In mathematics, the PoincarĂ©âBendixson theorem is a statement about the long-term behaviour of orbits of continuous dynamical systems on the plane, cylinder, or two-sphere. [1]
Given a differentiable real dynamical system defined on an open subset of the plane, every non-empty compact Ï-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.
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 Ï-limit sets that defy the three possible cases under the PoincarĂ©âBendixson 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]
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 PoincarĂ©âBendixson theorem says that C is not a strange attractor at allâit is either a limit cycle or it converges to a limit cycle.
It is important to note that PoincarĂ©âBendixson theorem does not apply to discrete dynamical systems, where chaotic behaviour can arise in two- or even one-dimensional systems.
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