We study dynamical systems for which at most $n$ orbits can accompanya given arbitrary orbit.For simplicity we call them $n$-expansive(or positively $n$-expansive if positive orbits are considered instead).We prove that these systems can satisfy properties of expansive systems or not.For instance, unlike positively expansive maps [3],positively $n$-expansive homeomorphisms may exist on certain infinite compact metric spaces.We also provethat a map (resp. bijective map) is positively $n$-expansive (resp. $n$-expansive) if and only if it isso outside finitely many points.Finally, we prove that a homeomorphism on a compact metric spaceis $n$-expansive if and only if it is so outside finitely many orbits.These last resuls extends previous ones for expansive systems [2],[11],[12].