$u_t  =   \Delta u + h(u,\nabla u),     t>0, x \in \Omega,     \quad  (1)$

on  a smooth bounded  domain $\Omega \subset  R^N$, $N \ge  2$.
The nonlinearity  $h:  R  \times   R^N  \to  R$  is  assumed  continuously
differentiable and spatially homogeneous  (that is, independent of x).
Applying the method of realization  of vector fields, we show that (1)
can generate very complicated dynamics.  For example, choosing $h$
and $\Omega$ suitably, one achieves that (1) has an invariant manifold
$W$ the  flow on which  has a  transverse homoclinic  orbit to  a periodic
orbit.  Another choice of $h$  and $\Omega$ yields a transitive Anosov
flow on $W$.  The invariant manifold $W$ can have any dimension
$n \ge 3$ while  $N= \dim\Omega$ is  fixed (greater then 1).   In particular,
the solutions of (1) can  have $\omega$-limit sets of arbitrarily high
dimensions even though $\Omega$ has low dimension.

We  describe the  method of  realization of  vector fields  in detail,
first in an abstract context and  then for the above specific class of
equations.  A large  piece of our analysis is devoted  to the study of
perturbations of multiple eigenvalues and corresponding eigenfunctions
of the Laplacian under perturbation of the domain.