Theorem

Poincare-Bendixson Theorem

Analysis

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

Statement

Every 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

Proof Year

1901 1

Classification

Statement Form

Classification Theorem 1

Connections

Has Statement Form

Classification Theorem

In Branch

Dynamical Systems

Source Poincare-Bendixson Theorem (Wikipedia)

Proved By

Henri Poincaré

Sources

1. Poincare-Bendixson Theorem (Wikipedia)

Wikimedia Foundation

Theorem 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.
Discussion
A weaker version of the theorem was originally conceived by Henri Poincaré (1892), although he lacked a complete proof which was later given by Ivar Bendixson (1901).
In Branch: Dynamical Systems, Lead sentence
t the long-term behaviour of orbits of continuous dynamical systems on the plane, cylinder, or two-sphere.

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Poincare-Bendixson Theorem

Has Statement Form

In Branch

Proved By