Editing Logistic map (section)

== Characterization of the logistic map ==
[[File:Logistic map with parameter from 0.02 to 4 t from 0 to 200.gif|class=skin-invert-image|thumb|284x284px|The sequence behaviour from r=0.02 to r=4, one can visualize the horizontal coordinate as time, and the vertical coordinate either as a position in space at time t or as the population size at time t]]
The animation shows the behaviour of the sequence <math>x_n</math> over different values of the parameter r. A first observation is that the sequence does not diverge and remains finite for r between 0 and 4. It is possible to see the following qualitative phenomena in order of time: 
* exponential convergence to zero
* convergence to a non-zero fixed value (see [[Exponential function]] or [[Characterizations of the exponential function#Characterizations|Characterizations of the exponential function]] point 4)
* initial oscillation and then convergence (see [[Damping]] and [[Harmonic oscillator#Damped harmonic oscillator|Damped harmonic oscillator]])
* stable oscillations between two values (see [[Resonance]] and [[Harmonic oscillator#Simple harmonic oscillator|Simple harmonic oscillator]])
* growing oscillations between a set of values which are multiples of two such as 2,4,8,16 etc. (see [[Period-doubling bifurcation]])
* [[Intermittency]] (i.e. sprouts of oscillations at the onset of chaos)
* fully developed [[Chaos theory|chaotic oscillations]]
* [[topological mixing]] (i.e. the tendency of oscillations to cover the full available space).

The first four are also available in standard [[linear systems]], oscillations between two values are available too under [[resonance]], chaotic systems though have typically a large range of resonance conditions.
The other phenomena are peculiar to [[Chaos theory|chaos]]. This progression of stages is strikingly similar to the onset of [[turbulence]].
Chaos is not peculiar to non-linear systems alone and it can also be exhibited by infinite dimensional linear systems.<ref>{{cite journal
 |last1=Bonet |first1=J. |last2=Martínez-Giménez |first2=F. |last3=Peris |first3=A.
 |year=2001
 |title=A Banach space which admits no chaotic operator
 |journal=Bulletin of the London Mathematical Society
 |volume=33
 |issue=2 |pages=196–198
 |doi=10.1112/blms/33.2.196
|s2cid=121429354 }}</ref>

As mentioned above, the logistic map itself is an ordinary quadratic function.<!--[ 15 ]--> An important question in terms of dynamical systems is how the behavior of the trajectory changes when the parameter {{mvar|r}} changes.<!--[ 12 ]--> Depending on the value of {{mvar|r}}, the behavior of the trajectory of the logistic map can be simple or complex.<ref name=":1" group="Thompson & Stewart">{{harvnb|Thompson|Stewart|2002|p=162}}</ref> Below, we will explain how the behavior of the logistic map changes as {{mvar|r}} increases.

=== Domain, graphs and fixed points ===
[[File:Graph of logistic map.png|class=skin-invert-image|thumb|Graph of the logistic map (the relationship between <math>x_{n +1}</math> and <math>x_n</math>). The graph has the shape of a parabola, and the vertex of the parabola changes as the parameter r changes.]]
As mentioned above, the logistic map can be used as a model to consider the fluctuation of population size. In this case, the variable x of the logistic map is the number of individuals of an organism divided by the maximum population size, so the possible values of x are limited to 0 ≤ x ≤ 1. For this reason, the behavior of the logistic map is often discussed by limiting the range of the variable to the interval [0, 1].<ref name=":1" group="Hirsch,Smale & Devaney">{{harvnb|Hirsch| Smale|Devaney|2007|pp=344-345}}</ref>

If we restrict the variables to 0 ≤ x ≤ 1, then the range of the parameter r is necessarily restricted to 0 to 4 (0 ≤ r ≤ 4). This is because if <math>x_n</math> is in the range [0, 1], then the maximum value of <math>x_{n+1}</math> is
r/4. Thus, when r > 4, the value of <math>x_{n +1}</math> can exceed 1. On the other hand, when r is negative, x can take negative values.<ref name=":1" group="Hirsch,Smale & Devaney"/>

A graph of the map can also be used to learn much about its behavior.<!--[ 59 ]--> The graph of the logistic map <math>x_{n+1}= r x (1-x_n)</math> is the plane curve that plots the relationship between <math>x_n</math> and <math>x_{n+1}</math>, with <math>x_n</math> (or x) on the horizontal axis and <math>x_{n + 1}</math> (or f (x)) on the vertical axis.<!--[ 60 ]--> The graph  of the logistic map looks like this, except for the case r = 0:

It has the shape of a parabola with a vertex at<ref name=":4" group="Gulick">{{harvnb|Gulick|1995|p=16|loc=Example 3}}</ref><!--[ 61 ]--> 
{{NumBlk|:|<math>{\displaystyle (x_{n},x_{n+1})=\left(0.5,{\frac {r}{4}}\right)}</math>|{{EquationRef|2-1}}}} 
When r is changed, the vertex moves up or down, and the shape of the parabola changes<!--[62]-->. In addition, the parabola of the logistic map intersects with the horizontal axis (the line where <math>x_{n+1} = 0</math>) at two points<!--[63]-->. The two intersection points are <math>(x_n, x_{n + 1}) = (0,0)</math> and <math>(x_n, x_{n + 1}) = (1,0)</math>, and the positions of these intersection points are constant and do not depend on the value of r.<!--[ 63 ]-->
[[File:ロジスティック写像のグラフとクモの巣と不動点.png|class=skin-invert-image|thumb|An example of a spider web projection of a trajectory on the graph of the logistic map, and the locations of the fixed points <math>x_{f1}</math> and <math>x_{f2}</math> on the graph.]]
Graphs of maps, especially those of one variable such as the logistic map, are key to understanding the behavior of the map.<!--[ 64 ]--> One of the uses of graphs is to illustrate fixed points, called points.<!--[ 65 ]--> Draw a line y = x (a 45° line) on the graph of the map. If there is a point where this 45° line intersects with the graph, that point is a fixed point.<!--[ 66 ]--> In mathematical terms, a fixed point is

{{NumBlk|:|<math>f(x)=x</math>|{{EquationRef|2-2}}}}		

It means a point that does not change when the map is applied.<!--[ 67 ]--> We will denote the fixed point as <math>x_f</math>. In the case of the logistic map, the fixed point that satisfies equation (2-2) is obtained by solving <math>rx (1 - x)= x </math>.

{{NumBlk|:|<math>{\displaystyle x_{f1}=0}</math>|{{EquationRef|2-3}}}}
{{NumBlk|:|<math> x_{f2}=1-{\frac {1}{r}}</math>		|{{EquationRef|2-4}}}}
(except for r = 0).<!--[ 68 ]--> The concept of fixed points is of primary importance in discrete dynamical systems.<!--[ 69 ]-->

Another graphical technique that can be used for one-variable mappings is the [[cobweb diagram|spider web]] projection.<!--[ 70 ]--> After determining an initial value <math>x_0</math> on the horizontal axis, draw a vertical line from the initial value <math>x_0</math> to the curve of f(x). Draw a horizontal line from the  point where the curve of f(x) meets the 45° line of y = x, and then draw a vertical line from the point where the curve meets the 45° line to the curve of f(x). By repeating this process, a spider web or staircase-like diagram is created on the plane.<!--[ 71 ]--> This construction is in fact equivalent to calculating the trajectory graphically, and the [[cobweb diagram|spider web diagram]] created represents the trajectory starting from <math>x_0</math>.<!--[ 72 ]--> This projection allows the overall behavior of the trajectory to be seen at a glance.<!--[ 73 ]-->

===Behavior dependent on {{mvar|''r''}}===
The image below shows the [[amplitude]] and [[frequency]] content of a logistic map that iterates itself for parameter values ranging from 2&nbsp;to&nbsp;4. Again one can see  initial linear behaviours then chaotic behaviour not only in the [[time domain]] (left) but especially in the frequency domain or [[spectrum]] (right), i.e. chaos is present at all scales as it is in the case of [[Energy cascade]] of [[Kolmogorov]] and it even propagates from one scale to another.<ref name=":2" group="Thompson & Stewart">{{harvnb|Thompson|Stewart|2002|p=7}}</ref>

[[Image:Logistic map animation.gif|class=skin-invert-image]]

By varying the parameter {{mvar|r}}, the following behavior is observed:

==== Case when 0 ≤ r < 1 ====
First, when the parameter r = 0, <math>x_1 = 0</math>, regardless of the initial value <math>x_0</math>. In other words, the trajectory of the logistic map when a = 0 is a trajectory in which all values after the initial value are 0, so there is not much to investigate in this case.

