Editing Logistic map (section)

{{short description|Simple polynomial map exhibiting chaotic behavior}}
{{For|the sigmoid curve|Logistic function}}
[[File:LogisticCobwebChaos.gif|class=skin-invert-image|thumb|284x284px|The behavior of the logistic map is shown in [[Cobweb plot]] form. The animation shows the change in behavior as the parameter (r in the figure) is increased from 1 to 4, starting from an initial value of 0.2.)]]
The '''logistic map''' is a discrete [[dynamical system]] defined by the quadratic [[difference equation]]:
{{NumBlk|:|<math>x_{n+1} = r x_n (1 - x_n),</math>|{{EquationRef|1}}}}
Equivalently it is a [[recurrence relation]] and a [[polynomial]] [[map (mathematics)|mapping]] of [[Quadratic function|degree&nbsp;2]]. It is often referred to as an archetypal example of how complex, [[chaos theory|chaotic]] behaviour can arise from very simple [[Nonlinearity|nonlinear]] dynamical equations.

The map was initially utilized by [[Edward Norton Lorenz|Edward Lorenz]] in the 1960s to showcase properties of irregular solutions in climate systems.<ref>{{Cite journal |last=Lorenz |first=Edward N. |date=1964-02-01 |title=The problem of deducing the climate from the governing equations |url=https://a.tellusjournals.se/articles/10.3402/tellusa.v16i1.8893|journal=Tellus |volume=16 |issue=1 |pages=1–11 |doi=10.1111/j.2153-3490.1964.tb00136.x |bibcode=1964Tell...16....1L |issn=0040-2826}}</ref> It was popularized in a 1976 paper by the biologist [[Robert May, Baron May of Oxford|Robert May]],<ref name=":2" group="May, Robert M. (1976)">{{harvnb|May|1976}}</ref> in part as a discrete-time demographic model analogous to the [[Logistic function#Logistic differential equation|logistic equation]] written down by [[Pierre François Verhulst]].<ref>{{MathWorld | urlname=LogisticEquation | title= Logistic Equation}}</ref>
Other researchers who have contributed to the study of the logistic map include [[Stanisław Ulam]], [[John von Neumann]], [[Pekka Myrberg]], [[Oleksandr Sharkovsky]], [[Nicholas Metropolis]], and [[Mitchell Feigenbaum]].<ref>see [[#Research_history|Research history]]</ref>
{{Citation needed|reason=Ideally also Add one citation per author|date=December 2024}}