Abstract:In this paper the $\omega $-limit behaviour of trajectories of solutions of ordinary differential equations is studied by methods of an axiomatic theory of solution spaces. We prove, under very general assumptions, semi-invariance of $\omega $-limit sets and a Poincar\'{e}-Bendixon type theorem.
Keywords: $\omega $-limit sets, stationary points, the Poincar\'{e}-Bendixon theorem
AMS Subject Classification: Primary 34C05; Secondary 34C11