Delayed stochastic systems

Abstract

Noise and time delay are two elements that are associated with many natural systems, and often they are sources of complex behaviors. Understanding of this complexity is yet to be explored, particularly when both elements are present. As a step to gain insight into such complexity for a system with both noise and delay, we investigate such delayed stochastic systems both in dynamical and probabilistic perspectives. A Langevin equation with delay and a random-walk model whose transition probability depends on a fixed time-interval past (delayed random walk model) are the subjects of in depth focus. As well as considering relations between these two types of models, we derive an approximate Fokker-Planck equation for delayed stochastic systems and compare its solution with numerical results.

Balancing with positive feedback: the case for discontinuous control

“…Also, f(x)Cg(x)Z1. Thus, the repulsive delayed random walk that corresponds to (4.1) is given by (4.2) together with these choices of f, g. The equation for the non-stationary time dependence of the correlation function C(D, t)ZhX(t)X(tKD)i, and hence the change in the variance C(0, t), can be derived from the equation for the joint probability distribution (Ohira & Yamane 2000;Ohira & Milton in press) Pðn; t C 1; l; t C 1KDÞ Z X m gðmÞPðn K1; t; l; t C 1KD; m; tKtÞ C X m f ðmÞPðn C 1; t; l; t C 1KD; m; tK tÞ; ð4:4Þ…”

“…Since (4.1) describes an unstable time-delayed dynamical system, it is difficult to obtain expressions for the transient variance and correlation using standard methods. A useful trick for the analysis is to recast (4.1) in terms of a delayed random walk (Ohira & Milton 1995, in press;Ohira & Yamane 2000): the walker takes a discrete step of unit length per unit time in a direction determined by the conditional probabilities that depend on the position of the walker at some time, t, in the past. In particular, when the delayed random walk evolves in a quadratic potential, the Fokker-Planck equation (Milton et al 2008;Ohira & Milton in press) is identical to that obtained starting from (4.1) (Frank 2005 and where g and D are constants and P(x, t; y, tKt) is the joint probability that the walker is at position x at time t and position y at time tKt.…”

Correlation times in stochastic equations with delayed feedback and multiplicative noise

“…The correlation function of the linear counterpart of Eq. (1) but with additive noise is known [10,11]. The correlation function and correlation time of Eq.…”

Predictive dynamical and stochastic systems

“…Finally, in the same way that we have considered random walks with a delay (delayed random walks) [19,20], random walks with a prediction (predictive random walk) can also be considered. Even with a small delay, delayed random walks give rather complex analytical expressions for statistical quantities, such as variance.…”