Editing Butterfly effect (section)

==Theory and mathematical definition==
{{See also|Chaos theory#Lorenz's pioneering contributions to chaotic modeling}}

[[File:Lorenz attractor yb.svg|thumb|upright=1.25|A plot of Lorenz' [[strange attractor]] for values ρ=28, σ = 10, β = 8/3. The butterfly effect or sensitive dependence on initial conditions is the property of a [[dynamical system]] that, starting from any of various arbitrarily close alternative [[initial condition]]s on the attractor, the [[iteration#Mathematics|iterated points]] will become arbitrarily spread out from each other.]]

[[Poincaré recurrence theorem|Recurrence]], the approximate return of a system toward its initial conditions, together with sensitive dependence on initial conditions, are the two main ingredients for chaotic motion. They have the practical consequence of making [[complex system]]s, such as the [[weather]], difficult to predict past a certain time range (approximately a week in the case of weather) since it is impossible to measure the starting atmospheric conditions completely accurately.

A [[dynamical system]] displays sensitive dependence on initial conditions if points arbitrarily close together separate over time at an exponential rate. The definition is not topological, but essentially metrical. Lorenz<ref name=":5">{{cite book |last=Lorenz |first=Edward N. |url=https://www.worldcat.org/oclc/56620850 |title=The essence of chaos |date=1993 |publisher=UCL Press |isbn=0-203-21458-7 |location=London |oclc=56620850}}</ref> defined sensitive dependence as follows:

''The property characterizing an orbit (i.e., a solution) if most other orbits that pass close to it at some point do not remain close to it as time advances.''

If ''M'' is the [[state space (dynamical system)|state space]] for the map <math>f^t</math>, then <math>f^t</math> displays sensitive dependence to initial conditions if for any x in ''M'' and any δ&nbsp;>&nbsp;0, there are y in ''M'', with distance {{math|''d''(. , .)}} such that <math>0 < d(x, y) < \delta </math> and such that
:<math>d(f^\tau(x), f^\tau(y)) > \mathrm{e}^{a\tau} \, d(x,y)</math>

for some positive parameter ''a''. The definition does not require that all points from a neighborhood separate from the base point ''x'', but it requires one positive [[Lyapunov exponent]]. In addition to a positive Lyapunov exponent, boundedness is another major feature within chaotic systems.<ref>{{cite book |last=W. |first=Jordan, Dominic |url=https://www.worldcat.org/oclc/772641393 |title=Nonlinear ordinary differential equations: an introduction for scientists and engineers |date=2011 |publisher=Oxford Univ. Press |isbn=978-0-19-920825-8 |oclc=772641393}}</ref>

The simplest mathematical framework exhibiting sensitive dependence on initial conditions is provided by a particular parametrization of the [[logistic map]]:
:<math>x_{n+1} = 4 x_n (1-x_n) , \quad 0\leq x_0\leq 1,</math>

which, unlike most chaotic maps, has a [[closed-form solution]]:
:<math>x_{n}=\sin^{2}(2^{n} \theta \pi)</math>

where the [[initial condition]] parameter <math>\theta</math> is given by <math>\theta = \tfrac{1}{\pi}\sin^{-1}(x_0^{1/2})</math>. For rational <math>\theta</math>, after a finite number of [[iterated function|iterations]] <math>x_n</math> maps into a [[periodic point|periodic sequence]]. But [[almost all]] <math>\theta</math> are irrational, and, for irrational <math>\theta</math>, <math>x_n</math> never repeats itself – it is non-periodic. This solution equation clearly demonstrates the two key features of chaos – stretching and folding: the factor 2<sup>''n''</sup> shows the exponential growth of stretching, which results in sensitive dependence on initial conditions (the butterfly effect), while the squared sine function keeps <math>x_n</math> folded within the range&nbsp;[0,&nbsp;1].