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.
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.
be a metric space, and let
be a continuous function.
ω
, is the set of cluster points of the forward orbit
of the iterated function
if and only if there is a strictly increasing sequence of natural numbers
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.
α ( x , f ) = ω ( x ,
is compact, they are compact and nonempty.
Given a real dynamical system
, we call a point y an
if there exists a sequence
-limit point of some point on the orbit.
Analogously we call
if there exists a sequence
-limit point of some point on the orbit.
-limit points) for a given orbit
-limit set) is disjoint from the orbit
) a ω-limit cycle (α-limit cycle).
Alternatively the limit sets can be defined as and
This article incorporates material from Omega-limit set on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.