Degenerate bifurcations near a double eigenvalue in the Brusselator

The steady state bifurcations near a double zero eigenvalue of the reaction diffusion equation associated with a tri-molecular chemical reaction (the Brusselator) are analysed. Special emphasis is put on three degeneracies where previous results of Schaeffer and Golubitsky do not apply. For these degeneracies it is shown by means of a LiapunovSchmidt reduction that the steady state bifurcations are determined by codimension-three normal forms. They are of types (9)31, (8)221 and (6a)ρ,κ in a recent classification of Z(2)-equivariant imperfect bifurcations with corank two. Each normal form couples an ordinary corank-1 bifurcation in the sense of Golubitsky and Schaeffer to a degenerate Z(2)-equivariant corank-1 bifurcation of Golubitsky and Langford in a specific way.

Imperfect Bifurcations and Spatial Structures in Dissipative Systems

One-dimensional reaction-diffusion equations associated with the trimolecular model of Prigogine and Lefever ("Brusselator") are analyzed. A physical description of possibilities of keeping con-centrations of initial components constant is discussed. It is shown that the problem considering diffusion of initial components gives rise to an imperfect bifurcation problem. The diffusion equa-tions have been solved numerically by a continuation procedure for the fixed and zero flux boundary conditions. The analysis indicates that the models including diffusion of all reacting components do not admit an occurence of trivial solutions. These models, as a result, also exclude the pos-sibility of primary bifurcations. The models which consider diffusion of the initial components suppress the number of possible solutions of governing equations. These models may also predict both symmetric and asymmetric states. Apparently this type of models is more suitable for predic-tion of patterns of spatial organization in growth. Since the number of possible profiles is strongly reduced this principle may lead to a more deterministic way of an evolution process. Symmetric profiles occuring on an isola cannot be reached by an evolution process unless a large perturbation is imposed on the system.