Poincaré–Bendixson theorem

Summary

In mathematics, the Poincaré–Bendixson theorem is a statement about the long-term behaviour of orbits of continuous dynamical systems on the plane, cylinder, or two-sphere. Theorem Given a differentiable real dynamical system defined on an open subset of the plane, 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.

Moreover, there is at most one orbit connecting different fixed points in the same direction. However, there could be countably many homoclinic orbits connecting one fixed point. A weaker version of the theorem was originally conceived by , although he lacked a complete proof which was later given by . Discussion The condition that the dynamical system be on the plane is necessary to the theorem. On a torus, for example, it is p

Official source

https://en.wikipedia.org/wiki/Poincar%C3%A9%E2%80%93Bendixson_theorem

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