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.
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.

Contents

History

Poincaré–Bendixson theorem

Notes

Applications

History

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.

Poincaré–Bendixson 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.

Notes

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.

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.