  This paper is concerned with the dynamics of an infinite-dimensional gradient
system under small almost periodic perturbations. Under the assumption that the
original autonomous system has a global attractor given as the union of
unstable manifolds of a finite number of hyperbolic equilibrium solutions, we
prove that the perturbed non-autonomous system has exactly the same number of
almost periodic solutions. As a consequence, the pullback attractor of the
perturbed system is given by the union of unstable manifolds of these finitely
many almost periodic solutions. An application of the result to the
Chafee-Infante equation is discussed.