Next, when the parameter r is in the range 0 < r < 1, <math>x_n</math> decreases monotonically for any value of <math>x_0</math> between 0 and 1. That is, <math>x_n</math> converges to 0 in the limit n → ∞. 
<ref name=":1" group="Gulick">{{harvnb|Gulick|1995|p=36}}</ref> The point to which <math>x_n</math> converges is the fixed point <math>x_{f1}</math> shown in equation (2-3).<!--[ 78 ]--> Fixed points of  this type, where orbits around them converge, are called asymptotically stable, stable, or attractive. Conversely, if orbits around <math>x_f</math> move away from <math>x_f</math> as time n increases, the fixed point <math>x_f</math> is called unstable or repulsive. <ref name=":2" group="Gulick">{{harvnb|Gulick|1995|p=9}}</ref>

[[File:Logistic map cobweb and time evolution a=0.9.png|class=skin-invert-image|thumb|284x284px| Spider plot (left) and time series (n vs. x n) (right) for parameter r = 0.9. The trajectory converges monotonically to 0.]]
A common and simple way to know whether a fixed point is asymptotically stable is to take the derivative of the map f.<ref name=":3" group="Gulick">{{harvnb|Gulick|1995|p=10}}</ref> This derivative is expressed as <math>f'(x)</math>,
<math>x_f</math> is asymptotically stable if the following condition is satisfied.<!--[ 80 ]-->

{{NumBlk|:|<math> \left|f'(x_{f})\right|<1</math>|{{EquationRef|3-1}}}}

[[File:1次写像の不動点と安定性とクモの巣_その2.svg|class=skin-invert-image|thumb|284x284px|Tangent slopes of an asymptotically stable fixed point (left) and an unstable fixed point (right) and the state of the surrounding orbits]]

We can see this by graphing the map: if the slope of the tangent to the curve at <math>x_f</math> is between −1 and 1, then <math>x_f</math> is stable and the orbit around it is attracted to <math>x_f</math>.<!--[ 81 ]--> The derivative of the logistic map is

{{NumBlk|:|<math>f'(x)=r(1-2x)</math>|{{EquationRef|3-2}}}}

Therefore, for x = 0 and 0 < r < 1, 0 < f  '(0) < 1, so the fixed point <math>x_{f1}</math> = 0  satisfies equation (3-1).<!--[ 82 ]-->

However, the discrimination method using equation (3-1) does not know the range of orbits from <math>x_f</math> that are attracted to <math>x_f</math>.<!--[ 83 ]--> It only guarantees that x within a certain neighborhood of <math>x_f</math> will converge.<!--[ 83 ]--> In this case, the domain of initial values that converge to 0 is the entire domain [0, 1], but to know this for certain, a separate study is required.<!--[ 77 ]-->

The method for determining whether a fixed point is unstable can be found by similarly differentiating the map.<!--[ 80 ]--> For r<1 if a fixed point <math>x_f</math> is unstable if

{{NumBlk|:|<math>\left|f'(x_{f})\right|>1</math>|{{EquationRef|3-3}}}}

If the parameter lies in the range 0 < r < 1, then the other fixed point <math>x_{f2} = 1 - 1/a</math>
is negative and therefore does not lie in the range [0, 1], but it does exist as an unstable fixed point.<!--[ 84 ]-->

==== Case when 1 ≤ r ≤ 2 ====
In the general case with {{mvar|r}} between 1 and 2, the population will quickly approach the value {{math|{{sfrac|''r'' − 1|''r''}}}}, independent of the initial population.

[[Image:ロジスティック写像のトランスクリティカル分岐.png|class=skin-invert-image|thumb|center|650px|[[Transcritical bifurcation]] of the logistic map occurring at r = 1. For r < 1, <math>x_{f2}</math> exists outside [0, 1] as an unstable fixed point, but for r = 1, the two fixed points collide, and for r > 1, <math>x_{f2}</math> appears between [0, 1] as a stable fixed point.]]

When the parameter r = 1, the trajectory of the logistic map converges to 0 as before,<!--[ 85 ]--> but the convergence speed is slower at r = 1.<!--[ 86 ]--> The fixed point 0 at r = 1 is asymptotically stable, but does not satisfy equation (3-1).<!--[ 87 ]--> In fact, the discrimination method based on equation (3-1) works by approximating the map to the first order near the fixed point.<!--[ 88 ]--> When r = 1, this approximation does not hold, and stability or instability is determined by the quadratic (square) terms of the map, or in order words the second order perturbation.<!--[ 86 ]-->

When r = 1 is graphed, the curve is tangent to the 45° diagonal at x = 0.<!--[ 62 ]--> In this case, the fixed point <math>x_{f2} = 1 - 1/r</math>, which exists in the negative range for <math>0 < r < 1</math>, is  <math>x_{f2}=0</math>.
For <math>x_{f2} = 0</math>,<!--[ 89 ]--> that is, as r increases, the value of <math>x_{f2}</math> approaches 0, and just at r = 1  , <math>x_{f2}</math> collides with  <math>x_{f1} = 0</math>.<!--[ 89 ]--> This collision gives rise to a phenomenon known as a [[transcritical bifurcation]].<!--[ 90 ]-->
Bifurcation is a term used to describe a qualitative change in the behavior of a dynamical system. In this case, transcritical bifurcation is when the stability of fixed points alternates between each other<!--[ 91]-->. That is, when r is less than 1, <math>x_{f1}</math> is stable and <math>x_{f2}</math> is unstable, but when r is greater than 1, <math>x_{f1}</math> is unstable and <math>x_{f2}</math> is stable.<!--[ 84  ]--> The parameter  values at which bifurcation occurs are called bifurcation points.<!--[ 92 ]--> In this case, r = 1 is the bifurcation point.<!--[ 90 ]-->

<div class=skin-invert-image>{{multiple image
 | align     = center
 | direction = horizontal
 | total_width     = 590
 | image1    = Logistic map cobweb plot a=1.2.png
 | caption1  = Fixed point <math>x_{f2}=1-1/r</math> Example of monotonically decreasing convergence to (r = 1.2, x 0 = 0.6)
 | image2    = Logistic map cobweb plot a=1.8.png
 | caption2  = Fixed point <math>x_{f2}=1-1/a</math> Example of monotonically increasing convergence to (r = 1.8, x 0 = 0.2)
}}</div>

As a result of the bifurcation, the orbit of the logistic map converges to the limit point <math>x_{f2} = 1 - 1/r</math> instead of <math>x_{f1} = 0</math>.<!--[ 93 ]--> In particular, if the parameter <math>1 < r \le 2</math>, then the trajectory starting from a value <math>x_0</math>in the interval (0, 1), exclusive of 0 and 1, converges to <math>x_{f2}</math> by increasing or decreasing monotonically.<!--[ 93 ]--> The difference in the convergence pattern depends on the range of the initial value.<!--[ 94 ]--> <math>0 < x_0 < 1 - 1/r</math>

In the case of <math>1 - 1/r< x_0 < 1/r</math>
Then, it converges monotonically, 
<math>1/r< x_0 < 1</math>, the function converges monotonically except for the first step.<!--[ 94 ]-->

Furthermore, the fixed point <math>x_{f1} = 0</math> becomes unstable due to bifurcation, but continues to exist as a fixed point even after r > 1.<!--[95]--> This does not mean that there is no initial value other than <math>x_{f1}</math> itself that  can reach this unstable fixed point  <math>x_{f1}</math>.<!--[ 96 ]--> This is <math>x_0 = 1</math>, and since the logistic map satisfies f (1) = 0 regardless of the value of r,  applying the map once to <math>x_0 = 1</math> maps it to <math>x_{f1} = 0</math>.<!--[ 54 ]--> A point such as x = 1 that can be reached directly as a fixed point by a  finite number of iterations of the map is called a final fixed point.<!--[ 97 ]-->

==== Case when 2 ≤ r ≤ 3 ====
With {{mvar|r}} between 2 and 3, the population will also eventually approach the same value {{math|{{sfrac|''r'' − 1|''r''}}}}, but first will fluctuate around that value for some time. The [[rate of convergence]] is linear, except for {{math|''r'' {{=}} 3}}, when it is dramatically slow, less than linear (see [[Bifurcation memory]]).

