Lagrange stability and asymptotic periods

In this article we describe selected topological and dynamical properties of asymptotically periodic motions in continuous dynamical systems (flows). The main result is to show Lagrange stability (i.e. the closure of positive orbit is compact) of such motion with the aid of topological properties of limit sets. Two sufficient conditions for this kind of stability are provided: the value of asymptotic period (our approach to this notion differs from the usual one) and proper metric space property.