Spontaneous linearization and periodic solutions in Hopf and symmetric bifurcations

We show that for a rather large class of symmetries, symmetric polynomial evolution equations — and therefore symmetric bifurcation equations — exhibit spontaneous linearization; i.e. although the equations are nonlinear, their asymptotic solutions are governed by linear equations. The same mechanism leads to periodicity of such solutions. This is exemplified in the cases of SO(2) and SU(2) symmetries, corresponding to standard Hopf and quaternionic bifurcations.