When the parameter 2 < r < 3, except for the initial values 0 and 1, the fixed point <math>x_{f2} = 1 - 1/r</math> is the same as when 1 < r ≤ 2.<!--[ 98 ]-->
However, in this case the convergence is not monotonically.<!--[ 99 ]--> As the variable approaches <math>x_{f2}</math>, it becomes larger and smaller than <math>x_{f2}</math> repeatedly, and follows a convergent trajectory that oscillates around <math>x_{f2}</math>.<!--[ 99 ]-->

<!-- TODO: obscure translation to be fixed, maybe using a ref
Tentative:
In this parameter range, <math>1/2<x_{f2}<1</math>,
The oscillations around the fixed point of the orbit are bounded and given it is an attractor, they are convergent. <ref name=":1" group="Devaney 1989"/>
The value that is mapped to <math>x_{f2}</math> by applying the mapping once is <math>f(\tilde{x}_{f2}) = \tilde{\tilde{x}}_{f2}</math>

Original:
The oscillations around the fixed point of the orbit vary over the following range: In this parameter range, <math>1/2<x_{f2}<1</math><!/--[ 100 ]-->. The value that is mapped to <math>x_{f2}</math> by applying the mapping once is <math>f(\tilde{x}_{f2}) = x_{f2}</math>
-->
<!-- ~x
Let us write it as f  2. That is, <math>f(\tilde{x}_{f 2})= x_{f2}</math> [ 101 ].
~x f  2, x f  2), the orbit starts to oscillate around a fixedpoint [ 102 ].
~x f  2, x f  2)is(x f  2, a/4] to (x f  2, a/4] teeth [ 1/2, x f  2) and oscillates in the form [ 103 ].
-->
[[File:CobwebConstruction.gif|class=skin-invert-image|thumb|Animation of the spider projection at a = 2.8, converging around a fixed point.]]
In general, [[bifurcation diagram]]s are useful for understanding bifurcations.<!--[ 104 ]--> These diagrams are graphs of fixed points (or periodic points, as described below) x as a function of a parameter a, with a on the horizontal axis and x on the vertical axis.<!--[ 104 ]--> To distinguish between stable and unstable fixed points, the former curves are sometimes drawn as solid lines and the latter as dotted lines.<!--[ 105 ]--> When drawing a bifurcation diagram for the logistic map, we have a straight line representing the fixed point <math>x_{f1} = 0</math>  and a straight line representing the fixed point <math>x_{f2} = 1-1/a</math>
It can be seen that the curves representing a and b intersect at r = 1, and that stability is switched between the two.<!--[ 95 ]-->

[[Image:パラメータ0から3までのロジスティック写像の分岐図.png|class=skin-invert-image|thumb|center|480px|Bifurcation diagram of the logistic map for parameters 0 to 3. The blue line represents the fixed point <math>x_{f1} = 0</math>, and the red line represents the fixed point <math>x_{f2} = 1 - 1/r</math> Represents.]]
==== Case when 3 ≤ r ≤ 3.44949 ====
In the general case With {{mvar|r}} between 3 and 1&nbsp;+&nbsp;{{sqrt|6}} ≈ 3.44949 the population will approach permanent oscillations between two values. These two values are dependent on {{mvar|r}} and given by<ref name="Takashi Tsuchiya, Daisuke Yamagishi, 1997"/> <math>x_{\pm}=\frac{1}{2r}\left(r+1\pm\sqrt{(r-3)(r+1)}\right)</math>.

When the parameter is exactly r = 3, the orbit also has a fixed point <math>x_{f2} = 1-1/r</math>.<!--[ 106 ]--> 
However, the variables converge more slowly than when <math>2 < r < 3</math>.<!--[ 107 ]--> When <math>r = 3</math>, the derivative <math>f'(x_{f2})</math> reaches −1 and no longer satisfies equation (3-1).<!--[ 108 ]--> When r exceeds 3, <math>f'(x_{f2})<-1</math>, and <math>x_{f2}</math> becomes  an unstable fixed point.<!--[ 108 ]--> That is, another bifurcation occurs at <math>r = 3</math>.<!--[  108 ]-->

For <math>r = 3</math> a type of bifurcation known as a [[period doubling bifurcation]] occurs.<!--[ 109 ]--> For <math>r > 3</math>, the orbit no longer converges to a single point, but instead alternates between large and small values even after a sufficient amount of time has passed.<!--[109]--> For example, for <math>r = 3.3</math>, the variable alternates between the values 0.4794... and 0.8236....<!--[ 110 ]-->
<div class=skin-invert-image>{{multiple image
 | align     = center
 | direction = horizontal
 | total_width     = 590
 | footer  = Spider diagram and time series for a = 3.3. The orbit is attracted to a stable 2-periodic point.
 | image1    = ロジスティック写像、安定2周期軌道、クモの巣図.png
 | image2    = ロジスティック写像、安定2周期軌道、時系列.png
}}</div>
An orbit that cycles through the same values periodically is called a periodic orbit.<!--[ 111 ]--> In this case, the final behavior of the variable as n → ∞ is a periodic orbit with two periods.<!--[ 112 ]-->  Each value (point) that makes up a periodic orbit is called a periodic point.<!--[ 111 ]--> In the example where a = 3.3, 0.4794... and 0.8236... are periodic points.<!--[ 113 ]--> If a certain x is a periodic point, then in the case of two periodic points, applying the map twice to x will return it to its original state, so
{{NumBlk|:|<math> f(f(x))=f^{2}(x)=x</math>|{{EquationRef|3-4}}}}
<!--[ 114 ]-->
If we apply the logistic map equation (1-4) to this equation, we get
{{NumBlk|:|<math>r^{2}x(1-x)(1-ax(1-x))=x</math>|{{EquationRef|3-5}}}}

This gives us the following fourth-order equation.<!--[ 115 ]--> The solutions of this equation are the periodic points.<!--[ 116 ]--> In fact, there are two fixed points <math>x_{f1} = 0</math> and <math>x_{f2} = 1 - 1/a</math>
also satisfies equation (3-4).<!--[ 117 ]--> Therefore, of the solutions to equation (3-5), two correspond to <math>x_{f1}</math> and <math>x_{f2}</math>, and the remaining two solutions are 2-periodic points.<!--[ 117 ]--> Let the 2-periodic points be denoted as <math>x^{(2)}_{f1}</math> and <math>x^{(2)}_{f2}</math>, respectively. By solving equation (3-5), we can obtain them as follows<!--[ 116 ]-->

{{NumBlk|:|<math> x_{f1}^{(2)},\ x_{f2}^{(2)}={\frac {r+1\pm {\sqrt {(r+1)(r-3)} }}{2r}}</math>|{{EquationRef|3-6}}}}

A similar theory about the stability of fixed points can also be applied to periodic points.<!--[ 114 ]--> That is, a periodic point that attracts surrounding orbits is called an asymptotically stable periodic point, and a periodic point where the surrounding orbits move away is called an unstable periodic point.<!--[ 118 ]--> It is possible to determine the stability of periodic points in the same way as for fixed points.<!--[ 119 ]--> In the general case, consider <math>f^k(x)</math> after k iterations of the map. 
Let <math>(f^k)'(x)</math> be the derivative  <math>df^k(x)/dx</math> of the k-periodic point <math>x^{(k)}_f</math>. If <math>x^{(k)}_f</math> satisfies<!--[ 120 ]-->:
{{NumBlk|:|<math> \left|(f^{k})'(x_{f}^{(k)})\right|<1 </math>|{{EquationRef|3-7}}}}
then <math>x^{(k)}_f</math> is asymptotically stable.
{{NumBlk|:|<math> \left|(f^{k})'(x_{f}^{(k)})\right|>1</math>|{{EquationRef|3-8}}}}
then <math>x^{(k)}_f</math> is unstable.<!--[ 120 ]-->

The above discussion of the stability of periodic points can be easily understood by drawing a graph, just like the fixed points.<!--[ 121 ]--> In this diagram, the horizontal axis is xn and the vertical axis is <math>x_{n+2}</math>, and a curve is drawn that shows the relationship between <math>x_{n+2}</math> and <math>x_n</math>.<!--[ 122 ]--> The intersections of this curve and the 45° line are points that satisfy equation (3-4), so the intersections represent fixed points and 2-periodic points.<!--[122]--> If we draw a graph of the logistic map <math>f^2(x)</math>, we can observe that the slope of the tangent at the fixed point <math>x_{f2}</math> exceeds  1 at the boundary <math>r=3</math> and becomes unstable.<!--[ 122 ]--> At the same time, two new intersections appear, which are the periodic points <math>x^{(2)}_{f1}</math> and <math>x^{(2)}_{f2}</math>.<!--[ 122 ]-->
<div class=skin-invert-image>{{multiple image
 | align     = center
 | direction = horizontal
 | total_width     = 620
 | image1    = ロジスティック写像2回反復グラフの周期倍化分岐の様（a = 2.7）.png
 | caption1=The relationship between <math>x_{n+2}</math> and <math>x_n</math> when r = 2.7, before the period doubling bifurcation occurs. The orbit converges to a fixed point <math>x_{f2}</math>.
 | image2    =ロジスティック写像2回反復グラフの周期倍化分岐の様（a = 3）.png
 | caption2=The relationship between <math>x_{n+2}</math> and <math>x_n</math> when r = 3. The tangent slope at the fixed point <math>x_{f2}</math> is exactly 1, and a period doubling bifurcation occurs.
 | image3    = ロジスティック写像2回反復グラフの周期倍化分岐の様（a = 3.3）.png
 | caption3  = The relationship between <math>x_{n+2}</math> and <math>x_n</math> when r = 3.3. <math>x_{f 2}</math> becomes unstable and the orbit converges to the periodic points <math>x^{(2)}_{f1}</math> and <math>x^{(2)}_{f2}</math>.
}}</div>

When we actually calculate the differential coefficients of two periodic points for the logistic map, we get

