Hyperbolic almost periodic solutions and toroidal limit sets

We consider an almost periodic solution φ(t ) of an autonomous differential equationx ′ =f (x ). Letk denote the topological dimension of the hullH (φ). By considering the linearized equationx ′ =A (t )x alone [whereA (t ) is the Jacobian matrix off evaluated along φ(t )], one can derive a sufficient condition in order that φ(t ) be quasi-periodic and that the hullH (φ) be diffeomorphic to ak -dimensional torusTk . The proof is based upon an extension of the center manifold theorem to nonautonomous nonperiodic differential systems.