynamics is far from understood. In particular, a statement on the exact role played by non-homogeneous components on the stability of solutions in delayed feedback models is still missing. Here, the authors propose a novel approach to handle that problem using center manifold theory formulated for delay equations, where an explicit time-dependence is taken into account. Their method makes possible to capture the effect of forcing on the stability of an delayed feedback system in the vicinity of a bifurcation, and serves as new strategy to highlight novel non-linear phenomena.

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Section: Insights About Stochastic Systems Stabilitysupporting

confidence: 61%

“…similar to the single frequency case (48). By virtue of the linearity of the problem, the solution of (51) represents a linear superposition of the solutions (49) for different input strengths and frequencies.…”

Section: Driving At Multiple Frequenciesmentioning

confidence: 99%

Reduced dynamics for delayed systems with harmonic or stochastic forcing

Lefebvre

Hutt

LeBlanc

et al. 2012

Chaos

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“…An approximate expression has been given in [6] in the limit of small delay. We directly integrate Eq.…”

Section: Correlation Timementioning

confidence: 99%

“…It was shown that the location of the bifurcation threshold depends on the intensity of the parametric noise. Analytical approximations for the bifurcation lines have been given in [6] under the assumption that the delay time is small compared to other characteristic time scales of the system. The pitchfork bifurcation line separates exponentially decaying and growing solutions when bτ > −1.…”

Section: Introductionmentioning

confidence: 99%

Correlation times in stochastic equations with delayed feedback and multiplicative noise

2011

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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.

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“…Up to now, analytical researches have been concerned with the stability and performance in nonlinear stochastic DDSs, 19 the approximate solutions in nonlinear stochastic systems with small delays, [20][21][22][23] and the dynamical behaviors including resonance and bifurcations. [24][25][26][27][28] Besides, stochastic DDSs are widely used in other fields. For instance, Frank and Beek 29 were dedicated on exploring the biological systems, with the adoption of linear stochastic delay differential equations.…”

Section: Introductionmentioning

confidence: 99%

Delay-induced stochastic bifurcations in a bistable system under white noise

Sun

Xiao

et al. 2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

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In this paper, the effects of noise and time delay on stochastic bifurcations are investigated theoretically and numerically in a time-delayed Duffing-Van der Pol oscillator subjected to white noise. Due to the time delay, the random response is not Markovian. Thereby, approximate methods have been adopted to obtain the Fokker-Planck-Kolmogorov equation and the stationary probability density function for amplitude of the response. Based on the knowledge that stochastic bifurcation is characterized by the qualitative properties of the steady-state probability distribution, it is found that time delay and feedback intensity as well as noise intensity will induce the appearance of stochastic P-bifurcation. Besides, results demonstrated that the effects of the strength of the delayed displacement feedback on stochastic bifurcation are accompanied by the sensitive dependence on time delay. Furthermore, the results from numerical simulations best confirm the effectiveness of the theoretical analyses.

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Analytical determination of the bifurcation thresholds in stochastic differential equations with delayed feedback

Cited by 13 publications

References 20 publications

Reduced dynamics for delayed systems with harmonic or stochastic forcing

Reduced dynamics for delayed systems with harmonic or stochastic forcing

Correlation times in stochastic equations with delayed feedback and multiplicative noise

Delay-induced stochastic bifurcations in a bistable system under white noise

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