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[2]
Moreover, there is at most one orbit connecting different fixed points in the same direction. However, there could be countably many homoclinic orbits connecting one fixed point.
Discussion[edit]
A weaker version of the theorem was originally conceived by Henri Poincaré (1892), although he lacked a complete proof which was later given by Ivar Bendixson (1901).
Continuous dynamical systems that are defined on two-dimensional manifolds other than the plane (or cylinder or two-sphere), 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,[3] 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.[4][5]
Applications[edit]
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.
It is important to note that Poincaré–Bendixson theorem does not apply to discrete dynamical systems, where chaotic behaviour can arise in two- or even one-dimensional systems.