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Inmathematics, especially in the study ofdynamical systems, alimit 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 atequilibrium.

• fixed points
• periodic orbits
• limit cycles
• attractors

In general, limits sets can be very complicated as in the case ofstrange attractors, but for 2-dimensional dynamical systems thePoincaré–Bendixson theorem provides a simple characterization of all nonempty, compact${\displaystyle \omega }$-limit sets that contain at most finitely many fixed points as a fixed point, a periodic orbit, or a union of fixed points andhomoclinic orheteroclinic orbits connecting those fixed points.

Let${\displaystyle X}$ be ametric space, and let${\displaystyle f:X\to X}$ be acontinuous function. The${\displaystyle \omega }$-limit set of${\displaystyle x\in X}$, denoted by${\displaystyle \omega (x,f)}$, is the set of cluster points of the forward orbit${\displaystyle \{f^n(x){\}}_{n\in \mathbb{N}}}$ of theiterated function${\displaystyle f}$.^[1] Hence,${\displaystyle y\in \omega (x,f)}$if and only if there is a strictly increasing sequence of natural numbers${\displaystyle \{n_k{\}}_{k\in \mathbb{N}}}$ such that${\displaystyle f^{n_k}(x)\to y}$ as${\displaystyle k\to \mathrm{\infty }}$. Another way to express this is

${\displaystyle \omega (x,f)=\bigcap _{n\in \mathbb{N}}\overline{\{f^k(x):k>n\}},}$

where${\displaystyle \overline{S}}$ denotes theclosure of set${\displaystyle S}$. The points in the limit set are non-wandering (but may not berecurrent points). This may also be formulated as the outer limit (limsup) of a sequence of sets, such that

${\displaystyle \omega (x,f)=\bigcap _{n=1}^{\mathrm{\infty }}\overline{\bigcup _{k=n}^{\mathrm{\infty }}\{f^k(x)\}}.}$

If${\displaystyle f}$ is ahomeomorphism (that is, a bicontinuous bijection), then the${\displaystyle \alpha }$-limit set is defined in a similar fashion, but for the backward orbit;i.e.${\displaystyle \alpha (x,f)=\omega (x,f^{-1})}$.

Both sets are${\displaystyle f}$-invariant, and if${\displaystyle X}$ iscompact, they are compact and nonempty.

Given areal dynamical system${\displaystyle (T,X,\phi )}$ withflow${\displaystyle \phi :\mathbb{R}\times X\to X}$, a point${\displaystyle x}$, we call a point${\displaystyle y}$ an${\displaystyle \omega }$-limit point of${\displaystyle x}$ if there exists a sequence${\displaystyle (t_n{)}_{n\in \mathbb{N}}}$ in${\displaystyle \mathbb{R}}$ so that

${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{t}_n=\mathrm{\infty }}$

${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\phi (t_n,x)=y}$.

For anorbit${\displaystyle \gamma }$ of${\displaystyle (T,X,\phi )}$, we say that${\displaystyle y}$ is an${\displaystyle \omega }$-limit point of${\displaystyle \gamma }$, if it is an${\displaystyle \omega }$-limit point of some point on the orbit.

Analogously we call${\displaystyle y}$ an${\displaystyle \alpha }$-limit point of${\displaystyle x}$ if there exists a sequence${\displaystyle (t_n{)}_{n\in \mathbb{N}}}$ in${\displaystyle \mathbb{R}}$ so that

${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{t}_n=-\mathrm{\infty }}$

${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\phi (t_n,x)=y}$.

For anorbit${\displaystyle \gamma }$ of${\displaystyle (T,X,\phi )}$, we say that${\displaystyle y}$ is an${\displaystyle \alpha }$-limit point of${\displaystyle \gamma }$, if it is an${\displaystyle \alpha }$-limit point of some point on the orbit.

The set of all${\displaystyle \omega }$-limit points (${\displaystyle \alpha }$-limit points) for a given orbit${\displaystyle \gamma }$ is called${\displaystyle \omega }$-limit set (${\displaystyle \alpha }$-limit set) for${\displaystyle \gamma }$ and denoted${\displaystyle \underset{\omega }{lim}\gamma }$ (${\displaystyle \underset{\alpha }{lim}\gamma }$).

If the${\displaystyle \omega }$-limit set (${\displaystyle \alpha }$-limit set) is disjoint from the orbit${\displaystyle \gamma }$, that is${\displaystyle \underset{\omega }{lim}\gamma \cap \gamma =\varnothing }$ (${\displaystyle \underset{\alpha }{lim}\gamma \cap \gamma =\varnothing }$), we call${\displaystyle \underset{\omega }{lim}\gamma }$ (${\displaystyle \underset{\alpha }{lim}\gamma }$) aω-limit cycle (α-limit cycle).

Alternatively the limit sets can be defined as

${\displaystyle \underset{\omega }{lim}\gamma :=\bigcap _{s\in \mathbb{R}}\overline{\{\phi (x,t):t>s\}}}$

and

• For anyperiodic orbit${\displaystyle \gamma }$ of a dynamical system,${\displaystyle \underset{\omega }{lim}\gamma =\underset{\alpha }{lim}\gamma =\gamma }$
• For anyfixed point${\displaystyle x_0}$ of a dynamical system,${\displaystyle \underset{\omega }{lim}{x}_0=\underset{\alpha }{lim}{x}_0=x_0}$

• ${\displaystyle \underset{\omega }{lim}\gamma }$ and${\displaystyle \underset{\alpha }{lim}\gamma }$ areclosed
• if${\displaystyle X}$ is compact then${\displaystyle \underset{\omega }{lim}\gamma }$ and${\displaystyle \underset{\alpha }{lim}\gamma }$ arenonempty,compact andconnected
• ${\displaystyle \underset{\omega }{lim}\gamma }$ and${\displaystyle \underset{\alpha }{lim}\gamma }$ are${\displaystyle \phi }$-invariant, that is${\displaystyle \phi (\mathbb{R}\times \underset{\omega }{lim}\gamma )=\underset{\omega }{lim}\gamma }$ and${\displaystyle \phi (\mathbb{R}\times \underset{\alpha }{lim}\gamma )=\underset{\alpha }{lim}\gamma }$

• Julia set
• Stable set
• Limit cycle
• Periodic point
• Non-wandering set
• Kleinian group

• Teschl, Gerald (2012).Ordinary Differential Equations and Dynamical Systems.Providence:American Mathematical Society.ISBN 978-0-8218-8328-0.

This article incorporates material from Omega-limit set onPlanetMath, which is licensed under theCreative Commons Attribution/Share-Alike License.

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