the Poincare-Bendixson theorem

In a flat, two-dimensional world, motion has remarkably few options for what it can do forever. A trajectory cannot cross itself, and the plane is too cramped to wander chaotically — so in the long run it must either come to rest, or settle into a loop. The Poincare-Bendixson theorem makes this folk intuition into a precise, powerful classification of long-term behaviour in the plane.

The statement: consider a smooth planar system whose trajectory stays inside a closed, bounded region for all future time and does not run into an equilibrium. Then its long-term limit (its omega-limit set, the set of points it keeps returning near) must be a closed orbit — a periodic cycle. So a bounded, equilibrium-avoiding planar trajectory has only one fate left: it spirals onto a limit cycle. The usual way to use this is the 'trapping region' trick: find a ring (an annulus) that the flow enters but never exits, containing no equilibrium inside; the theorem then guarantees a limit cycle lives in that ring.

This is the single most useful tool for proving periodic motion exists in nonlinear planar systems, where you can rarely write the orbit down. Its power comes entirely from being two-dimensional — it rests on the Jordan curve theorem (a closed loop in the plane separates inside from outside), which is what forbids more complicated wandering. The honest limit: it is false in three or more dimensions. In 3D a bounded trajectory can avoid both rest and periodicity and instead fill out a strange attractor — which is exactly how chaos becomes possible once you leave the plane.