Subordination of the fast-relaxing degree of freedom in the center manifold of the Belousov-Zhabotinsky system

A deterministic-phenomenological model for the Belousov-Zhabotinsky system is coupled with Gaussian additive noise to analyze the role of fluctuations when a dissipative structure emerges. We define a parameter $\overline{k}$ which measures the degree to which the fast-relaxing degree of freedom is "enslaved" (in Haken's terminology) to the two order parameters. We derive the center-manifold equation which gives the fast degree of freedom as a function of the slow variables. All the recurrent and locally attractive behavior of the system lies in this surface. We then derive an equation relating the relaxation time of the fast-relaxing degree of freedom, the intensity of the Gaussian noise, and the parameter $\overline{k}$. The relaxation time is taken as the reciprocal of the damping constant which depends on the bifurcation parameters of the system. The maxima in the probability density factor for the order variables coincide with the macroscopically observed states.