Editing Attractor (section)

== Mathematical definition ==
Let <math>t</math> represent time and let <math>f(t,\cdot)</math> be a function which specifies the dynamics of the system. That is, if <math>a</math> is a point in an <math>n</math>-dimensional phase space, representing the initial state of the system, then <math>f(0,a)=a</math> and, for a positive value of <math>t</math>, <math>f(t,a)</math> is the result of the evolution of this state after <math>t</math> units of time. For example, if the system describes the evolution of a free particle in one dimension then the phase space is the plane <math>\R^2</math> with coordinates <math>(x,v)</math>, where <math>x</math> is the position of the particle, <math>v</math> is its velocity, <math>a=(x,v)</math>, and the evolution is given by

[[File:Julia immediate basin 1 3.png|right|thumb|Attracting period-3 cycle and its immediate basin of attraction for a certain parametrization of the [[Julia set]], which [[Complex quadratic polynomial#iteration|iterates]] the function ''f''(''z'')&nbsp;=&nbsp;''z''<sup>2</sup>&nbsp;+&nbsp;''c''. The three darkest points are the points of the 3-cycle, which lead to each other in sequence, and iteration from any point in the basin of attraction leads to (usually asymptotic) convergence to this sequence of three points.]]

: <math> f(t,(x,v))=(x+tv,v).\ </math>

An attractor is a [[subset]] <math>A</math> of the [[phase space]] characterized by the following three conditions:
* <math>A</math> is ''forward invariant'' under <math>f</math>: if <math>a</math> is an element of <math>A</math> then so is <math>f(t,a)</math>, for all <math>t>0</math>.
* There exists a [[Neighbourhood (mathematics)|neighborhood]] of <math>A</math>, called the ''basin of attraction'' for <math>A</math> and denoted <math>B(A)</math>, which consists of all points <math>b</math> that "enter" <math>A</math> in the limit <math>t\to\infty</math>. More formally, <math>B(A)</math> is the set of all points <math>b</math> in the phase space with the following property:
:: For any open neighborhood <math>N</math> of <math>A</math>, there is a positive constant <math>T</math> such that <math>f(t,b)\in N</math> for all real <math>t>T</math>.
* There is no proper (non-empty) subset of <math>A</math> having the first two properties.

Since the basin of attraction contains an [[open set]] containing <math>A</math>, every point that is sufficiently close to <math>A</math> is attracted to <math>A</math>. The definition of an attractor uses a [[metric space|metric]] on the phase space, but the resulting notion usually depends only on the topology of the phase space. In the case of <math>\R^n</math>, the Euclidean norm is typically used.

Many other definitions of attractor occur in the literature. For example, some authors require that an attractor have positive [[measure (mathematics)|measure]] (preventing a point from being an attractor), others relax the requirement that <math>B(A)</math> be a neighborhood.<ref>{{cite journal | author=John Milnor | author-link=John Milnor | title= On the concept of attractor | journal=Communications in Mathematical Physics | year=1985 | volume=99 | pages=177–195| doi= 10.1007/BF01212280 | issue=2| bibcode=1985CMaPh..99..177M | s2cid=120688149 }}</ref>