Editing Attractor (section)

==Attractors characterize the evolution of a system==

[[File:Logistic Map Bifurcation Diagram, Matplotlib.svg|350px|thumb|right|Bifurcation diagram of the [[logistic map]]. The attractor(s) for any value of the parameter <math>r</math> are shown on the ordinate in the domain <math>0<x<1</math>. The colour of a point indicates how often the point <math>(r, x)</math> is visited over the course of 10<sup>6</sup> iterations: frequently encountered values are coloured in blue, less frequently encountered values are yellow. A [[period-doubling bifurcation|bifurcation]] appears around <math>r\approx3.0</math>, a second bifurcation (leading to four attractor values) around <math>r\approx3.5</math>. The behaviour is increasingly complicated for <math>r>3.6</math>, interspersed with regions of simpler behaviour (white stripes).]]

The parameters of a dynamic equation evolve as the equation is iterated, and the specific values may depend on the starting parameters.  An example is the well-studied [[logistic map]],  <math>x_{n+1}=rx_n(1-x_n)</math>, whose basins of attraction for various values of the parameter <math>r</math> are shown in the figure. If <math>r=2.6</math>, all starting <math>x</math> values of <math>x<0</math> will rapidly lead to function values that go to negative infinity; starting <math>x</math> values of <math>x>1</math> will also go to negative infinity. But for <math>0<x<1</math> the <math>x</math> values rapidly converge to <math>x\approx0.615</math>, i.e. at this value of <math>r</math>, a single value of <math>x</math> is an attractor for the function's behaviour. For other values of <math>r</math>, more than one value of <math>x</math> may be visited: if <math>r</math> is 3.2, starting values of <math>0<x<1</math> will lead to function values that alternate between <math>x\approx0.513</math> and <math>x\approx0.799</math>. At some values of <math>r</math>, the attractor is a single point (a [[#Fixed_point|"fixed point"]]), at other values of <math>r</math> two values of <math>x</math> are visited in turn (a [[period-doubling bifurcation]]), or, as a result of further doubling, any number <math>k\times 2^n</math> values of <math>x</math>; at yet other values of <math>r</math>, any given number of values of <math>x</math> are visited in turn; finally, for some values of <math>r</math>, an infinitude of points are visited. Thus one and the same dynamic equation can have various types of attractors, depending on its parameters.