The simplest pattern forming system

The simple continuous thermostat model will be considered:

   u_t = u_xx   0 < x <L

  u_x(0,t) = 0,   u_x(L,t) = -alpha F(u(0,t))

where F(0)=0, F'(0)=1.

In joint work with William Ouligian, we show that the trivial state, u(x,t)=0 loses stability and can generate spatio-temporal wave-like patterns.  This instability arises because the boundary conditions of the associated linear problem are not self-adjoint. The generalized Fredholm alternative is used to compute the normal form for the Hopf bifurcation