Limit set

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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. A system that has reached its limiting set is said to be at equilibrium.

Types

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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 functions

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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 flows

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Given a real dynamical system   with flow  , a point  , we call a point y an  -limit point of   if there exists a sequence   in   so that

 
 .

For an orbit   of  , we say that   is an  -limit point of  , if it is an  -limit point of some point on the orbit.

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

 
 .

For an orbit   of  , we say that   is an  -limit point of  , if it is an  -limit point of some point on the orbit.

The set of all  -limit points ( -limit points) for a given orbit   is called  -limit set ( -limit set) for   and denoted   ( ).

If the  -limit set ( -limit set) is disjoint from the orbit  , that is   ( ), we call   ( ) a ω-limit cycle (α-limit cycle).

Alternatively the limit sets can be defined as

 

and

 

Examples

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  • For any periodic orbit   of a dynamical system,  
  • For any fixed point   of a dynamical system,  

Properties

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  •   and   are closed
  • if   is compact then   and   are nonempty, compact and connected
  •   and   are  -invariant, that is   and  

See also

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References

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

Further reading

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