Small delay approximation of stochastic delay differential equations

Guillouzic, Steve; L’Heureux, Ivan; Longtin, André

doi:10.1103/physreve.59.3970

Cited by 327 publications

(282 citation statements)

References 20 publications

Supporting

Mentioning

273

Contrasting

Order By: Relevance

“…The same expression has been obtained from a Taylor expansion of the conditional probability distribution function p(x τ ,t − τ |x,t) in [2], based on methods introduced in [20,21] under the Ito interpretation of stochastic differential equation. It can also be obtained from a Taylor expansion of the stochastic process defined byẋ (t,t − τ ) [22,23]. All methods assume that the time delay is small and lead to, under stationary conditions,…”

Section: Correlation Timementioning

confidence: 99%

Correlation times in stochastic equations with delayed feedback and multiplicative noise

2011

View full text Add to dashboard Cite

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.

show abstract

Section: Correlation Timementioning

confidence: 99%

Correlation times in stochastic equations with delayed feedback and multiplicative noise

2011

View full text Add to dashboard Cite

show abstract

“…[3]. With this approximation, we first transform the SDDEs to stochastic non-delayed DEs, and then to deterministic DEs with the use of DMA [21].…”

Section: Conclusion and Discussionmentioning

confidence: 99%

“…In order to get some insight on this issue, we have performed AMM calculations for our FN model with larger ensemble sizes of N = 100 and 1000, and obtained again a suppression of the oscillation by noises [34]. It is not clear for us how ZV took into account the non-Markovian property of SDDE within their FPE method [3] [5].…”

Section: B Effects Of Noise (β)mentioning

confidence: 99%

“…In recent years, linear SDDEs of Langevin equation are beginning to gain much attention [1]- [6]. The parameter range for the stationary solutions of the Langevin equation has been examined with the use of the step by step method [1], the moment mothod [2] and the Fokker-Planck equation (FPE) method [3] [4].…”

Section: Introductionmentioning

confidence: 99%

“…Since the time to simulate networks by conventional methods grows as N 2 with N, the size of the ensemble, it is rather difficult to simulate realistic neuron clusters. Although FPE is a powerful method in dealing with the stochastic DE, a simple FPE application to SDDE fails because of its non-Markovian property: an evaluation of the probability density at t requires prior knowledge of the conditional probability density between t and t − τ , τ being the delay time [3] [5].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Augmented moment method for stochastic ensembles with delayed couplings. II. FitzHugh-Nagumo model

Hasegawa

2004

Phys. Rev. E

View full text Add to dashboard Cite

Dynamics of FitzHugh-Nagumo (FN) neuron ensembles with time-delayed couplings subject to white noises, has been studied by using both direct simulations and a semi-analytical augmented moment method (AMM) which has been proposed in a recent paper [H. Hasegawa, E-print: cond-mat/0311021]. For N -unit FN neuron ensembles, AMM transforms original 2N -dimensional stochastic delay differential equations (SDDEs) to infinite-dimensional deterministic DEs for means and correlation functions of local and global variables. Infinite-order recursive DEs are terminated at the finite level m in the level-m AMM (AMMm), yielding 8(m + 1)-dimensional deterministic DEs. When a single spike is applied, the oscillation may be induced if parameters of coupling strength, delay, noise intensity and/or ensemble size are appropriate. Effects of these parameters on the emergence of the oscillation and on the synchronization in FN neuron ensembles have been studied. The synchronization shows the fluctuation-induced enhancement at the transition between nonoscillating and oscillating states. Results calculated by AMM5 are in fairly good agreement with those obtained by direct simulations.

show abstract

Stochastic ecological kinetics of regime shifts in a time‐delayed lake eutrophication ecosystem

et al. 2017

View full text Add to dashboard Cite

Abstract. Lake eutrophication ecosystems tend to exhibit bistable kinetics with two preferential configurations of oligotrophic and eutrophic states, and perturbations in the environmental parameters can cause regime shifts between two alternative stable states, yielding catastrophic regime shifts. In this article, we study stochastic kinetics of regime shifts in a time-delayed lake eutrophication ecological system, where the environmental parameters are assumed to be disturbed by both input and loss external noises. The dynamical behavior of stochastic delay on the regime shifts between oligotrophic and eutrophic states is analyzed using numerical simulations. Time delays in loss and recycling processes are considered. It is shown from time series and probability distribution that noises and delays can induce regime shifts between two stable states. Theoretical analyses are used to support our numerical simulations. By using the mean first passage time (MFPT) technique, we also discuss the escape time problem between two stable states in time-delayed ecosystems. A detailed study of the MFPT depending on various noise characteristics is performed, and the main results are compared for different cases of time delays.

show abstract

Small delay approximation of stochastic delay differential equations

Cited by 327 publications

References 20 publications

Correlation times in stochastic equations with delayed feedback and multiplicative noise

Correlation times in stochastic equations with delayed feedback and multiplicative noise

Augmented moment method for stochastic ensembles with delayed couplings. II. FitzHugh-Nagumo model

Stochastic ecological kinetics of regime shifts in a time‐delayed lake eutrophication ecosystem

Contact Info

Product

Resources

About