Poincaré-Bendixson theorem
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Poincaré-Bendixson theorem is one of the most important theorems in dynamical systesms. It is a statement about the long-term behaviour of orbits of continuous dynamical systems on a plane.
Given a differentiable real dynamical system defined on an open subset of the plane, then every non-empty compact $\omega$-limit set of an orbit, which contains only finitely many fixed points, is either
- a fixed point.
- a periodic orbit.
- 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 countably 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).
A direct consequence of this theorem is that 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-dim or even one-dimensional systems.
See also
Papers
- Bendixson, I. (1901). Sur les courbes définies par des équations différentielles. Acta Mathematica, 24(1), 1-88.