Editing Logistic map (section)

== Two introductory examples ==
=== Dynamical Systems example ===
{{see also|Logistic_map#Characterization_of_the_logistic_map|label 1=Characterization of the logistic map}}
In the logistic map, x is a variable, and r is a parameter. It is a [[map (mathematics)|map]] in the sense that it maps a configuration or [[phase space]] to itself (in this simple case the space is one dimensional in the variable x)

<math display="block">{\displaystyle f:x\mapsto ax(1-x)}</math>

It can be interpreted as a tool to get next position in the configuration space after one time step. The difference equation is a discrete version of the [[Logistic function#Logistic differential equation|logistic differential equation]], which can be compared to a time evolution equation of the system. 

Given an appropriate value for the parameter r and performing calculations starting from an initial condition <math>x_0</math>, we obtain the sequence <math>x_0</math>, <math>x_1</math>, <math>x_2</math>, .... which can be interpreted as a sequence of time steps in the evolution of the system. 

In the field of [[dynamical systems]], this sequence is called an [[Orbit (dynamics)|orbit]], and the orbit changes depending on the value given to the parameter. When the parameter is changed, the orbit of the logistic map can change in various ways, such as settling on a single value, repeating several values periodically, or showing [[Aperiodic|non-periodic]] fluctuations known as [[Chaos theory|chaos]].<ref name=":1" group="Devaney 1989">{{harvnb|Devaney|1989|p=27}}</ref><ref name="Gleick"/>

Another way to understand this [[sequence]] is to iterate the logistic map (here represented by <math>f(x)</math>) to the initial state <math>x_0</math><ref name=":3" group="Devaney 1989">{{harvnb|Devaney|1989|p=2}}</ref>

<math display="block">
\begin{aligned}
x_{1}&=f(x_{0})\\
x_{2}&=f(x_1)=f(f(x_{0}))\\
x_{3}&=f(x_2)=f(f(f(x_{0})))\\
x_{4}&=...\\
\end{aligned}
</math>

Now this is important given this was the initial approach of [[Henri Poincaré]] to study [[dynamical systems]] and ultimately chaos starting from the study of [[fixed point (mathematics)|fixed points]]  or in other words states that do not change over time (i.e. when <math>x_n=...=x_1=x_0=f(x_0)</math>). Many chaotic systems such as the [[Mandelbrot set]] emerge from iteration of very simple quadratic nonlinear functions such as the logistic map.<ref>{{cite book|first=Benoit B. |last=Mandelbrot |author-link=Benoit Mandelbrot| year=2004|title= Fractals and Chaos, The Mandelbrot Set and Beyond |doi=10.1007/978-1-4757-4017-2 |isbn=978-1-4419-1897-0 |url=https://link.springer.com/book/10.1007/978-1-4757-4017-2 |pages=259–267}}</ref>

=== Demographic model example ===
Taking the biological [[Lotka–Volterra equations|population model]] as an example {{mvar|x<sub>n</sub>}} is a number between zero and one, which represents the ratio of existing [[population]] to the [[carrying capacity|maximum possible population]]. <ref name=":1" group="May, Robert M. (1976)" >{{harvnb|May|1976|loc=formula 2 and 3}}</ref>
This nonlinear difference equation is intended to capture two effects:
* ''reproduction'', where the population will increase at a rate [[Proportionality (mathematics)|proportional]] to the current population when the population size is small,
* ''starvation'' (density-dependent mortality), where the [[Population growth|growth rate]] will decrease at a rate proportional to the value obtained by taking the theoretical "carrying capacity" of the environment less the current population.

The usual values of interest for the parameter {{mvar|r}} are those in the interval {{math|[0,&nbsp;4]}}, so that {{mvar|x<sub>n</sub>}} remains bounded on {{math|[0,&nbsp;1]}}. The {{math|''r'' {{=}} 4}} case of the logistic map is a nonlinear transformation of both the [[bit-shift map]] and the {{math|''μ'' {{=}} 2}} case of the [[tent map]]. If {{math|''r'' > 4}}, this leads to negative population sizes. (This problem does not appear in the older [[Ricker model]], which also exhibits chaotic dynamics.) One can also consider values of {{mvar|r}} in the interval {{math|[−2,&nbsp;0]}}, so that {{mvar|x<sub>n</sub>}} remains bounded on {{math|[−0.5,&nbsp;1.5]}}.<ref name="Takashi Tsuchiya, Daisuke Yamagishi, 1997"/>

{{expand Japanese}}
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