The Brusselator with diffusion

We will write a system of equations with diffusion for the mechanism of reaction (6.38). According to principles presented in Section 4.4 the respective equations for a reaction with diffusion are as follows  

At present, we will examine the problem of the stability of the spatially homogeneous stationary solution, x = a, y = b/a, of the Brusselator in terms of the possibile loss of stability of this state and the generation of dissipative structures. Introducing into (6.154) the substitutions 

When n = 0 or Dx = Dy = 0 we obtain the characteristic equation of the problem without diffusion 

Consider a situation wherein the stationary state (a, b/a) of the equation without diffusion is stable, that is both the roots of equation (6.160) have negative real parts. It follows from the Routh-Hurwitz criterion (Appendix A5.8) that in this case the control parameters must satisfy the relationships 

The state (a, b/a) will lose stability if one of the eigenvalues becomes positive. Hence, the sensitive state corresponds to the vanishing of one of the eigenvalues. One of the numbers /i1 2 vanishes when the free term in equation (6.159) is equal to zero 

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The models of reactions with diffusion will be described below. The Brusselator with diffusion is a model system while the Oregonator with diffusion corresponds to the Belousov-Zhabotinskii reaction. First we shall describe the Fisher-Kolmogorov model with diffusion which, as will be shown later, may be regarded as a special case of the Oregonator with diffusion. 

Finally, we note that ci, C2- < as A oo. According to the discussion in Sect. 2.4, this means that the Brusselator with diffusion behaves like a nonlinear SchrOdinger equation slightly above the Hopf bifurcation point, provided A is sufficiently large. 

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