Poincaré–Bendixson theorem: Difference between revisions

Content deleted Content added

Inline

Revision as of 10:36, 16 September 2011

In mathematics, the Poincaré–Bendixson theorem is a statement about the long-term behaviour of orbits of continuous dynamical systems on the plane.

Theorem

Given a differentiable real dynamical system defined on an open subset of the plane, then every non-empty compact ω-limit set of an orbit, which contains only fintely many fixed points, is either^[1]

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.

Moreover, there is at most one orbit connecting different fixed points in the same direction. However, there could be countable many homoclinic orbits connecting one fixed point.

A weaker version of the theorem was originally conceived by Henri Poincaré, although he lacked a complete proof which was later given by Ivar Bendixson (1901).

Discussion

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. In particular, 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.

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

^ Teschl, Gerald. Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society.

Bendixson, Ivar (1901), "Sur les courbes définies par des équations différentielles", Acta Mathematica, 24 (1), Springer Netherlands: 1–88, doi:10.1007/BF02403068
Poincaré, H. (1892), "Sur les courbes définies par une équation différentielle", Oeuvres, vol. 1, Paris{{citation}}: CS1 maint: location missing publisher (link)

[1] Teschl, Gerald. Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society.

[1]

@@ Line 1: / Line 1: @@
 In [[mathematics]], the '''Poincaré–Bendixson theorem''' is a statement about the long-term behaviour of [[orbit (dynamics)|orbit]]s of [[continuous dynamical system]]s on the plane.
-==Overview==
+==Theorem==
+Given a [[differentiable real dynamical system]] defined on an [[open set|open]] subset of the plane, then every [[non-empty]] [[compact space|compact]] [[ω-limit set]] of an [[orbit (dynamics)|orbit]], which contains only fintely many fixed points, is either<ref>{{cite book
-{{Expert-subject|Mathematics|statement|date=November 2009}}
+ | surname = Teschl
-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 point (mathematics)|fixed point]]s are [[isolated point|isolated]], approaches its [[limit set|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 orbit|homoclinic]] and [[heteroclinic orbit|heteroclinic]] orbits connecting these. Thus [[chaos theory|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 system]]s, where chaotic behaviour can arise in two or even one dimensional systems.
+ | given = Gerald
+|authorlink=Gerald Teschl | title = Ordinary Differential Equations and Dynamical Systems
+ | publisher=[[American Mathematical Society]]
+ | place = [[Providence, Rhode Island|Providence]]
+ | year =
+ | url = http://www.mat.univie.ac.at/~gerald/ftp/book-ode/}}</ref>
+* a [[critical point|fixed point]],
+* a [[periodic orbit]], or
+* a [[connected set]] composed of a finite number of fixed points together with [[homoclinic orbit|homoclinic]] and [[heteroclinic orbit|heteroclinic]] orbits connecting these.
+Moreover, there is at most one orbit connecting different fixed points in the same direction. However, there could be countable many homoclinic orbits connecting one fixed point.
-A weaker version of the theorem was originally conceived by [[Henri Poincaré]], although he lacked a complete proof.  {{harvs|txt|first=Ivar |last= Bendixson|authorlink=Ivar Otto Bendixson|year=1901}} gave a rigorous proof of the full theorem.
+A weaker version of the theorem was originally conceived by [[Henri Poincaré]], although he lacked a complete proof which was later given by  {{harvs|txt|first=Ivar |last= Bendixson|authorlink=Ivar Otto Bendixson|year=1901}}.
-== Theorem ==
-Given a [[differentiable real dynamical system]] defined on an [[open set|open]] and [[simply connected]] subset of the plane, then every [[non-empty]] [[compact space|compact]] [[α-limit set]] (or [[ω-limit set]]) of an [[orbit (dynamics)|orbit]], which contains no fixed points, is a periodic orbit.<ref>Meiss, J. D.. Differential dynamical systems . Philadelphia: Society for Industrial and Applied Mathematics, 2007. </ref>
 == Discussion ==
 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.
+In particular, [[chaos theory|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 system]]s, where chaotic behaviour can arise in two or even one dimensional systems.
 ==Applications==