Abstract
In this paper, we mainly study the stochastic stability and stochastic bifurcation of Brusselator system with multiplicative white noise. Firstly, by a polar coordinate transformation and a stochastic averaging method, the original system is transformed into an Itô averaging diffusion system. Secondly, we apply the largest Lyapunov exponent and the singular boundary theory to analyze the stochastic local and global stability. Thirdly, by means of the properties of invariant measures, the stochastic dynamical bifurcations of stochastic averaging Itô diffusion equation associated with the original system is considered. And we investigate the phenomenological bifurcation by analyzing the associated Fokker–Planck equation. We will show that, from the view point of random dynamical systems, the noise “destroys” the deterministic stability. Finally, an example is given to illustrate the effectiveness of our analyzing procedure.
Highlights
The determined Brusselator is a coupled differential equation written as ⎧⎨u = A – (B + 1)u + u2v, ⎩v = Bu – u2v, (1.1)where u(t) and v(t) are the concentrations of reactants at time t, respectively
As a theoretical model for a type of autocatalytic reaction, the Brusselator model has attracted the attention of many scholars since it was proposed by Prigogine and Lefever [1]
It is generally recognized that the effects of “external fluctuations” are inevitable in dynamical systems due to various factors, such as possible changes of system parameters, variations in excitations, errors in modeling schemes
Summary
Where u(t) and v(t) are the concentrations of reactants at time t, respectively. A > 0 and B > 0 are external system parameters describing the (constant) supply of “reservoir” chemicals. In order to study the effect of environmental fluctuation, the deterministic model system (1.1) can be extended to a stochastic differential equation system as follows:. By applying polar coordinate transformation and stochastic averaging method, we obtain a stochastic averaging Itô diffusion equation. 4, the stochastic dynamical and phenomenological bifurcations of stochastic averaging Itô diffusion equation associated with system (1.2) will be discussed by means of the properties of invariant measures and by considering the Fokker–Planck equation, the phenomenological bifurcation of system (1.2) is shown. We will apply polar coordinate transformation and stochastic averaging method to obtain an averaging Itô diffusion equation from system (1.2). It is efficient to consider the averaging amplitude equation of system (2.5) to obtain the critical point of stochastic stability and bifurcation phenomena of system (1.2).
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