Poincaré–Bendixson theorem

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

[edit] Overview

Basically the theorem states that any orbit which stays in a bounded region of the state space of the 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 French mathematician Henri Poincaré, although he lacked a complete proof. In 1901 Swedish mathematician Ivar Otto Bendixson 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 critical to the theorem. On a torus, for example, it is possible to have a recurrent non-periodic orbit, as in the suspension of an irrational rotation of the circle.

[edit] 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.

[edit] References

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