In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, especially in the study of
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. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...
s, 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.
Types
*
fixed points
*
periodic orbit In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time.
Iterated functions
Given a ...
s
*
limit cycle
In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity o ...
s
*
attractor
In the 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 values remain ...
s
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
Let
be a
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
, and let
be a
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
. The
-limit set of
, denoted by
, is the set of
cluster points of the
forward orbit of the
iterated function
In mathematics, an iterated function is a function (that is, a function from some set to itself) which is obtained by composing another function with itself a certain number of times. The process of repeatedly applying the same function is ...
.
Hence,
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bic ...
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 point In mathematics, a recurrent point for a function ''f'' is a point that is in its own limit set by ''f''. Any neighborhood containing the recurrent point will also contain (a countable number of) iterates of it as well.
Definition
Let X be a Haus ...
s''). This may also be formulated as the outer limit (
limsup
In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a ...
) of a sequence of sets, such that
:
If
is a
homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isom ...
(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
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...
, they are compact and nonempty.
Definition for flows
Given a
real dynamical system (''T'', ''X'', φ) with
flow
Flow may refer to:
Science and technology
* Fluid flow, the motion of a gas or liquid
* Flow (geomorphology), a type of mass wasting or slope movement in geomorphology
* Flow (mathematics), a group action of the real numbers on a set
* Flow (psyc ...
, 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
In celestial mechanics, an orbit 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 object or position in space such as ...
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
In celestial mechanics, an orbit 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 object or position in space such as ...
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 lim
ω γ (lim
α γ).
If the ω-limit set (α-limit set) is disjoint from the orbit γ, that is lim
ω γ ∩ γ = ∅ (lim
α γ ∩ γ = ∅), we call lim
ω γ (lim
α γ) a
ω-limit cycle
In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity ...
(
α-limit cycle
In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity ...
).
Alternatively the limit sets can be defined as
:
and
:
Examples
* For any
periodic orbit In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time.
Iterated functions
Given a ...
γ of a dynamical system, lim
ω γ = lim
α γ = γ
* For any
fixed point of a dynamical system, lim
ω = lim
α =
Properties
* lim
ω γ and lim
α γ are
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
* if ''X'' is compact then lim
ω γ and lim
α γ are
nonempty,
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...
and
connected
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
* lim
ω γ and lim
α γ are φ-invariant, that is φ(
× lim
ω γ) = lim
ω γ and φ(
× lim
α γ) = lim
α γ
See also
*
Julia set
In the context of complex dynamics, a branch of mathematics, the Julia set and the Fatou set are two complementary sets (Julia "laces" and Fatou "dusts") defined from a function. Informally, the Fatou set of the function consists of values wi ...
*
Stable set
*
Limit cycle
In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity o ...
*
Periodic point
*
Non-wandering set
*
Kleinian group
In mathematics, a Kleinian group is a discrete subgroup of the group of orientation-preserving isometries of hyperbolic 3-space . The latter, identifiable with , is the quotient group of the 2 by 2 complex matrices of determinant 1 by their ...
References
Further reading
*
{{PlanetMath attribution, id=4316, title=Omega-limit set