$ u_t = \Delta u + f(x,u,\nabla u) $

on a bounded domain in $R^N$, $N>1$ under Dirichlet boundary
condition. We say that a given  ODE is realizable in this class of
PDEs if for some $f$ the corresponding PDE has an invariant  manifold,
the flow on which is equivalent to the flow of the given ODE.

We prove that any ODE has an arbitrarily small perturbation that
is realizable. Also, any linear ODE is realizable (this is in fact
an inverse eigenvalue result for an elliptic operator).

These theorems guarantee that chaotic dynamics can be found in
such PDEs. Also, solutions of such equations  have their
$\omega$-limit sets of arbitrarily high dimensions (even
when  $\Omega$ has  dimension 2).