The Poincare-Bendixson Theorem describes the possible long-term behavior of trajectories of a continuous dynamical system in the plane, showing that a nonempty, bounded limit set containing no fixed point must be a periodic orbit. Named for Henri Poincare and Ivar Bendixson, it rules out chaotic behavior for continuous flows confined to two dimensions, a restriction that does not extend to three or more dimensions.
Facts
StatementEvery non-empty compact omega-limit set of an orbit that contains only finitely many fixed points is either a fixed point, a periodic orbit, or a connected set of finitely many fixed points joined by homoclinic and heteroclinic orbits. 1 Classification
Statement Form Connections
Sources
1. Poincare-Bendixson Theorem (Wikipedia)
Wikimedia FoundationTheorem section, statement sentence
every non-empty compact ω-limit set of an orbit, which contains only finitely many fixed points, is either a fixed point, a periodic orbit, or a connected set composed of a finite number of fixed points together with homoclinic and heteroclinic orbits connecting these.
Lead section, history sentence
A weaker version of the theorem was originally conceived by Henri Poincare (1892), although he lacked a complete proof which was later given by Ivar Bendixson (1901).
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