Regular polyhedra and bifurcations of symmetric equilibria of ordinary differential equations

All local 1-parameter bifurcations of symmetric equilibrium states corresponding to triple eigenvalue 0 are considered. In each case the corresponding “bifurcation group” the restriction of the full symmetry group of the differential equations to the centre manifold, is associated with symmetries of a regular (3-dimensional) polyhedron. It is shown that in all cases but one the bifurcation event is just a version of equilibrium branching. The proofs are based on the existence of functions (similar to Lyapunov functions) whose derivative by virtue of the equations has constant sign. These functions do not depend on the bifurcation parameter and are homogeneous of degree zero.