{{NumBlk|:|<math> (f^{2})'(x_{f}^{(2)})=4+2r-r^{2}</math>|{{EquationRef|3-9}}}}
<!--[ 123 ]--> 
When this is applied to equation (3-7), the parameter a becomes:
{{NumBlk|:|<math>\left|4+2r-r^{2}\right|<1</math>|{{EquationRef|3-10}}}}
It can be seen that the 2-periodic points are asymptotically stable when<!--[ 123 ]--> this range is <math>3 < r < 1 + \sqrt{6}</math>, i.e., when r exceeds <math>1 + \sqrt{6} = 3.44949...</math>, the 2-periodic points are no longer asymptotically stable and their behavior changes.<!--[ 124 ]-->

Almost all initial values in [0, 1] are attracted to the 2-periodic points, but <math>x_{f1}= 0</math> and <math>x_{f2} = 1 - 1/a</math>
remains as an unstable fixed point in [0,1].<!--[ 125 ]--> These unstable fixed points continue to remain in [0,1] even if r is increased.<!--[ 126 ]--> Therefore, when the initial value is exactly <math>x_{f1}</math> or <math>x_{f2}</math>, the orbit  does not attract to a 2-periodic point.<!--[ 127 ]--> Moreover, when the initial value is the final fixed point for <math>x_{f1}</math> or the final fixed point for <math>x_{f2}</math>, the orbit does not attract to a 2-periodic point.<!--[ 128 ]--> There are an infinite number of such final fixed points in [0, 1].<!--[ 127 ]--> However, the number of such points is negligibly small compared to the set of real numbers [ 0, 1].<!--[ 128 ]-->

==== Case when 3.44949 ≤ r ≤ 3.56995 ====
With {{mvar|r}} between 3.44949 and 3.54409 (approximately), from almost all initial conditions the population will approach permanent oscillations among four values. The latter number is a root of a 12th degree polynomial {{OEIS|id=A086181}}.

With {{mvar|r}} increasing beyond 3.54409, from almost all initial conditions the population will approach oscillations among 8 values, then 16, 32, etc. The lengths of the parameter intervals that yield oscillations of a given length decrease rapidly;  the ratio between the lengths of two successive bifurcation intervals approaches the [[Feigenbaum constant]] {{math|''δ'' ≈ 4.66920}}. This behavior is an example of a [[period-doubling bifurcation|period-doubling cascade]].

When the parameter r exceeds <math>1 + \sqrt{6} = 3.44949...</math>, the previously stable 2-periodic points become unstable, stable 4-periodic points are generated, and the orbit gravitates toward a 4-periodic oscillation.<!--[ 129 ]--> That is, a period-doubling bifurcation occurs again at <math>r = 3.44949...</math>.<!--[ 130 ]--> The value of x at the 4-periodic point is also

{{NumBlk|:|<math>f(f(f(f(x)))) = f^{4}(x)=x</math>|{{EquationRef|3-11}}}}

satisfies, so that solving this equation allows the values of x at the 4-periodic points to be found.<!--[ 131 ]--> However, equation (3-11) is a 16th-order equation, and even if we factor out the four solutions for the fixed points and the 2-periodic points, it is still a 12th-order equation.<!--[ 131 ]--> Therefore, it is no longer possible to solve this equation to obtain an explicit function of a that represents the values of the 4-periodic points in the same way as for the 2-periodic points.<!--[ 132 ]-->

{| class="wikitable floatright" style="text-align:center;"
|+ Examples of branching points up to 256 periods<!--[ 133 ]-->
|-
|-
| The kth branch || Period 2<sup>''k''</sup> || Branch point ''a<sub>k</sub>''
|-
| 1 || 2 || 3.0000000
|-
| 2 || 4 || 3.4494896
|-
| 3 || 8 || 3.5440903
|-
| 4 || 16 || 3.5644073
|-
| 5 || 32 || 3.5687594
|-
| 6 || 64 || 3.5696916
|-
| 7 || 128 || 3.5698913
|-
| 8 || 256 || 3.5699340
|}

As a becomes larger, the stable 4-periodic point undergoes another period doubling, resulting in a stable 8-periodic point.<!--[ 134 ]--> As an increases, period doubling bifurcations occur infinitely: 16, 32, 64, ..., and so on, until an infinite period, i.e., an orbit that never returns to its original value.<!--[ 134 ]--> This infinite series of period doubling bifurcations is called a cascade.<!--[ 135 ]--> While these period doubling bifurcations occur infinitely, the intervals between a at which they occur decrease in a geometric progression.<!--[ 136 ]--> Thus, an infinite number of period doubling bifurcations occur before the parameter a reaches a finite value.<!--[ 134 ]--> Let the bifurcation from period 1 to period 2 that occurs at r = 3 be counted as the first period doubling bifurcation. Then, in this cascade of period doubling bifurcations, a stable 2k-periodic point occurs at the k-th bifurcation point. Let the k-th bifurcation point a be denoted as a k. In this case, it is known that <math>r_k</math> converges to the following value as k → ∞.<!--[ 137 ]-->

{{NumBlk|:|<math>r_{\infty }=\lim _{k\rightarrow \infty }r_{k}=3.56994...</math>|{{EquationRef|3-12}}}}

Furthermore, it is known that the rate of decrease of a k reaches a constant value in the limit, as shown in the following equation.<!--[ 138 ]-->

{{NumBlk|:|<math>\delta =\lim_{k\to \infty }{\frac {r_{k}-r_{k-1}}{r_{k+1}-r_{k}}}=4.66920...</math>|{{EquationRef|3-13}}}}

This value of δ is called the Feigenbaum constant because it was discovered by mathematical physicist Mitchell Feigenbaum.<!--[ 139 ]--> a∞ is called the Feigenbaum point.<!--[ 136 ]--> In the period doubling cascade, <math>f^m</math>  and <math>f^{2m}</math> have the property that they are locally identical after an appropriate scaling transformation.<!--[ 140 ]--> The Feigenbaum constant can be found by a technique called renormalization that exploits this self-similarity.<!--[ 140 ]--> The properties that the logistic map exhibits in the period doubling cascade are also universal in a broader class of maps, as will be discussed later.<!--[ 141 ]--> 

To get an overview of the final behavior of an orbit for a certain parameter, an approximate bifurcation diagram, orbital diagram, is useful.<!--[ 142 ]--> In this diagram, the horizontal axis is the parameter r and the vertical axis is the variable x, as in the bifurcation diagram.<!--[ 143 ]--> Using a computer, the parameters are determined and, for example, 500 iterations are performed.<!--[ 144 ]--> Then, the first 100 results are ignored and only the results of the remaining 400 are plotted.<!--[ 144 ]--> This allows the initial transient behavior to be ignored and the asymptotic behavior of the orbit remains.<!--[ 144 ]--> For example, when one point is plotted for r, it is a fixed point, and when m points are plotted for r, it corresponds to an m-periodic orbit.<!--[ 145 ]--> When an orbital diagram is drawn for the logistic map, it is possible to see how the branch representing the stable periodic orbit splits, which represents a cascade of period-doubling bifurcations.<!--[ 146 ]-->

[[File:ロジスティック写像、周期倍分岐カスケードの軌道図.png|class=skin-invert-image|thumb|center|520px|[[Bifurcation diagram]] of the period-doubling bifurcation cascade occurring between parameters <math>r_1 = 3</math> and <math>a_{\infty} = 3.56994...</math>. After 64 periods (<math>a_5</math>), the spacing becomes very narrow and almost collapses.]]
When the parameter <math>r = r_{\infty}</math> is exactly the accumulation point of the period-doubling cascade, the variable <math>x_n</math> is attracted to aperiodic orbits that never close.<!--[147]--> In other words, there exists a periodic point with infinite period at <math>r_{\infty}</math>.<!--[ 148 ]--> This aperiodic orbit is called the Feigenbaum attractor.<!--[ 149]--> The critical <math>2^{\infty}</math> attractor.<!--[ 150 ]--> An attractor is a term used to refer to a region that has the property of attracting surrounding orbits, and is the orbit that is eventually drawn into and continues.<!--[ 151 ]--> The attractive fixed points and periodic points mentioned above are also members of the attractor family.<!--[ 152 ]-->

The structure of the Feigenbaum attractor is the same as that of a fractal figure called the Cantor set.<!--[ 153 ]--> The number of points that compose the Feigenbaum attractor is infinite and their cardinality is equal to the real numbers.<!--[ 154 ]--> However, no matter which two of the points are chosen, there is always an unstable periodic point between them, and the distribution of the points is not continuous.<!--[ 155 ]--> The fractal dimension of the Feigenbaum attractor, the Hausdorff dimension or capacity dimension, is known to be approximately 0.54.<!--[ 156 ]-->
[[File:Cantor set in seven iterations.svg|class=skin-invert-image|thumb|center|520px|An example of the construction of a Cantor set: if you keep removing the central third of a line segment infinitely, you will end up with a shape that appears to have zero length but has an uncountably infinite number of points, each of which has an infinitesimal neighborhood of other points.<!--[ 157 ]-->]]

