Poincaré–Bendixson theorem
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In mathematics, the Poincaré–Bendixson theorem is a statement about the behaviour of trajectories of continuous dynamical systems on the plane.
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[edit] Theorem
The theorem can be stated in several equivalent ways. One statement is:
- Let F be a dynamical system on the real plane defined by
- where f and g are continuous differentiable functions of x and y. Let S be a closed bounded subset of the two-dimensional phase space of F that does not contain a stationary point of F and let C be a trajectory of F that never leaves S. Then C is either a limit-cycle or C converges to a limit-cycle.
[edit] Topological considerations
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.
[edit] Implications
The Poincaré–Bendixson theorem limits the types of long term behaviour that can be exhibited by continuous planar dynamical systems.
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 says that chaotic behaviour can only arise in continuous dynamical systems whose phase space has 3 or more dimensions. However, this restriction does not apply to discrete dynamical systems, where chaotic behaviour can arise in two or even one dimensional systems.
[edit] 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.
