NORMAL-FORM FOR HOPF-BIFURCATION OF PARTIAL-DIFFERENTIAL EQUATIONS ONTHE SQUARE

We derive and analyse a normal form governing dynamics of Hopf bifurca tions of partial differential evolution equations on a square domain. We assume that the differential operator for the linearized problem de composes into two one-dimensional self-adjoint operators and a local ' reaction' operator; this gives a basis of i.e. of the form u(x(1),x(2) ) = f(1)(x(1))f(2)(x(2)). The normal form reduces to that investigated by Swift [23] for bifurcation of modes with odd parity but is new for modes with even parity where the centre eigenspace carries a reducibl e action of D-4 x S-1. We consider the Brusselator equations as an exa mple and discover that a separable linearization introduces a degenera cy which causes the three` new third order terms in the normal form to be related in an unexpected but simple way.