==== Case when 3.56995 < r < 4 ====
===== Qualitative Summary =====
[[File:Feigenbaum Tree.gif|class=skin-invert-image|thumb|284x284px|Evolution of different initial conditions as a function of {{mvar|r}} (The parameter k from the figure corresponds to the parameter r from the definition in the article.)]]
[[File:Feigenbaum tree with bias.gif|class=skin-invert-image|thumb|Evolution of different initial conditions as a function of ''{{mvar|r}}'' with bias (The parameter k from the figure corresponds to the parameter r from the definition in the article.)|283x283px]]

* At {{math|''r'' ≈ 3.56995}} {{OEIS|id=A098587}} is the onset of chaos, at the end of the period-doubling cascade. From almost all initial conditions, we no longer see oscillations of finite period.  Slight variations in the initial population yield dramatically different results over time, a prime characteristic of chaos.
* This number shall be compared and understood as the equivalent of the [[Reynolds number]] for the onset of other chaotic phenomena such as [[turbulence]] and similar to the [[critical temperature]] of a [[phase transition]]. In essence the [[phase space]] contains a full subspace of cases with extra dynamical variables to characterize the microscopic state of the system, these can be understood as [[Eddies]] in the case of turbulence and [[order parameters]] in the case of [[phase transitions]].
* Most values of {{mvar|r}} beyond 3.56995 exhibit chaotic behaviour, but there are still certain isolated ranges of {{mvar|r}} that show non-chaotic behavior; these are sometimes called ''islands of stability''. For instance, beginning at 1&nbsp;+&nbsp;{{sqrt|8}}<ref>{{cite journal|last=Zhang |first=Cheng |title=Period three begins |journal=[[Mathematics Magazine]] |volume=83 |date=October 2010 |issue=4 |pages=295–297 |doi=10.4169/002557010x521859|s2cid=123124113 }}</ref> (approximately 3.82843) there is a range of parameters {{mvar|r}} that show oscillation among three values, and for slightly higher values of {{mvar|r}} oscillation among 6 values, then 12 etc.
* At <math>r = 1 + \sqrt 8 = 3.8284...</math>, the stable period-3 cycle emerges.<ref>{{Cite journal |last=Bechhoefer |first=John |date=1996-04-01 |title=The Birth of Period 3, Revisited |url=https://doi.org/10.1080/0025570X.1996.11996402 |journal=Mathematics Magazine |volume=69 |issue=2 |pages=115–118 |doi=10.1080/0025570X.1996.11996402 |issn=0025-570X}}</ref>
* The development of the chaotic behavior of the logistic sequence as the parameter {{mvar|r}} varies from approximately 3.56995 to approximately 3.82843 is sometimes called the [[Pomeau–Manneville scenario]], characterized by a periodic (laminar) phase interrupted by bursts of aperiodic behavior. Such a scenario has an application in semiconductor devices.<ref name="carson82">{{cite journal|first1=Carson |last1=Jeffries|first2=José |last2=Pérez |journal=[[Physical Review A]]|year=1982|title=Observation of a Pomeau–Manneville intermittent route to chaos in a nonlinear oscillator|volume=26 |issue=4 |pages=2117–2122|doi=10.1103/PhysRevA.26.2117|bibcode = 1982PhRvA..26.2117J |s2cid=119466337 |url=http://www.escholarship.org/uc/item/2dm2k8mm}}</ref> There are other ranges that yield oscillation among 5 values etc.; all oscillation periods occur for some values of {{mvar|r}}.  A ''period-doubling window'' with parameter {{mvar|c}} is a range of {{mvar|r}}-values consisting of a succession of subranges.  The {{mvar|k}}th subrange contains the values of {{mvar|r}} for which there is a stable cycle (a cycle that attracts a set of initial points of unit measure) of period {{math|2<sup>''k''</sup>''c''}}.  This sequence of sub-ranges is called a ''cascade of harmonics''.<ref name=":2" group="May, Robert M. (1976)"/> In a sub-range with a stable cycle of period {{math|2<sup>''k''*</sup>''c''}}, there are unstable cycles of period {{math|2<sup>''k''</sup>''c''}} for all {{math|''k'' < ''k''*}}.  The {{mvar|r}} value at the end of the infinite sequence of sub-ranges is called the ''point of accumulation'' of the cascade of harmonics.  As {{mvar|r}} rises there is a succession of new windows with different {{mvar|c}} values.  The first one is for {{math|''c'' {{=}} 1}}; all subsequent windows involving odd {{mvar|c}} occur in decreasing order of {{mvar|c}} starting with arbitrarily large {{mvar|c}}.<ref name=":2" group="May, Robert M. (1976)"/><ref>{{cite journal |last1=Baumol |first1=William J. |author-link=William Baumol |last2=Benhabib |first2=Jess |author-link2=Jess Benhabib |title=Chaos: Significance, Mechanism, and Economic Applications |journal=[[Journal of Economic Perspectives]] |date=February 1989 |volume=3 |issue=1 |pages=77–105 |doi=10.1257/jep.3.1.77 |doi-access=free }}</ref>  
* At <math>r = 3.678..., x = 0.728...</math>, two chaotic bands of the bifurcation diagram intersect in the first [[Misiurewicz point]] for the logistic map. It satisfies the equations <math>r^3 - 2r^2 - 4r -8 = 0, x = 1-1/r</math>.<ref>{{Cite web |title=Misiurewicz Point of the Logistic Map |url=https://sprott.physics.wisc.edu/chaos/mispoint.htm |access-date=2023-05-08 |website=sprott.physics.wisc.edu}}</ref>  
* Beyond {{math|''r'' {{=}} 4}}, almost all initial values eventually leave the interval {{math|[0,1]}} and diverge. The set of initial conditions which remain within {{math|[0,1]}} form a [[Cantor set]] and the dynamics restricted to this Cantor set is chaotic.<ref>{{cite book |last1=Teschl |first1=Gerald | author-link1=Gerald Teschl |title=Ordinary Differential Equations and Dynamical Systems |url=https://www.mat.univie.ac.at/~gerald/ftp/book-ode/index.html |url-access=registration |publisher=Amer. Math Soc. |year=2012 |isbn=978-0-8218-8328-0 }}</ref>

For any value of {{mvar|r}} there is at most one stable cycle.  If a stable cycle exists, it is globally stable,  attracting almost all points.<ref>{{cite book |last1=Collet |first1=Pierre |first2=Jean-Pierre |last2=Eckmann |author-link2=Jean-Pierre Eckmann |title=Iterated Maps on the Interval as Dynamical Systems |url=https://archive.org/details/iteratedmapsonin0000coll |url-access=registration |publisher=Birkhauser |year=1980 |isbn=978-3-7643-3026-2 }}</ref>{{rp|13}}  Some values of {{mvar|r}} with a stable cycle of some period have infinitely many unstable cycles of various periods.

[[File:Logistic Bifurcation map High Resolution.png|class=skin-invert-image|thumb|right|[[Bifurcation diagram]] for the logistic map.
The [[attractor]] for any value of the parameter {{mvar|r}} is shown on the vertical line at that {{mvar|r}}.]]

The [[bifurcation diagram]] at right summarizes this. The horizontal axis shows the possible values of the parameter {{mvar|r}} while the vertical axis shows the set of values of {{mvar|x}} visited asymptotically from almost all initial conditions by the iterates of the logistic equation with that {{mvar|r}} value.

The bifurcation diagram is a [[self-similar]]: if we zoom in on the above-mentioned value {{math|''r'' ≈ 3.82843}} and focus on one arm of the three, the situation nearby looks like a shrunk and slightly distorted version of the whole diagram. The same is true for all other non-chaotic points. This is an example of the deep and ubiquitous connection between [[chaos (mathematics)|chaos]] and [[fractal]]s.


