Limit set

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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. Examples of limit sets include fixed points, periodic orbits, limit cycles and attractors.

In general limits sets can be very complicated as in the case of strange attractors, but for 2-dimensional dynamical systems the Poincare-Bendixson theorem provides a simple characterization of all possible limit sets as a union of fixed points and periodic orbits.

1 Definition for iterated functions
2 Definition for flows
- 2.1 Examples
- 2.2 Properties
3 See also

Definition for iterated functions

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

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

The points in the limit set are called recurrent points.

If $f$ is a homeomorphism (that is, a bicontinuous bijection), then the $a$ -limit set is defined in a similar fashion, but for the backward orbit; i.e. $a(x, f) = ?(x, f - 1)$ .

Both sets are $f$ -invariant, and if $X$ is compact, they are compact and nonempty.

Definition for flows

Given a real dynamical system (T, X, f) with flow $\varphi:\mathbb{R}\times X\to X$ , a point x and an orbit ? through x, we call a point y an ?-limit point of ? if there exists a sequence $(t_n)_{n \in \mathbb{N}}$ in R so that

$\lim_{n \to \infty} t_n = \infty$

$\lim_{n \to \infty} \varphi(t_n, x) = y$ .

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

$\lim_{n \to \infty} t_n = -\infty$

$\lim_{n \to \infty} \varphi(t_n, x) = y$ .

The set of all ?-limit points (a-limit points) for a given orbit ? is called ?-limit set (a-limit set) for ? and denoted lim_? ? (lim_a ?).

If the ?-limit set (a-limit set) is disjunct from the orbit ?, that is lim_? ? n ? = Ø (lim_a ? n ? = Ø) , we call lim_? ? (lim_a ?) a ?-limit cycle (a-limit cycle).

Alternatively the limit sets can be defined as

$\lim_\omega \gamma := \bigcap_{n\in \mathbb{R}}\overline{\{\varphi(x,t):t>n\}}$

and

$\lim_\alpha \gamma := \bigcap_{n\in \mathbb{R}}\overline{\{\varphi(x,t):t<n\}}.$

Examples

For any periodic orbit ? of a dynamical system, lim_? ? = lim_a = ?

Properties

lim_? ? and lim_a ? are closed
if X is compact then lim_? ? and lim_a ? are nonempty, compact and simply connected
lim_? ? and lim_a ? are f-invariant, that is f(R × lim_? ?) = lim_? ? and f(R × lim_a ?) = lim_a ?

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Limmern lake	Limmernsee	Limmie & the Family Cookin
Limmie & the Family Cookin'	Limmie & the Family Cooking	Limmie Funk Unlimited
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Limmie and the Family Cookin'	Limmie and the Family Cooking	Limmu
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Limmu Sakka	Limmud	Limmy
Limn	Limnades	Limnades Caza
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Limnochorion (Achaia), Greece	Limnochromis	Limnochromis abeelei
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Limnohorio (Achaia), Greece	Limnohorion (Achaia), Greece	Limnohóri (Achaía), Greece
Limnohório (Achaía), Greece	Limnohórion (Achaía), Greece	Limnokhori (Achaia), Greece
Limnokhorio (Achaia), Greece	Limnokhorion (Achaia), Greece	Limnological

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Contents

Definition for iterated functions

Definition for flows

Examples

Properties

See also

Index Of Related Pages