Suppose we have a smooth dynamical system on $R^n$ (defined by a system of ODEs). Assume that:
(1) The system has an absorbing ball, that is every trajectory eventually enters this ball and stays in it.
(2) The system has a unique stationary point, and this stationary point is locally asymptotically stable.
(2) The system has no period orbits.
Can we conclude that the stationary point is in fact globally stable?