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

on a bounded domain in $R^N$, under Dirichlet boundary
condition. We prove that any ODE on $R^{N+1}$ has a realization
in the above PDE, with a suitably chosen $f$. More specifically,
the PDE has an invariant manifold of dimension $N+1$, the flow
on which is equivalent to the flow of the given ODE.

Compared with other realization results (e.g. those in
[Dancer and Polacik, Mem. Amer. Math. Soc.  140 (1999)],
or in [Polacik, J. Differential  Equations. 119 (1995), 24-53],
see the list), the present result shows a precise  realization
of any ODE, whereas the other papers show the realization of an
arbitrarily small perturbation of the given ODE. On the other
hand, the present realization result is limited to ODEs on
$R^{N+1}$ which is not the case in the other papers.

The purpose of the realization results is to demonstrate that
scalar parabolic equations in  space dimension two and higher
can have very complicated dynamics.