Stochastic dynamics and the noisy Brusselator behaviour

TL;DR

This paper investigates how stochastic noise influences the oscillatory behavior of the Brusselator model, deriving analytical expressions and analyzing bifurcations in noisy chemical reaction systems.

Contribution

It provides new analytical formulas for noise-averaged orbits and variances, and examines the modification of Hopf bifurcations under stochastic influences.

Findings

Noise alters the Hopf bifurcation in the Brusselator.

Derived analytical expressions for noise average orbits and variances.

Established a diffusion relationship in thermal noise scenarios.

Abstract

In non-linear dynamics there are several model systems to study oscillations. One iconic example is the "Brusselator", which describes the dynamics of the concentration of two chemical species in the non-equilibrium phase. In this work we study the Brusselator dynamics as a stochastic chemical reaction without diffusion analysing the corresponding stochastic differential equations with thermal or multiplicative noise. In both stochastic scenarios we investigate numerically how the Hopf bifurcation of the non-stochastic system is modified. Furthermore, we derive analytical expressions for the noise average orbits and variance of general stochastic dynamics, a general diffusion relationship in the thermal noise framework, and an asymptotic expression for the noise average quadratic deviations. Hence, besides the impact of these results on the noisy Brusselator's dynamics, our findings are also relevant for general stochastic systems.