formulasearchengine - User contributions [en]

Foucault pendulum

2015-01-01T08:55:46Z

123.2.117.167: /* References */

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2014-02-28T17:16:41Z

123.2.117.167: /* Atari Inc. (1972–1984) */

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Available energy (particle collision)

2013-07-31T08:29:28Z

123.2.81.102: /* Derivation */ Changed 'simplifying' to 'expanding' - the term was used in the wrong context.

In [[mathematics]], especially in the study of [[dynamical system]]s, 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.

==Types ==
* [[Fixed point (mathematics)|fixed point]]s
* [[periodic orbit]]s
* [[limit cycle]]s
* [[attractor]]s.

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

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

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

where <math>\overline{S}</math> denotes the ''closure'' of set <math>S</math>. 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 point]]s'''). This may also be formulated as the outer limit ([[limsup]]) of a sequence of sets, such that

:<math>\omega(x,f) = \bigcap_{n=1}^\infty \overline{\bigcup_{k=n}^\infty \{f^k(x)\}}.</math>

If <math>f</math> is a [[homeomorphism]] (that is, a bicontinuous bijection), then the <math>\alpha</math>-limit set is defined in a similar fashion, but for the backward orbit; ''i.e.'' <math>\alpha(x,f)=\omega(x,f^{-1})</math>.

Both sets are <math>f</math>-invariant, and if <math>X</math> is [[compact space|compact]], they are compact and nonempty.

==Definition for flows==
Given a [[real dynamical system]] (''T'', ''X'', φ) with [[flow (mathematics)|flow]] <math>\varphi:\mathbb{R}\times X\to X</math>, a point ''x'' and an [[orbit (dynamics)|orbit]] γ through ''x'', we call a point ''y'' an ω-'''limit point''' of γ if there exists a sequence <math>(t_n)_{n \in \mathbb{N}}</math> in '''R''' so that
:<math>\lim_{n \to \infty} t_n = \infty</math>
:<math>\lim_{n \to \infty} \varphi(t_n, x) = y </math>.

Analogously we call ''y'' an α-'''limit point''' if there exists a sequence <math>(t_n)_{n \in \mathbb{N}}</math> in '''R''' so that
:<math>\lim_{n \to \infty} t_n = -\infty</math>
:<math>\lim_{n \to \infty} \varphi(t_n, x) = y </math>.

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 limω γ (limα γ).

If the ω-limit set (α-limit set) is disjoint from the orbit γ, that is limω γ ∩ γ = ∅ (limα γ ∩ γ = ∅), we call limω γ (limα γ) a '''[[ω-limit cycle]]''' ('''[[α-limit cycle]]''').

Alternatively the limit sets can be defined as
:<math>\lim_\omega \gamma := \bigcap_{n\in \mathbb{R}}\overline{\{\varphi(x,t):t>n\}} </math>
and
:<math>\lim_\alpha \gamma := \bigcap_{n\in \mathbb{R}}\overline{\{\varphi(x,t):t<n\}}.</math>

=== Examples ===
* For any [[periodic orbit]] γ of a dynamical system, limω γ = limα γ = γ
* For any [[Fixed point (mathematics)|fixed point]] <math>x_0</math> of a dynamical system, limω <math>x_0</math> = limα <math>x_0</math> = <math>x_0</math>

=== Properties ===
* limω γ and limα γ are [[closed set|closed]]
* if ''X'' is compact then limω γ and limα γ are [[nonempty]], [[compact set|compact]] and [[Connected space|connected]]
* limω γ and limα γ are φ-invariant, that is φ('''R''' × limω γ) = limω γ and φ('''R''' × limα γ) = limα γ

==See also==
* [[Julia set]]
* [[Stable manifold|Stable set]]
* [[Limit cycle]]
* [[Periodic point]]
* [[Non-wandering set]]
* [[Kleinian group]]

==References==
* {{cite book| last = Teschl| given = Gerald|authorlink=Gerald Teschl| title = Ordinary Differential Equations and Dynamical Systems| publisher=[[American Mathematical Society]]| place = [[Providence, Rhode Island|Providence]]| year = 2012| isbn= 978-0-8218-8328-0| url = http://www.mat.univie.ac.at/~gerald/ftp/book-ode/}}

