Limit set

• fixed points
• periodic orbits
• limit cycles
• attractors

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 ${\displaystyle \omega }$ -limit sets that contain at most finitely many fixed points as a fixed point, a periodic orbit, or a union of fixed points and homoclinic or heteroclinic orbits connecting those fixed points.

Let ${\displaystyle X}$  be a metric space, and let ${\displaystyle f:X\rightarrow X}$  be a continuous 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 the iterated 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 ${\displaystyle f^{n_{k}}(x)\rightarrow y}$  as ${\displaystyle k\rightarrow \infty }$ . 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 the closure of set ${\displaystyle S}$ . 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

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

If ${\displaystyle f}$  is a homeomorphism (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 ${\displaystyle X}$  is compact, they are compact and nonempty.

Given a real dynamical system ${\displaystyle (T,X,\varphi )}$  with flow ${\displaystyle \varphi :\mathbb {R} \times X\to X}$ , a point ${\displaystyle x}$ , we call a point 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

${\displaystyle \lim _{n\to \infty }t_{n}=\infty }$ 

${\displaystyle \lim _{n\to \infty }\varphi (t_{n},x)=y}$ .

For an orbit ${\displaystyle \gamma }$  of ${\displaystyle (T,X,\varphi )}$ , 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

${\displaystyle \lim _{n\to \infty }t_{n}=-\infty }$ 

${\displaystyle \lim _{n\to \infty }\varphi (t_{n},x)=y}$ .

For an orbit ${\displaystyle \gamma }$  of ${\displaystyle (T,X,\varphi )}$ , 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 ${\displaystyle \lim _{\omega }\gamma }$  (${\displaystyle \lim _{\alpha }\gamma }$ ).

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

Alternatively the limit sets can be defined as

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

and

${\displaystyle \lim _{\alpha }\gamma :=\bigcap _{s\in \mathbb {R} }{\overline {\{\varphi (x,t):t<s\}}}.}$ 

• For any periodic orbit ${\displaystyle \gamma }$  of a dynamical system, ${\displaystyle \lim _{\omega }\gamma =\lim _{\alpha }\gamma =\gamma }$ 
• For any fixed point ${\displaystyle x_{0}}$  of a dynamical system, ${\displaystyle \lim _{\omega }x_{0}=\lim _{\alpha }x_{0}=x_{0}}$ 

• ${\displaystyle \lim _{\omega }\gamma }$  and ${\displaystyle \lim _{\alpha }\gamma }$  are closed
• if ${\displaystyle X}$  is compact then ${\displaystyle \lim _{\omega }\gamma }$  and ${\displaystyle \lim _{\alpha }\gamma }$  are nonempty, compact and connected
• ${\displaystyle \lim _{\omega }\gamma }$  and ${\displaystyle \lim _{\alpha }\gamma }$  are ${\displaystyle \varphi }$ -invariant, that is ${\displaystyle \varphi (\mathbb {R} \times \lim _{\omega }\gamma )=\lim _{\omega }\gamma }$  and ${\displaystyle \varphi (\mathbb {R} \times \lim _{\alpha }\gamma )=\lim _{\alpha }\gamma }$ 

• 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 on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.