In mathematics, the Poincaré–Bendixson theorem is a statement about the long-term behaviour of orbits of continuous dynamical systems on the plane, cylinder, or two-sphere.[1]

Theorem

Given a differentiable real dynamical system defined on an open subset of the plane, every non-empty compact ω-limit set of an orbit, which contains only finitely many fixed points, is either[2]

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.

Discussion

A weaker version of the theorem was originally conceived by Henri Poincaré (1892), although he lacked a complete proof which was later given by Ivar Bendixson (1901).

Continuous dynamical systems that are defined on two-dimensional manifolds other than the plane (or cylinder or two-sphere), as well as those defined on higher-dimensional manifolds, may exhibit ω-limit sets that defy the three possible cases under the Poincaré–Bendixson theorem. On a torus, for example, it is possible to have a recurrent non-periodic orbit,[3] and three-dimensional systems may have strange attractors. Nevertheless, it is possible to classify the minimal sets of continuous dynamical systems on any two-dimensional compact and connected manifold due to a generalization of Arthur J. Schwartz.[4][5]

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.

It is important to note that Poincaré–Bendixson theorem does not apply to discrete dynamical systems, where chaotic behaviour can arise in two- or even one-dimensional systems.

See also

References

  1. ↑ Coddington, Earl A.; Levinson, Norman (1955). "The Poincaré–Bendixson Theory of Two-Dimensional Autonomous Systems". Theory of Ordinary Differential Equations. New York: McGraw-Hill. pp. 389–403. ISBN 978-0-89874-755-3.
  2. ↑ Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0.
  3. ↑ D'Heedene, R.N. (1961). "A third order autonomous differential equation with almost periodic solutions". Journal of Mathematical Analysis and Applications. Elsevier. 3 (2): 344–350. doi:10.1016/0022-247X(61)90059-2.
  4. ↑ Schwartz, Arthur J. (1963). "A Generalization of a Poincare-Bendixson Theorem to Closed Two-Dimensional Manifolds". American Journal of Mathematics. 85 (3): 453. doi:10.2307/2373135.
  5. ↑ Katok, Anatole; Hasselblatt, Boris (1995-04-28). Introduction to the Modern Theory of Dynamical Systems (1 ed.). Cambridge University Press. doi:10.1017/cbo9780511809187. ISBN 978-0-521-34187-5.
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