Poincaré–Bendixson theorem

In mathematics, the Poincaré - Bendixson theorem is a theorem about the behavior of trajectories in two-dimensional continuous dynamic systems.

Theorem

The theorem exists in some equivalent formulations. A common version is the following:

  • A critical point,
  • A periodic orbit or
  • A connected set consisting of a finite number of critical points together with homoclinic or heteroclinic orbits that connect these.

Note that this sentence is false in higher dimensions. This is mainly due to the application of the jordan curve theorem in the proof.

History

A weaker form of this theorem was originally written by the French mathematician Henri Poincaré, even though he knew no complete proof. In 1901, finally proved the Swedish mathematician Ivar Bendixson the full set.

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