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

under    Dirichlet  boundary condition  on   $\partial  \Omega$.  Here
$\Omega$ is a two-dimensional open disk. We show that for any positive
integer $k$ there is a $C^k$ function $f$ such that the above equation
has a  bounded solution  whose  $\omega$-limit set is  homeomorphic to
$S^1$.  This shows that  nonconvergent bounded solutions can occur, in
contrast to one-dimensional equations, or higher-dimensional equations
with  analytic nonlinearities,  where  bounded   solutions are  always
convergent.