<div class=skin-invert-image>{{multiple image
| align             = center
| direction         = horizontal
| total_width       = 900
| image1            = Subsection Bifurcation Diagram Logistic Map.png
| caption1          = Magnification of the chaotic region of the map
| image2            = Logistic map bifurcation diagram magnifications.png
| caption2          = Stable regions within the chaotic region, where a tangent bifurcation occurs at the boundary between the chaotic and periodic attractor, giving intermittent trajectories as described in the [[Pomeau–Manneville scenario]]
}}</div>

We can also consider negative values of {{mvar|r}}:
* For {{mvar|r}} between -2 and -1 the logistic sequence also features chaotic behavior.<ref name="Takashi Tsuchiya, Daisuke Yamagishi, 1997">{{cite journal|last1=Tsuchiya|first1=Takashi|last2=Yamagishi|first2=Daisuke|date=February 11, 1997|title=The Complete Bifurcation Diagram for the Logistic Map|journal=Z. Naturforsch.|volume=52a|issue=6–7 |pages=513–516|doi=10.1515/zna-1997-6-708 |bibcode=1997ZNatA..52..513T |s2cid=101491730 |doi-access=free}}</ref>
* With {{mvar|r}} between -1 and 1&nbsp;-&nbsp;{{sqrt|6}} and for {{mvar|x}}<sub>0</sub> between 1/{{mvar|r}} and 1-1/{{mvar|r}}, the population will approach permanent oscillations between two values, as with the case of {{mvar|r}} between 3 and 1&nbsp;+&nbsp;{{sqrt|6}}, and given by the same formula.<ref name="Takashi Tsuchiya, Daisuke Yamagishi, 1997"/>
=====The Emergence of Chaos=====
<div class=skin-invert-image>{{multiple image
| align             = center
| direction         = horizontal
| total_width       = 620
| image1            = パラメータ3.82のロジスティック写像における微差の初期値から出発する2つの軌道.png
| caption1          = Chaotic orbits of the logistic map when r = 3.82. The orange squares are orbits starting from <math>x_0 = 0.1234</math>, and the blue-green circles are orbits starting from <math>\hat{x}_0 = 0.1234 + 10^{-9}</math>.
| image2            = パラメータ3.82のロジスティック写像におけるズレの進展.png
| caption2          = The trajectory starting from x 0 = 0.1234 and ˆx The difference in orbits starting from <math>x_0 = 0.1234 + 10^{-9}</math> grows exponentially. The vertical axis is <math>\Delta x_n = | x_n - \hat{x}_n |</math>,shown on a [[logarithmic scale]].
}}</div>

When the parameter r exceeds <math>r_{\infty} = 3.56994...</math>, the logistic map exhibits chaotic behavior.<!--[ 145 ]--> Roughly speaking, chaos is a complex and irregular behavior that occurs despite the fact that the difference equation describing the logistic map has no probabilistic ambiguity and the next state is completely and uniquely determined.<!--[ 158 ]--> The range of <math>r > r_{\infty}</math> of the logistic map is called the chaotic region.<!--[ 159 ]-->

One of the properties of chaos is its unpredictability, symbolized by the term [[butterfly effect]].<!--[ 160 ]--> This is due to the property of chaos that a slight difference in the initial state can lead to a huge difference in the later state.<!--[ 160 ]--> In terms of a discrete dynamical system, if we have two initial values <math>x_0</math> and <math>\hat{x}_0</math>
No matter how close they are, once time n has progressed to a certain extent, each destination <math>x_n</math> and <math>\hat{x}_n</math> can vary significantly.<!--[ 161 ]--> For example, use <math>r = 3.95, x_0 = 0.1, \hat{x}_0=x_0+10^{-9}</math>
If the orbits are calculated using two very similar initial values, 0 = 0.1000000001, the difference grows to macroscopic values that are clearly visible on the graph after about 29 iterations.<!--[ 162 ]-->

This property of chaos, called initial condition sensitivity, can be quantitatively expressed by the [[Lyapunov exponent]]. For a one-dimensional map, the Lyapunov exponent λ can be calculated as follows<!--[ 163 ]-->:

{{NumBlk|:|<math>{\displaystyle \lambda =\lim _{N\to \infty }{\frac {1}{N}}\sum _{i=0}^{N-1}\log \left|f^{\prime }(x_{i})\right\vert }</math>|{{EquationRef|3-14}}}}

Here, log means natural logarithm. This λ is the distance between the two orbits (<math>x_n</math> and
<math>\hat{x}_n</math>).
A positive value of λ indicates that the system is sensitive to initial conditions, while a zero or negative value indicates that the system is not sensitive to initial conditions.<!--[ 164 ]--> When calculating λ of numerically, it can be confirmed λ remains in the range of zero or negative values in the range <math>r < r_{\infty}</math>, and that λ can take positive values in the range <math>r > r_{\infty}</math>.<!--[ 165 ]-->

'''Window, intermittent'''

Even beyond <math>r_{\infty}</math>, the behavior does not depend simply on the parameter r.<!--[ 166 ]--> Many sophisticated mathematical structures lurk in the chaotic region for <math>r > r_{\infty}</math>.<!--[ 167 ]--> In this region, chaos does not persist forever; stable periodic orbits reappear.<!--[ 166 ]--> The behavior for <math>r_{\infty}< a \le 4</math> can be broadly divided into two types:<!--[ 168 ]-->

* Stable periodic point: In this case, the Lyapunov exponent is negative.
* Aperiodic orbits: In this case, the Lyapunov exponent is positive.
The region of stable periodic points that exists for r <math>r_{\infty} < r \le 4</math> is called a periodic window, or simply a window.<!--[ 169 ]--> If one looks at a chaotic region in an orbital diagram, the region of nonperiodic orbits looks like a cloud of countless points, with the windows being the scattered blanks surrounded by the cloud.<!--[ 170 ]-->

{{Image frame|width=620|content=[[File:Logistic orbit map 3.55 to 4.png|class=skin-invert-image|620px]]
|caption=Orbit diagram of the logistic map from r = 3.55 to r = 4 (parameter is denoted as r in the diagram)|align=center}}

In each window, the cascade of [[period-doubling]] bifurcations that occurred before <math>r_{\infty} = 3.56994...</math> occurs again.<!--[ 171 ]--> However, instead of the previous stable periodic orbits of 2 k, new stable periodic orbits such as 3×2 k and 5×2 k are generated.<!--[ 172 ]--> The first window has a period of p, and the windows from which the period-doubling cascade occurs are called windows of period p, etc..<!--[ 173 ]--> For example, a window of period 3 exists in the region around 3.8284 < a < 3.8415, and within this region the period doublings are: 3, 6, 12, 24, ..., 3×2 k, ....<!--[ 174 ]-->

[[File:ロジスティック写像の過渡カオス.png|class=skin-invert-image|thumb|Transient chaos at a = 3.8285. The system behaves chaotically until it is attracted into a periodic 3 orbit.]]

In the window region, chaos does not disappear but exists in the background.<!--[ 175 ]--> However, this chaos is unstable, so only stable periodic orbits are observed.<!--[ 175 ]--> In the window region, this potential chaos appears before the orbit is attracted from its initial state to a stable periodic orbit.<!--[ 176]--> Such chaos is called transient chaos.<!--[177 ]--> In this potential presence of chaos, windows differ from the periodic orbits that appeared before a∞.<!--[ 175 ]-->

There are an infinite number of windows in the range a∞ < a < 4.<!--[ 178 ]--> The windows have various periods, and there is a window with a period for every natural number greater than or equal to three.<!--[ 179 ] --> However, each window does not occur exactly once.<!-- [ 180 ]--> The larger the value of p, the more often a window with that period occurs.<!--[ 181 ]--> A window with period 3 occurs only once, while a window with period 13 occurs 315 times.<!--[ 182 ]--> When a periodic orbit of 3 occurs in the window with period 3, the Szarkovsky order is completed, and all orbits with all periods have been seen.<!--[ 183 ]-->

If we restrict ourselves to the case where p is a prime number, the number of windows with period p is

{{NumBlk|:|<math>{\displaystyle N_{p}={\frac {2^{p-1}-1}{p}}}</math>|{{EquationRef|3-15}}}}

<!--[ 184 ]--> 

This formula was derived for p to be a prime number, but in fact it is possible to calculate with good accuracy the number of stable p- periodic points for non-prime p as well.<!--[ 181 ]-->

The window width (the difference between a where the window begins and a where the window ends) is widest for windows with period 3 and narrows for larger periods.<!--[ 185 ]--> For example, the window width for a window with period 13 is about 3.13 × 10−6.<!--[ 186 ]--> Rough estimates suggest that about 10% of <math>[ r_{\infty}, 4]</math> is in the window region, with the rest dominated by chaotic orbits.<!--[ 187 ]-->

The change from chaos to a window as r is increased is caused by a tangent bifurcation,<!--[ 188 ]--> where the map curve is tangent to the diagonal of y = x at the moment of bifurcation, and further parameter changes result in two fixed points where the curve and the line intersect.<!--[ 189 ]--> For a window of period p, the iterated map <math>f^p(x)</math> exhibits tangent bifurcation, resulting in stable p-periodic orbits.<!--[ 168 ]--> The exact value of the bifurcation point for a window of period 3 is known, and if the value of this bifurcation point r is <math>r_3</math>, then <math>r_3 = 1 + \sqrt{8} = 3.828427...</math>.<!--[ 190 ]--> The outline of this bifurcation can be understood by considering the graph of <math>f^3(x)</math> (vertical axis <math>x_{n+3}</math>, horizontal axis <math>x_n</math>).<!--[ 191 ]-->  

<div class=skin-invert-image>{{multiple image
| align             = center
| direction         = horizontal
| total_width       = 620
| image1            = ロジスティック写像3回反復グラフの接線分岐の様子 (0.99xa).png
| caption1          = Graph of <math>f^3(x)</math> when r is slightly less than 3. The graph is not tangent except at the fixed points, and there are no 3-periodic points.
| image2            =ロジスティック写像3回反復グラフの接線分岐の様子 (1.00xa).png
| caption2          = When a is exactly 3, the graph touches the diagonal at exactly three points, resulting in three periodic points.
| image3            = ロジスティック写像3回反復グラフの接線分岐の様子 (1.01xa).png
| caption3          = When a is slightly greater than 3, the graph passes the diagonal and splits into stable and unstable 3-periodic points.}}</div>

