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, 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.
Discussion
A weaker version of the theorem was originally conceived by, although he lacked a complete proof which was later given by.Continuous dynamical systems that are defined on two-dimensional manifolds other than the plane, as well as those defined on higher-dimensional manifolds, may exhibit ω-limit sets that defy the three possible cases under the Poincaré–Bendixson theorem. On a torus, for example, it is possible to have a recurrent non-periodic orbit, and three-dimensional systems may have strange attractors. Nevertheless, it is possible to classify the minimal sets of continuous dynamical systems on any two-dimensional compact and connected manifold due to a generalization of Arthur J. Schwartz.
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.The Poincaré–Bendixson theorem does not apply to discrete dynamical systems, where chaotic behaviour can arise in two- or even one-dimensional systems.