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 closure is here needed, since we have not assumed that the underlying metric space of interest to be a complete metric space. 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 (T, X, φ) with flow ${\displaystyle \varphi$ :\mathbb {R} \times X\to X} , a point x, we call a point y an ω-limit point of x if there exists a sequence ${\displaystyle (t_{n})_{n\in \mathbb {N} }}$ in R so that

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

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

For an orbit 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 ${\displaystyle (t_{n})_{n\in \mathbb {N} }}$ in R so that

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

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

For an orbit 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 (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\}}}.}

• 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.