Poincaré–Bendixson theorem

Poincaré–Bendixson theorem

In mathematics, the Poincaré–Bendixson theorem is a statement about the long term behaviour of orbits of continuous dynamical systems on the plane.

Overview

The Poincaré–Bendixson theorem states that any orbit which stays in a compact region of the state space of a 2-dimensional planar continuous dynamical system either approaches a fixed point or a periodic orbit. Thus chaotic behaviour can only arise in continuous dynamical systems whose phase space has 3 or more dimensions. However the theorem does not apply to discrete dynamical systems, where chaotic behaviour can arise in two or even one dimensional systems.

A weaker version of the theorem was originally conceived by Henri Poincaré, although he lacked a complete proof. harvs|txt|first=Ivar |last= Bendixson|authorlink=Ivar Otto Bendixson|year=1901 gave a rigorous proof of the full theorem.

Given a differentiable real dynamical system defined on an open and simply connected subset of the plane, then every non empty compact α-limit set ( or ω-limit set) of an orbit, which contains no fixed points, is a periodic orbit.

The condition that the dynamical system be on the plane is necessary to the theorem. On a torus, for example, it is possible to have a recurrent non-periodic orbit.

Applications

One important implication is that a two-dimensional continuous dynamical system cannot give rise to a strange attractor. If a strange attractor C did exist in such a system, then it could be enclosed in a closed and bounded subset of the phase space. By making this subset small enough, any nearby stationary points could be excluded. But then the Poincaré–Bendixson theorem says that C is not a strange attractor at all - it is either a limit-cycle or it converges to a limit-cycle.

References

*citation|last=Bendixson|first=Ivar |title= Sur les courbes définies par des équations différentielles
journal= Acta Mathematica
publisher= Springer Netherlands
ISSN =1871-2509
volume =24|issue=1 |year= 1901
DOI =10.1007/BF02403068
pages =1-88

*citation|last=Poincaré|first=H.|chapter=Sur les courbes définies par une équations différentielle|volume=1|title=Oeuvres|place=Paris|year=1892


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