A Mean-Field Model for Extended Stochastic Systems with Distributed Time Delays

TL;DR

This paper extends a mean-field model to analyze the collective dynamics of noisy bistable elements with nonuniform time delays, revealing how delay distribution affects system stability and oscillatory behavior.

Contribution

The study generalizes existing models to include nonuniform delays, showing that delay variability influences stability and oscillations in stochastic systems.

Findings

Delay-independent stationary states exist for strong couplings.

Increasing delay distribution width stabilizes the trivial equilibrium.

Oscillatory states depend only on the mean delay for symmetric distributions.

Abstract

A network of noisy bistable elements with global time-delayed couplings is considered. A dichotomous mean field model has recently been developed describing the collective dynamics in such systems with uniform time delays near the bifurcation points. Here the theory is extended and applied to systems with nonuniform time delays. For strong enough couplings the systems exhibit delay-independent stationary states and delay-dependent oscillatory states. We find that the regions of oscillatory states in the parameter space are reduced with increasing width of the time delay distribution function; that is, nonuniformity of the time delays increases the stability of the trivial equilibrium. However, for symmetric distribution functions the properties of the oscillatory states depend only on the mean time delay.