The Oregonator with diffusion

we shall write equations for a reaction with diffusion assuming that at the front of a chemical wave z = 0 (z = [Ce4+]) which implies that the cerium ion is in the Ce3+ state and the first two reactions of the mechanism 

Note that if we neglect the y variable in (6.142), y = 0, then the Fisher-Kolmogorov equation (6.123b), in which a = 1, D = 1, is obtained for the x variable. 

We shall now proceed to the examination of the splution of a travelling wave type. Accordingly, we will substitute in (6.142) for 

Inspection of the autonomous system (6.145) will being with finding the stationary points, z1 = 0.z4 = 0. The system (6.145) has the following stationary points having the non-negative coordinates zx, z3  

We will linearize the system nearby both the stationary points 

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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. 

This model also appears in a description of the Belousov-Zhabotinskii reaction as a special case of the Oregonator with diffusion, see Section 6.3.2.3. 

The roots Ali2(1,2) in this model are identical with the eigenvalues, for a = D = 1, in the Fisher-Kolmogorov model. As the second pair of the roots does not lead to a generation of the sensitive states, the nature of a catastrophe for the Oregonator with diffusion is the same as for the Fisher-Kolmogorov model. In other words, the waves with the velocities v centre manifold theorem the sensitive state is associated only with the zlt z2 variables. 

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