Poincaré–Bendixson theorem
Encyclopedia
In mathematics
, the Poincaré–Bendixson theorem is a statement about the long-term behaviour of orbit
s of continuous dynamical systems on the plane.
subset of the plane, then every non-empty compact
ω-limit set of an orbit
, which contains only finitely many fixed points, is either
Moreover, there is at most one orbit connecting different fixed points in the same direction. However, there could be countable 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 .
, for example, it is possible to have a recurrent non-periodic orbit.
In particular, 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 or even one dimensional systems.
or it converges to a limit-cycle.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Poincaré–Bendixson theorem is a statement about the long-term behaviour of orbit
Orbit (dynamics)
In mathematics, in the study of dynamical systems, an orbit is a collection of points related by the evolution function of the dynamical system. The orbit is a subset of the phase space and the set of all orbits is a partition of the phase space, that is different orbits do not intersect in the...
s of continuous dynamical systems on the plane.
Theorem
Given a differentiable real dynamical system defined on an openOpen set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
subset of the plane, then every non-empty compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
ω-limit set of an orbit
Orbit (dynamics)
In mathematics, in the study of dynamical systems, an orbit is a collection of points related by the evolution function of the dynamical system. The orbit is a subset of the phase space and the set of all orbits is a partition of the phase space, that is different orbits do not intersect in the...
, which contains only finitely many fixed points, is either
- a fixed pointCritical pointCritical point may refer to:*Critical point *Critical point *Critical point *Construction point of a ski jumping hill-See also:*Brillouin zone*Percolation thresholds...
, - a periodic orbit, or
- a connected set composed of a finite number of fixed points together with homoclinicHomoclinic orbitIn mathematics, a homoclinic orbit is a trajectory of a flow of a dynamical system which joins a saddle equilibrium point to itself. More precisely, a homoclinic orbit lies in the intersection of the stable manifold and the unstable manifold of an equilibrium.Homoclinic orbits and homoclinic points...
and heteroclinicHeteroclinic orbitIn mathematics, in the phase portrait of a dynamical system, a heteroclinic orbit is a path in phase space which joins two different equilibrium points...
orbits connecting these.
Moreover, there is at most one orbit connecting different fixed points in the same direction. However, there could be countable many homoclinic orbits connecting one fixed point.
A weaker version of the theorem was originally conceived by Henri Poincaré
Henri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
, although he lacked a complete proof which was later given by .
Discussion
The condition that the dynamical system be on the plane is necessary to the theorem. On a torusTorus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
, for example, it is possible to have a recurrent non-periodic orbit.
In particular, chaotic
Chaos theory
Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...
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 or even one dimensional systems.
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-cycleLimit-cycle
In mathematics, in the area of dynamical systems, a limit-cycle on a plane or a two-dimensional manifold is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity or as time approaches negative infinity. Such...
or it converges to a limit-cycle.