Limit set

.com .net .org .info .mobi
.biz .us .co.uk .in
.eu .ws .bz .cc .tv Etc.
Domain Names

Website Development
Web Hosting
Email Hosting
Digital Certificate Etc.

@ Best Prices From

www.DomainsUAE.com

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 ?

Limit set	Limit state	Limit state design
Limit states	Limit states design	Limit stop
Limit superior	Limit superior and limit inferior	Limit superior of sets
Limit switch	Limit theorem	Limit theorems
Limit topology	Limitanei	Limitation
Limitation Act	Limitation Act 1980	Limitation and revocation procedures
Limitation and revocation procedures before the EPO	Limitation and revocation procedures before the European Patent Office	Limitation clause
Limitation of actions	Limitation of religious freedom in Saudi Arabia	Limitation of size
Limitation or revocation procedure before the European Patent Office	Limitation period	Limitation procedure before the European Patent Office
Limitations	Limitations, Statutes of	Limitations and exceptions to copyright
Limitations clause	Limitations on exclusive rights: Computer programs	Limitchik
Limite	Limite (film)	Limite 21
Limited	Limited-Liability Limited Partnership	Limited-access
Limited-access highway	Limited-access road	Limited-animation
Limited-edition candy	Limited-liability	Limited-liability company
Limited-overs cricket	Limited-run series	Limited-slip differential
Limited-stop	Limited (disambiguation)	Limited (solitaire)
Limited 2	Limited Brands	Limited Brands, Inc.
Limited Brands Inc.	Limited Brands Incorporated	Limited By Guarantee
Limited Company	Limited Distance Modem	Limited Duty Officer
Limited Edition	Limited Edition (The Concretes EP)	Limited Edition (album)
Limited Edition - EP	Limited Edition Art Print	Limited Edition Bonus EP
Limited Edition Bonus EP (Given away with first release of FEOS)	Limited Edition Live 12-6-2000 (album)	Limited Edition Tour CD
Limited Edition Tour EP	Limited Edition Tour EP (Phantom Planet album)	Limited Edition book
Limited Edition books	Limited Edition candy	Limited Editions 1990-1994
Limited Editions 1990-94	Limited English Proficiency	Limited English proficiency
Limited Excess	Limited Express	Limited Express (Has Gone?)
Limited Express (has gone?)	Limited Forum	Limited Geography Model (Book of Mormon)
Limited Good	Limited Inc	Limited Inc.
Limited Liability	Limited Liability Act 1855	Limited Liability Company
Limited Liability Partnership	Limited Liability Partnerships Act (Northern Ireland) 2002	Limited Liability Partnerships Act 2000
Limited Live & Rare	Limited Live And Rare	Limited Live and Rare
Limited Offensive Unit	Limited Overs International	Limited Papers
Limited Partnership	Limited Price Indexation	Limited Public Forum
Limited Series	Limited Slip Differential	Limited Systems Vehicle
Limited Test Ban Treaty	Limited Too	Limited Tour EP
Limited Voting	Limited access	Limited access highway
Limited access highways	Limited access road	Limited animation
Limited appearance	Limited atonement	Limited availability
Limited companies	Limited company	Limited depravity
Limited disclosure	Limited donor information	Limited duty officer
Limited duty officers	Limited edition	Limited edition art print
Limited edition book	Limited edition books	Limited editions
Limited energy source	Limited english proficiency	Limited express
Limited first strike	Limited forum	Limited function
Limited function (mathematics)	Limited geography model	Limited geography model (Book of Mormon)
Limited good	Limited government	Limited hang-out
Limited hangout	Limited jurisdiction	Limited liability
Limited liability companies	Limited liability company	Limited liability corporation
Limited liability limited partnership	Limited liability organization	Limited liability partnership
Limited liability partnerships	Limited liabilty company	Limited monarchy
