  We study a non-autonomous parabolic equation with almost-periodic, rapidly
oscillating principal part and nonlinear interactions. We associate to the
equation a skew-product semiflow and, for a special class of nonlinearities, we
define the Conley index of an isolated invariant set. As the frequency of the
oscillations tends to infinity, we prove that every isolated invariant set of
the averaged autonomous equation can be continued to an isolated invariant set
of the skew-product semiflow associated to the non-autonomous equation.
Finally, we illustrate some examples in which the Conley index can be
explicitely computed and can be exploited to detect the existence of recurrent
dynamics in the equation.
