In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Poincaré–Bendixson theorem is a statement about the long-term behaviour of
orbit
In celestial mechanics, an orbit (also known as orbital revolution) is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an ...
s of
continuous dynamical systems on the plane, cylinder, or two-sphere.
Theorem
Given a
differentiable real dynamical system defined on an
open
Open or OPEN may refer to:
Music
* Open (band), Australian pop/rock band
* The Open (band), English indie rock band
* ''Open'' (Blues Image album), 1969
* ''Open'' (Gerd Dudek, Buschi Niebergall, and Edward Vesala album), 1979
* ''Open'' (Go ...
subset of the plane, every
non-empty compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
''ω''-limit set of an
orbit
In celestial mechanics, an orbit (also known as orbital revolution) is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an ...
, which contains only finitely many fixed points, is either
* 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 countably many homoclinic orbits connecting one fixed point.
Discussion
A weaker version of the theorem was originally conceived by , although he lacked a complete proof which was later given by .
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
In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...
, for example, it is possible to have a recurrent non-periodic orbit,
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
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
and
connected manifold due to a generalization of Arthur J. Schwartz.
Applications
One important implication is that a two-dimensional continuous dynamical system cannot give rise to a
strange attractor
In the mathematics, mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor va ...
. 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.
The Poincaré–Bendixson theorem does not apply to
discrete dynamical system
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space, such as in a parametric curve. Examples include the mathematical models that describe the swinging of a clock ...
s, where chaotic behaviour can arise in two- or even one-dimensional systems.
See also
*
Rotation number
*
References
*
*
{{DEFAULTSORT:Poincare-Bendixson theorem
Theorems in dynamical systems