When we look at the behavior of <math>x_n</math> when r = 3.8282, which is slightly smaller than the branch point <math>r_3</math>, we can see that in addition to the irregular changes, there is also a behavior that changes periodically with approximately three periods, and these occur alternately.<!--[ 192 ]--> This type of periodic behavior is called a "laminar", and the irregular behavior is called a burst, in analogy with  fluids.<!--[ 193 ]--> There is no regularity in the length of the time periods of the bursts and laminars, and they change irregularly.<!--[ 194 ]--> However, when we observe the behavior at r = 3.828327, which is closer to <math>r_3</math>, the average length of the laminars is longer and the average length of the bursts is shorter than when r = 3.8282.<!--[194]--> If we further increase r, the length of the laminars becomes larger and larger, and at <math>r_3</math> it changes to a perfect three- period.<!--[ 195 ]-->

<div class=skin-invert-image>{{multiple image
| align             = center
| direction         = horizontal
| total_width       = 620
| image1            = Logistic map time evolution a=3.8282.png
| caption1          = Time series when r = 3.8282
| image2            = Logistic map time evolution a=3.828327.png
| caption2          = Time series when r = 3.828327
| footnote          = The intermittency that occurs just before <math>r_3 = 3.828427...</math> The part where the three almost identical values continue periodically is a laminar, and the part where chaotic irregular changes occur is a burst.
}}</div>

The phenomenon in which orderly motions called laminars and disorderly motions called bursts occur intermittently is called intermittency or intermittent chaos.<!--[ 196 ]--> If we consider the parameter a decreasing from a3, this is a type of emergence of chaos.<!--[ 197 ]--> As the parameter moves away from the window, bursts become more dominant, eventually resulting in a completely chaotic state.<!--[ 198 ]--> This is also a general route to chaos, like the period doubling bifurcation route mentioned above, and routes characterized by the emergence of intermittent chaos due to tangent bifurcations are called intermittency routes.<!--[ 199 ]-->

[[File:Laminar channel in three times iterated logistic map.png|class=skin-invert-image|thumb|Channel patterns appearing in the graph of f3 (x)]]

The mechanism of intermittency can also be understood from the graph of the map.<!--[ 194 ]--> When <math>r</math> is slightly smaller than <math>r_3</math>, there is a very small gap between the graph of <math>f^3(x)</math> and the diagonal<!--[ 190  ]-->. This gap is called a channel, and many iterations of the map occur as the orbit passes through the narrow channel.<!--[ 200 ]--> During the passage through this channel, <math>x_n</math> and <math>x_{n + 3}</math> become very close, and the variables change almost like a periodic three orbit.<!--[ 198 ]--> This corresponds to a laminar.<!--[ 201 ]--> The orbit eventually leaves the narrow channel, but returns to the channel again as a result of the global structure of the map.<!--[ 202 ]--> While leaving the channel, it behaves chaotically.<!--[ 202 ]--> This corresponds to a burst.<!--[ 201 ]-->

'''Band, window finish'''

Looking at the entire chaotic domain, whether it is chaotic or windowed, the maximum and minimum values on the vertical axis of the orbital diagram (the upper and lower limits of the attractor) are limited to a certain range.<!--[ 203 ]--> As shown in equation (2-1), the maximum value of the logistic map is given by r/4, which is the upper limit of the attractor.<!--[ 204 ]--> The lower limit of the attractor is given by the point f(r/4) where r/4 is mapped.<!--[ 204 ]--> Ultimately, the maximum and minimum values at which xn moves on the orbital diagram depend on the parameter r

{{NumBlk|:|<math>{\displaystyle {\frac {r^{2}(4-r)}{16}}\leq x_{n}\leq {\frac {r}{4}}}</math>|{{EquationRef|3-16}}}}

<!--[ 203 ]-->
Finally, for r = 4, the orbit spans the entire range [0, 1].<!--[ 205 ]-->

When observing an orbital map, the distribution of points has a characteristic shading.<!--[ 206 ]--> Darker areas indicate that the variable takes on values in the vicinity of the darker areas, whereas lighter areas indicate that the variable takes on values in the vicinity of the darker areas.{{clarify inline| date=March 2025}}<!--[ 206 ]--> These differences in the frequency of the points are due to the shape of the graph of the logistic map.<!--[ 206 ]--> The top of the graph, near r/4, attracts orbits with high frequency, and the area near f(r/4) that is mapped from there also becomes highly frequent, and the area near <math>f^2(r/4)</math> that is mapped from there also becomes  highly frequent, and so on.<!--[ 206 ]--> The density distribution of points generated by the map is characterized by a quantity called an invariant measure or distribution function, and the invariant measure of the attractor is reproducible regardless of the initial value.<!--[ 207 ]-->

Looking at the beginning of the chaotic region of the orbit diagram, just beyond the accumulation point <math>r_{\infty} = 3.56994</math> of the first period - doubling cascade, one can see that the orbit is divided into several subregions.<!--[ 208 ]--> These subregions are called bands.<!--[ 209 ]--> When there are multiple bands, the orbit moves through each band in a regular order, but the values within each band are irregular.<!--[ 210 ]--> Such chaotic orbits are called band chaos or periodic chaos, and chaos with k bands is called k -band chaos.<!--[ 211 ]--> Two-band chaos lies in the range 3.590 < r < 3.675, approximately.<!--[ 212 ]-->

{{Image frame|width=620|content=[[File:Bands of logistic map from 1 to 8.png|class=skin-invert-image|620px]]
|caption=Band structure. Because the <math>e_p</math> spacing rapidly decreases, it is not possible to show more than eight bands. The top and bottom lines, which contain the orbitals, are within the range of equation (3-16).|align=center}}

As the value of r is further decreased from the left-hand end of two-band chaos, r = 3.590, the number of bands doubles, just as in the period doubling bifurcation.<!--[ 212 ]--> Let <math>e_p</math> (for p = 1, 2, 4, ..., 2k, ...) denote the bifurcation points where p − 1 band chaos splits into p band chaos, or where p band chaos merges into p − 1 band chaos. Then, just as in the period doubling bifurcation, e p accumulates to a value as p → ∞.<!--[ 213 ]--> At this accumulation point <math>e_{\infty}</math>, the number of bands becomes infinite, and the value of <math>e_{\infty}</math> is equal to the value of <math>r_{\infty}</math>.<!--[ 214 ]-->

[[File:ロジスティック写像の全体の自己相似.png|class=skin-invert-image|thumb|Self-similar hierarchical structure of the entire trajectory map of the logistic map]]

Similarly, for the bifurcation points of the period-doubling bifurcation cascade that appeared before a∞, let a p (where p = 1, 2, 4, ..., 2k, ...) denote the bifurcation points where p stable periodic orbits branch into p + 1 stable periodic orbits. In this case, if we look at the orbital diagram from <math>r_2</math> to <math>e_2</math>, there are two reduced versions of the global orbital diagram from <math>r_1</math> to <math>e_1</math> in the orbital diagram from <math>r_2</math> to <math>e_2</math><!--[215]-->. Similarly, if we look at the orbital diagram from <math>r_4</math> to <math>e_4</math>, there are four reduced versions of the global orbital diagram from a1 to e1 in the orbital diagram from <math>r_4</math> to <math>e_4</math>.<!--[ 215 ]--> Similarly, there are p reduced versions of the global orbital diagram in the orbital diagram from ap to ep, and the branching structure of the logistic map has an infinite self-similar hierarchy.<!--[ 215 ]-->

[[File:ロジスティック写像の窓の自己相似.png|class=skin-invert-image|thumb|Self-similar hierarchical structure of windows of the logistic map]]

A self-similar hierarchy of bifurcation structures also exists within windows.<!--[ 216 ]--> The period-doubling bifurcation cascades within a window follow the same path as the cascades of period-2k bifurcations.<!--[ 217 ]--> That is, there are an infinite number of period-doubling bifurcations within a window, after which the behavior becomes chaotic again.<!--[ 217 ]--> For example, in a window of period 3, the cascade of stable periodic orbits ends at <math>a_{3\infty}</math> ≈ 3.8495.<!--[ 218 ]--> After <math>a_{3\infty}</math> ≈ 3.8495, the behavior becomes band chaos of multiples of three.<!--[ 218 ]--> As a increases from <math>a_{3\infty}</math>, these band chaos also merge by twos, until at the end of the window there are three bands.<!--[ 219 ]--> Within such bands within a window, there are an infinite number of windows.<!--[ 220 ]--> Ultimately, the window contains a miniature version of the entire orbital diagram for 1 ≤ a ≤ 4, and within the window there exists a self-similar hierarchy of branchings.<!--[ 221 ]-->

