Turing patterns with O(3) symmetry

We perform an explicit center manifold reduction for the general steady-state bifurcation of a chemical reaction–diffusion system in an arbitrary compact domain. In particular we focus on those systems with two chemical species in domains with O(3) symmetry. We illustrate with the Brusselator on the spherical shell and produce explicit computations of the solution branches and their stabilities for the low l bifurcations. Because of the phenomenon of parameter collapse we can create universal bifurcation diagrams for even l. This work is an extension of that done in [Physica D 132 (1999) 339–362].