Limit set

Summary

In mathematics, especially in the study of dynamical systems, a limit set is the state a dynamical system reaches after an infinite amount of time has passed, by either going forward or backwards in time. Limit sets are important because they can be used to understand the long term behavior of a dynamical system.

TypesEdit

In general, limits sets can be very complicated as in the case of strange attractors, but for 2-dimensional dynamical systems the Poincaré–Bendixson theorem provides a simple characterization of all nonempty, compact  -limit sets that contain at most finitely many fixed points as a fixed point, a periodic orbit, or a union of fixed points and homoclinic or heteroclinic orbits connecting those fixed points.

Definition for iterated functionsEdit

Let   be a metric space, and let   be a continuous function. The  -limit set of  , denoted by  , is the set of cluster points of the forward orbit   of the iterated function  .[1] Hence,   if and only if there is a strictly increasing sequence of natural numbers   such that   as  . Another way to express this is

 

where   denotes the closure of set  . The points in the limit set are non-wandering (but may not be recurrent points). This may also be formulated as the outer limit (limsup) of a sequence of sets, such that

 

If   is a homeomorphism (that is, a bicontinuous bijection), then the  -limit set is defined in a similar fashion, but for the backward orbit; i.e.  .

Both sets are  -invariant, and if   is compact, they are compact and nonempty.

Definition for flowsEdit

Given a real dynamical system (T, X, φ) with flow  , a point x, we call a point y an ω-limit point of x if there exists a sequence   in   so that

 
 .

For an orbit γ of (T, X, φ), we say that y is an ω-limit point of γ, if it is an ω-limit point of some point on the orbit.

Analogously we call y an α-limit point of x if there exists a sequence   in   so that

 
 .

For an orbit γ of (T, X, φ), we say that y is an α-limit point of γ, if it is an α-limit point of some point on the orbit.

The set of all ω-limit points (α-limit points) for a given orbit γ is called ω-limit set (α-limit set) for γ and denoted limω γ (limα γ).

If the ω-limit set (α-limit set) is disjoint from the orbit γ, that is limω γ ∩ γ = ∅ (limα γ ∩ γ = ∅), we call limω γ (limα γ) a ω-limit cycle (α-limit cycle).

Alternatively the limit sets can be defined as

 

and

 

ExamplesEdit

  • For any periodic orbit γ of a dynamical system, limω γ = limα γ = γ
  • For any fixed point   of a dynamical system, limω   = limα   =  

PropertiesEdit

  • limω γ and limα γ are closed
  • if X is compact then limω γ and limα γ are nonempty, compact and connected
  • limω γ and limα γ are φ-invariant, that is φ(  × limω γ) = limω γ and φ(  × limα γ) = limα γ

See alsoEdit

ReferencesEdit

  1. ^ Alligood, Kathleen T.; Sauer, Tim D.; Yorke, James A. (1996). Chaos, an introduction to dynamical systems. Springer.

Further readingEdit


This article incorporates material from Omega-limit set on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.