Poincaré-Bendixson theorem

In mathematics , the Poincaré – Bendixson theorem is a theorem about the behavior of trajectories in two-dimensional continuous dynamic systems . It is named after the French mathematician Henri Poincaré , who originally wrote a weaker form of the theorem, although he did not know a complete proof, and after the Swedish mathematician Ivar Bendixson , who proved the complete theorem in 1901.

He makes statements about the existence of periodic orbits or limit cycles in planar dynamic systems.

statement

The theorem exists in a few equivalent formulations. A general version is the following:

Given a differentiable dynamic system , which on an open subset is defined level: . Then every compact ω- limit set that contains only finitely many critical points is either ${\displaystyle {\dot {\vec {x}}}=f({\vec {x}})}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle {\vec {x}}\in \mathbb {R} ^{2}}$

• a critical point
• a periodic orbit (or limit cycle ) or
• a connected set, consisting of a finite number of critical points together with homoclinic or heteroclinic orbits that connect them. In this case there is at most one orbit connecting different critical points in the same direction; for a critical point, however, there can be more than one homoclinic orbit.

An orbit is called heteroclinic if it connects different fixed points and homoclinic if it starts and ends at the same fixed point (this is then a saddle point), i.e. it contains both a stable and an unstable manifold of the fixed point.

Another formulation of the theorem is that an orbit that remains forever in a closed, bounded subset R of the plane that does not contain any fixed points must be a periodic orbit or a limit cycle (that is, it asymptotically approaches a periodic orbit ). This is a more restricted version than the formulation above. For example, it does not follow that the limit sets (for ) limit cycles or fixed points; as mentioned above, they can also be connections of homoclinic or heteroclinic orbits (orbits) and fixed points. The sentence essentially excludes chaotic behavior as defined above in dynamic systems in the plane. ${\displaystyle t\to \infty }$

Note that the sentence is wrong in higher dimensions. This is mainly due to the application of the Jordanian curve theorem in the proof, which divides the plane into two regions. Even in three dimensions, an orbit can run in a closed, limited area without meeting a fixed point or a periodic orbit. There, for example, there is the chaotic phenomenon of the strange attractor .

The Poincaré-Bendixson theorem does not apply to two-dimensional areas with a different topology than the plane, for example the torus. Quasi-periodic movements can be constructed here relatively easily. It also does not apply to two-dimensional images, of which, for example, the baker's image shows highly chaotic behavior, or to the Hénon image , which has a strange attractor.

literature

1. ^ Poincaré, H. (1892), "Sur les courbes définies par une equation différentielle", Oeuvres, 1, Paris
2. ↑ Bendixson, Ivar (1901), “Sur les courbes définies par des equations différentielles”, Acta Mathematica (Springer Netherlands) 24 (1): 1-88, doi : 10.1007 / BF02403068 .
3. ^ Gerald Teschl : Ordinary Differential Equations and Dynamical Systems . American Mathematical Society, Providence 2012, ISBN 978-0-8218-8328-0 ( free online version ).
4. ↑ Poincaré-Bendixson theorem, WS Koon, Caltech, pdf
5. ^ WS Koon, Lectures on periodic orbits, pdf , Caltech 2009
6. ↑ Baker's illustration, spectrum lexicon of physics

Web links

• DV Anosov: Poincaré-Bendixson theory . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).