Extensive research in nonlinear system theory in recent years has shown that a number of system theoretic questions are closely related to the behaviour of the (controllability) orbits of the system. Thus orbit minimality, namely the property of the system having its state space identical to one orbit, arises naturally when dealing with controllability questions. This paper is concerned with the more general case where orbit minimality is not available, and attempts to explore various ways the orbits are patched together on the state space. Concepts like stability, attraction and asymptotic stability are defined and studied with the aid of certain sets naturally associated with the system like the limit set and the prolongational set. Since the related questions are topological in nature, the problems are set up using only the topological dynamics of the system.