Poincaré–Bendixson theorem
- 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.
Overview
The Poincaré–Bendixson theorem states that any orbit which stays in a compact region of the state space of a 2-dimensional planar continuous dynamical system either approaches a fixed point or a periodic orbit. Thus chaotic behaviour can only arise in continuous dynamical systems whose phase space has 3 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.
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.
Given a differentiable real dynamical system defined on an open and simply connected subset of the plane, then every non empty compact α-limit set ( or ω-limit set) of an orbit, which contains no fixed points, is a periodic orbit.
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.
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
*citation|last=Bendixson|first=Ivar |title= Sur les courbes définies par des équations différentielles
journal= Acta Mathematica
publisher= Springer Netherlands
ISSN =1871-2509
volume =24|issue=1 |year= 1901
DOI =10.1007/BF02403068
pages =1-88
*citation|last=Poincaré|first=H.|chapter=Sur les courbes définies par une équations différentielle|volume=1|title=Oeuvres|place=Paris|year=1892
Wikimedia Foundation.
2010.
Look at other dictionaries:
Poincaré-Bendixson-Theorem — In der Mathematik ist das Poincaré–Bendixson Theorem ein Satz über das Verhalten von Bahnkurven in zweidimensionalen stetigen dynamischen Systemen. Theorem Das Theorem existiert in einigen äquivalenten Formulierungen. Eine allgemeine Version ist… … Deutsch Wikipedia
Poincaré–Bendixson Theorem — In der Mathematik ist das Poincaré–Bendixson Theorem ein Satz über das Verhalten von Bahnkurven in zweidimensionalen stetigen dynamischen Systemen. Theorem Das Theorem existiert in einigen äquivalenten Formulierungen. Eine davon ist die folgende … Deutsch Wikipedia
Poincaré-Bendixson-Theorem — nach H. Poincaré (1854–1912) und I.O. Bendixson (1861–1935) benanntes mathematisches Theorem zum Nachweis der Existenz von ⇡ Grenzzyklen in zweidimensionalen nicht linearen ⇡ Differenzialgleichungssystemen. Das P. B. T. wird v.a. in der modernen… … Lexikon der Economics
Théorème de Poincaré-Bendixson — En mathématiques, le théorème de Poincaré Bendixson est un résultat qualitatif sur les équations différentielles. Il concerne les équations du type (1) : x = f(x) où f est une fonction continument dérivable, du plan réel dans lui… … Wikipédia en Français
Bendixson — Ivar Otto Bendixson (* 1. August 1861 in Stockholm; 29. November 1935 ebenda) war ein schwedischer Mathematiker. Bendixson ging in Stockholm zur Schule und studierte ab 1878 an der Königlich Technischen Hochschule Stockholm und ab 1879 an der… … Deutsch Wikipedia
Henri Poincaré — Infobox Scientist box width = 300px name = Henri Poincaré image size = 250px caption = Jules Henri Poincaré (1854 1912). Photograph from the frontispiece of the 1913 edition of Last Thoughts. birth date = birth date|df=yes|1854|4|29 birth place … Wikipedia
Henri Poincaré — «Poincaré» redirige aquí. Para el político francés, véase Raymond Poincaré. Henri Poincaré … Wikipedia Español
Ivar Otto Bendixson — (August 1, 1861 1935) was a Swedish mathematician. Biography Bendixson was born August 1, 1861 in Bergshyddan, Djurgardsbrunn, Sweden to a middle class family. His father Vilhelm Emanuel Bendixson was a merchant, and his mother was Tony Amelia… … Wikipedia
Ivar Bendixson — Ivar Otto Bendixson (* 1. August 1861 in Stockholm; † 29. November 1935 ebenda) war ein schwedischer Mathematiker. Bendixson ging in Stockholm zur Schule und studierte ab 1878 an der Königlich Technischen Hochschule Stockholm und ab 1879 an der… … Deutsch Wikipedia
Ivar Bendixson — Ivar Otto Bendixson (1er août 1861 29 novembre 1935) est un mathématicien suédois. Il a travaillé sur les infinis de Cantor, et est connu pour le théorème de Poincaré Bendixson, le théorème de Cantor Bendixson (en) et le… … Wikipédia en Français
