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

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The Poincaré–Bendixson theorem states that any orbit that stays in a compact region of the state space of a 2-dimensional planar continuous dynamical system, all of whose fixed points are isolated, approaches its ω-limit set, which is either a fixed point, a periodic orbit, or is a connected set composed of a finite number of fixed points together with homoclinic and heteroclinic orbits connecting these. Thus chaotic behaviour can only arise in continuous dynamical systems whose phase space has three 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. Ivar Bendixson (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

• Bendixson, Ivar (1901), "Sur les courbes définies par des équations différentielles", Acta Mathematica (Springer Netherlands) 24 (1): 1–88, doi:10.1007/BF02403068 
• Poincaré, H. (1892), "Sur les courbes définies par une équation différentielle", Oeuvres, 1, Paris