the omega-limit set
/ OH-meg-uh limit set /
Watch a single trajectory for a very long time and ignore the messy startup. Where does it eventually live? Maybe it homes in on one point. Maybe it laps around a loop forever. Maybe it wanders forever over an intricate set without ever settling. The omega-limit set of a trajectory is the precise answer to 'where does this particular path end up in the long run?' — the collection of all the places it keeps returning arbitrarily close to as time runs on.
Formally, the omega-limit set of a point x, written omega(x), is the set of all points q such that the trajectory through x comes within any chosen distance of q at arbitrarily large times — there is a sequence of times t_n going to infinity with phi_(t_n)(x) approaching q. (The matching alpha-limit set uses times going to minus infinity, capturing where the trajectory came from in the distant past.) Concretely: if the trajectory converges to an equilibrium p, then omega(x) = {p}, a single point. If it spirals onto a closed loop, omega(x) is that entire loop. If it fills out a strange attractor, omega(x) is the strange attractor.
The omega-limit set is the rigorous tool behind the loose phrase 'long-term behaviour'. It is always closed and invariant — whatever a trajectory limits onto is itself something the flow preserves — which is why limit sets are forced to be the structured objects (equilibria, cycles, attractors) we hunt for. The Poincare-Bendixson theorem is essentially a theorem about omega-limit sets in the plane: it says a bounded planar omega-limit set with no equilibria must be a closed orbit, sharply limiting how complicated planar long-term behaviour can be.
For the Van der Pol oscillator, take almost any starting point other than the unstable origin. The trajectory spirals outward (or inward) and approaches the single stable limit cycle. Its omega-limit set is exactly that closed loop — not one point on the loop, but the whole loop, since the trajectory keeps cycling through every point of it forever.
When a trajectory spirals onto a cycle, its omega-limit set is the entire cycle, the set of points it forever revisits.
An omega-limit set is a property of a trajectory, not necessarily an attractor: a trajectory sitting exactly on a repeller has that repeller as its omega-limit set even though nothing is attracted there. The two notions agree only for the well-behaved attracting cases.