Degenerate bifurcations near a double eigenvalue in the Brusselator

Abstract

AbstractThe 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.