We consider scalar parabolic PDEs $u_t = \Delta u + f(x,u,\nabla u)$
on  a  bounded,  at  least  two-dimensional  domain,  under  Dirichlet
boundary condition. We  are interested in ODEs that  are realizable in
PDEs of  this form.  We  say of  an ODE that  it is realizable  if its
dynamics is  equivalent to  the dynamics on  an invariant  manifold of
some PDE  in the  considered class.  The  main results state  that all
linear ODEs (in any dimension) are realizable, and any (nonlinear) ODE
has  an  arbitrarily small  realizable  perturbation.   We also  state
analogous results for periodically  forced equations of the form
$u_t = \Delta u + g(t,x,u)$.