{{PlanetMath attribution|id=4316|title=Omega-limit set}}

[[Category:Limit sets| ]]

List of Selby characters

2013-07-12T08:39:25Z

123.2.17.157: /* Inventions of Dr. Trifle */

A '''max-plus algebra''' is a [[semiring]] over the union of [[real number]]s and ε = <math>-\infty</math>, equipped with maximum and addition as the two binary operations.
It can be used appropriately to determine marking times within a given [[Petri net]] and a vector filled with marking state at the beginning.

==Operators==
=== Scalar operations ===

Let ''a'' and ''b'' be real scalars or ε. Then the operations maximum (implied by the '''max''' operator <math> \oplus</math>) and addition ('''plus''' operator <math> \otimes</math>) for these scalars are defined as

: <math>a \oplus b = \max(a,b) </math>
: <math>a \otimes b = a + b</math>

'''Watch''': Max-operator <math>\oplus</math> can easily be confused with the addition operation. Similar to the conventional algebra, all <math>\otimes</math> - operations have a higher [[Order of operations|precedence]] than <math>\oplus</math> - operations.

===Matrix operations===
Max-plus algebra can be used for matrix operands '''A''', '''B''' likewise, where the size of both matrices is the same. To perform the '''A''' <math>\oplus</math> '''B''' - operation, the elements of the resulting matrix at (row i, column j) have to be set up by the maximum operation of both corresponding elements of the matrices '''A''' and '''B''':

: <math>[A \oplus B]_{ij} = [A]_{ij} \oplus [B]_{ij} = \max([A]_{ij} , [B]_{ij})</math>

The <math> \otimes</math> - operation is similar to the algorithm of [[Matrix multiplication]], however, every "+" calculation has to be substituted by an <math>\oplus</math> - operation and every "<math>\cdot</math>" calculation by a <math>\otimes</math> - operation. More precisely, to perform the '''A''' <math>\otimes</math> '''B''' - operation, where '''A''' is a ''m''×''p'' matrix and '''B''' is a ''p''×''n'' matrix, the elements of the resulting matrix at (row i, column j) are determined by matrices '''A''' (row i) and '''B''' (column j):

: <math>[A \otimes B]_{ij} = \bigoplus_{k = 1}^p [A]_{ik} \otimes [B]_{kj} = \max([A]_{i1} + [B]_{1j}, \dots, [A]_{ip} + [B]_{pj})</math>

==Useful enhancement elements==
In order to handle marking times like <math>-\infty</math> which means "never before", the ε-element has been established by ε<math>=-\infty</math>. According to the idea of infinity, the following equations can be found:

: ε <math>\oplus</math> a = a
: ε <math>\otimes</math> a = ε
To point the zero number out, the element ''e'' was defined by <math>e=0</math>. Therefore:

: ''e'' <math>\otimes</math> ''a'' = ''a''

Obviously, ε is the neutral element for the <math>\oplus</math> - operation, as ''e'' is for the <math>\otimes</math> - operation

==Algebra properties==
*associativity:

:: <math>(a \oplus b) \oplus c = a \oplus (b \oplus c) </math>

:: <math>(a\otimes b) \otimes c = a \otimes (b \otimes c) </math>

*commutativity :

:: <math>a \oplus b = b \oplus a </math>

:: <math>a \otimes b = b \otimes a </math>

*distributivity:

:: <math> (a \oplus b) \otimes c = a \otimes c \oplus b \otimes c </math>

==See also==
* [[Tropical geometry]]

==Additional reading==
*{{citation|title=Max-linear Systems: Theory and Algorithms|first=Peter|last=Butkovič|series=Springer Monographs in Mathematics|doi=10.1007/978-1-84996-299-5|publisher=Springer-Verlag|year=2010}}

==External links==
*http://maxplus.org
*http://amadeus.inria.fr/gaubert/maxplus.html

[[Category:Algebras]]

[[fr:Algèbre max-plus]]