Dulac Criteria for Autonomous Systems Having an Invariant Affine Manifold

For a class of higher dimensional autonomous systems that have an invariant affine manifold, conditions are derived to preclude the existence of periodic solutions on the invariant manifold. It is established that these conditions are robust under certain types of local perturbations of the vector field. As a consequence, each bounded semitrajectory on the invariant manifold is shown to converge to a single equilibrium using a C1 closing lemma for this class. Applications to autonomous systems that are homogeneous of degree 1 are also consid - ered.