Limited overs	Limited overs cricket	Limited partner
Limited partners	Limited partnership	Limited partnerships
Limited public figure	Limited public forum	Limited purpose public figure
Limited release	Limited resource	Limited run manufacturing
Limited series	Limited series (comics)	Limited slip diff
Limited slip differential	Limited stop	Limited suffrage
Limited symptom attack	Limited vote	Limited voting
Limited war	Limiter	Limiter circuit
Limiting	Limiting Factor	Limiting amino acid
Limiting cases of Apollonius' problem	Limiting density of discrete points	Limiting factor
Limiting magnitude	Limiting molar conductivity	Limiting nutrient
Limiting point	Limiting reactant	Limiting reactants
Limiting reagent	Limiting recursive	Limiting resource
Limiting value	Limiting velocity	Limitless
Limitrophe Island	Limits	Limits.h
Limits (BDSM)	Limits (collection)	Limits (novel)
Limits and colimits	Limits in Kingdom Hearts II	Limits of Liberty
Limits of integration	Limits of the Alps	Limits of the continents
Limits on Depreciation Deduction	Limits on Marriage Amendment	Limits on Marriage amendment
Limits on marriage amendment	Limits to Arbitrage	Limits to Growth
Limits to arbitrage	Limits to computation	Limits to growth
Limité magazine	Limkokwing University College of Creative Technology	Limkokwing University Of Creative Technology
Limkokwing University of Creative Technology	Limla	Limlight
Limliht	Limlim:Example	Limlîht
Limma	Limmared	Limmat
Limmel	Limmen	Limmerance
Limmeren lake	Limmerence	Limmerensee
Limmerick	Limmern lake	Limmernsee
Limmie & the Family Cookin	Limmie & the Family Cookin'	Limmie & the Family Cooking
Limmie Funk Unlimited	Limmie Snell	Limmie and The Family Cookin'
Limmie and the Family Cookin	Limmie and the Family Cookin'	Limmie and the Family Cooking
Limmu	Limmu-Ennarea	Limmu (woreda)
Limmu Kosa	Limmu Sakka	Limmud
Limmy	Limn	Limnades
Limnades Caza	Limnaea alfredi hamiltoni	Limnaea ampulla
Limnaea arguta	Limnaea atomentosa	Limnaea tomentosa
Limnanthaceae	Limnanthes	Limnanthes alba
Limnanthes douglasii	Limnanthes vinculans	Limnatides
Limnea Museum	Limned	Limnephilidae
Limnephiloidea	Limner	Limnes
Limnes Lasithi	Limnetic zone	Limni
Limni, Euboea	Limni (Euboea), Greece	Limni (Evia), Greece
Limnia	Limnic	Limnic Eruption
Limnic eruption	Limnichthyidae	Limnichthys polyactis
Limnichthys rendahli	Limnidi	Limning
Limniris	Limnitis	Limnocharitaceae
Limnochori	Limnochori, Achaea	Limnochori (Achaia), Greece
Limnochorio (Achaia), Greece	Limnochorion (Achaia), Greece	Limnochromis
Limnochromis abeelei	Limnochromis auritus	Limnochromis staneri
Limnochóri (Achaía), Greece	Limnochório (Achaía), Greece	Limnochórion (Achaía), Greece
Limnocitrus	Limnocitrus littoralis	Limnoctites
Limnoctites rectirostris	Limnocythere	Limnocythere porphyretica
Limnodophyte	Limnodromus	Limnodromus griseus
Limnodromus scolopaceus	Limnodromus semipalmatus	Limnodynastes
Limnodynastes convexiusculus	Limnodynastes depressus	Limnodynastes dumerilii
Limnodynastes dumerilli	Limnodynastes fletcheri	Limnodynastes interioris
Limnodynastes lignarius	Limnodynastes ornatus	Limnodynastes peroni
Limnodynastes peronii	Limnodynastes salmini	Limnodynastes spenceri
Limnodynastes tasmaniensis	Limnodynastes terraereginae	Limnodynastidae
Limnodynastinae	Limnofregata	Limnogale
Limnogale mergulus	Limnognathia	Limnognathia maerski
Limnohori (Achaia), Greece	Limnohorio (Achaia), Greece	Limnohorion (Achaia), Greece
Limnohóri (Achaía), Greece	Limnohório (Achaía), Greece	Limnohórion (Achaía), Greece
Limnokhori (Achaia), Greece	Limnokhorio (Achaia), Greece	Limnokhorion (Achaia), Greece

Previous page (Limekiller) | Next page (Limnological)


BUILD YOUR WEB SITE WITH www.DomainsUAE.com

managed by www.butterflyportal.com

Contents

Definition for iterated functions

Definition for flows

Examples

Properties

See also

Index Of Related Pages