At the end of the window, the system reverts to widespread chaos. For a period 3 window, the final 3-band chaos turns into large-area 1-band chaos at a ≈ 3.857, ending the window.<!--[ 222 ]--> However, this change is discontinuous, and the 3-band chaotic attractor suddenly changes size and turns into a 1-band.<!--[ 223 ]--> Such discontinuous changes in attractor size are called crises.<!--[ 224 ]--> Crises of this kind, which occur at the end of a window, are also called internal crises.<!--[ 225 ]--> When a crisis occurs at the end of a window, a stable periodic orbit just touches an unstable periodic point that is not visible on the orbit diagram.<!--[ 226 ]--> This creates an exit point through which the periodic orbits can escape, resulting in an internal crisis.<!--[ 227 ]--> Immediately after the internal crisis, there are periods of widespread chaos, and periods of time when the original band chaotic behavior reoccurs, resulting in a kind of intermittency similar to that observed at the beginning of a window.<!--[ 197 ]-->

==== When r = 4 ====
{{Image frame|width=620|content=[[File:Logistic map cobweb and time evolution a=4.png|class=skin-invert-image|620px]]
|caption=Spider diagram of the logistic map with parameter r = 4 (left) and time series up to n = 500 (right) for the initial value <math>x_0</math> = 0.3.|align=center}}

When the parameter r = 4, the behavior becomes chaotic over the entire range [0, 1].<!--[ 205 ]--> At this time, the Lyapunov exponent λ is maximized, and the state is the most chaotic.<!--[ 228 ]--> The value of λ for the logistic map at r = 4 can be calculated precisely, and its value is λ = log 2.<!--[ 229 ]--> Although a strict mathematical definition of chaos has not yet been unified, it can be shown that the logistic map with r = 4 is chaotic on [0, 1] according to one well-known definition of chaos.<!--[ 231 ]-->

[[File:ロジスティック写像分布関数.png|class=skin-invert-image|thumb|Graph of the invariant measure ρ(x) for r = 4. The dot plot shows the actual frequency of points obtained over 10,000 iterations (with height scaled to ρ (x)).]]

The invariant measure of the density of points, ρ(x), can also be given by the exact function ρ(x) for r = 4<!--[ 232 ]-->:

{{NumBlk|:|<math>{\displaystyle \rho (x)={\frac {1}{\pi {\sqrt {x(1-x)}}}}}</math>|{{EquationRef|3-17}}}}

Here, ρ(x) means that the fraction of points xn that fall in the infinitesimal interval [x,x+dx] when the map is iterated is given by ρ(x) dx.<!--[ 233 ]--> The frequency distribution of the logistic map with r = 4 has high density near both sides of [0, 1] and is least dense at x = 0.5.<!--[ 234 ]-->

When r = 4, apart from chaotic orbits, there are also periodic orbits with any period.<!--[ 235 ]--> For a natural number n, the graph of <math>f_{r=4}^n(x)</math> is a curve with <math>2^{n-1}</math> peaks and <math>2^{n-1}-1</math> valleys, all of which are tangent to 0 and 1.<!--[ 180 ]--> Thus, the number of intersections between the diagonal and the graph is <math>2^n</math>, and there are <math>2^n</math> fixed points of <math>f^n(x)</math>.<!--[ 180 ]--> The n-periodic points are always included in these <math>2^n</math> fixed points, so any n-periodic orbit exists for <math>f_{r = 4}^n(x)</math>.<!--[ 235 ]--> Thus,when r = 4, there are an infinite number of periodic points on [0, 1], but all of these periodic points are unstable.<!--[ 145 ]--> Furthermore,the uncountably infinite set in the interval [0, 1], the number of periodic points is countably infinite, and so almost all orbits starting from initial values are not periodic but non-periodic.<!--[ 145 ]--> 

[[File:ロジスティック写像と記号力学系.png|class=skin-invert-image|thumb|If we convert the orbit of the logistic map <math>f_{r=4}</math> into a string of 0s and 1s, we can reproduce any string of symbols.]]

One of the important aspects of chaos is its dual nature: deterministic and stochastic.<!--[ 236]--> Dynamical systems are deterministic processes, but when the range of variables is appropriately coarse-grained, they become indistinguishable from stochastic processes.<!--[236 ]--> In the case of the logistic map with r = 4, the outcome of every coin toss can be described by the trajectory of the logistic map.<!--[ 236 ]--> This can be elaborated as follows.<!--[ 237 ]-->

Assume that a coin is tossed with a probability of 1/2 landing on heads or tails, and the coin is tossed repeatedly. If heads is 0 and tails is 1, then the result of heads, tails, heads, tails, etc. will be a symbol string such as 01001.... On the other hand, for the trajectory <math>x_0, x_1, x_2, ...</math> of the logistic map, values less than x = 0.5 are converted to 0 and values greater than x = 0.5 are converted to 1, and the trajectory is replaced with a symbol string consisting of 0s and 1s. For example, if the initial value is <math>x_0 = 0.2</math>, then <math>x_1 = 0.64</math>, <math>x_2 = 0.9216</math>, <math>x_3 = 0.28901</math>, ..., so the trajectory will be the symbol string 0110.... Let <math>S_C</math> be the symbol string resulting from the former coin toss, and <math>S_L</math> be the symbol string resulting from the latter logistic map. The symbols in the symbol string <math>S_C</math> were determined by random coin tossing, so any number sequence patterns are possible. So, whatever the string <math>S_L</math> of the logistic map, there is an identical one in <math>S_C</math>. And, what is "remarkable" is that the converse is also true: whatever the string of <math>S_C</math>, it can be realized by a logistic map trajectory <math>S_L</math> by choosing the appropriate initial values. That is, for any <math>S_C</math>, there exists a unique point <math>x_0</math> in [0, 1] such that <math>S_C = S_L</math>.<!--[ 237 ]-->


==== When r > 4 ====
[[File:A=4.5のロジスティック写像.png|class=skin-invert-image|thumb|For the logistic map with r = 4.5, trajectories starting from almost any point in [0, 1] go towards minus infinity.]]

When the parameter r exceeds 4, the vertex r /4 of the logistic map graph exceeds 1.<!--[ 238 ]--> To the extent that the graph penetrates 1, trajectories can escape [0, 1]<!--[ 238 ]--.. As a result, trajectories that start from almost any point in [0, 1] will at some point escape [0, 1] and eventually diverge to minus infinity <!/--[ 239 ]-->.

The bifurcation at r = 4 is also a type of crisis, specifically a boundary crisis.<!--[ 240 ]--> In this case, the attractor at [0, 1] becomes unstable and collapses, and since there is no attractor outside it, the trajectory diverges to infinity.<!--[ 240 ]-->

On the other hand, there are orbits that remain in [0, 1] even if r > 4.<!--[ 241 ]--> Easy-to-understand examples are fixed points and periodic points in [0, 1], which remain in [0, 1].<!--[ 241 ]--> However, there are also orbits that remain in [0, 1] other than fixed points and periodic points.<!--[ 242 ]-->

Let <math>A_0</math> be the interval of x such that f  (x) > 1. As mentioned above,once a variable <math>x_n</math> enters <math>A_0</math>, it diverges to minus infinity. There is also <math>r_n</math> x in [0, 1] that maps to <math>A_0</math> after one application of the map. This interval of x is divided into two, which are collectively called <math>A_1</math>. Similarly, there are four intervals that map to <math>A_1</math> after one application of the map, which are collectively called <math>A_2</math>. Similarly,there are 2n intervals <math>A_n</math> that reach <math>A_0</math> after n iterations.<!--[ 243 ]--> Therefore, the interval <math>\Lambda</math> obtained by removing <math>A_n</math> from [0, 1] an infinite number of times as follows is a collection of orbits that remain in I.<!--[ 244 ]-->

{{NumBlk|:|<math>{\displaystyle \Lambda =\left[0,\ 1\right]-\bigcup _{n=0}^{\infty }A_{n}}</math>|{{EquationRef|3-18}}}}

The process of removing <math>A_n</math> from [0, 1] is similar to the construction of the Cantor set mentioned above, and in fact Λ exists in [0, 1] as a Cantor set (a closed, completely disconnected, and complete subset of [0, 1]).<!--[ 245 ]--> Furthermore, on <math>\Lambda</math>, the logistic map <math>f_{r >4}</math> is chaotic.<!--[ 246 ]-->

==== When r < 0 ====

Since the logistic map has been often studied as an ecological model, the case where the parameter r is negative has rarely been discussed.<!--[ 58 ]--> As a decreases from 0, when −1 < r < 0, the map asymptotically approaches a stable fixed point of xf = 0, but when a exceeds −1, it bifurcates into two periodic points, and as in the case of positive values, it passes through a period doubling bifurcation and reaches chaos.<!--[ 58 ]--> Finally, when a falls below −2, the map diverges to plus infinity.<!--[ 58 ]-->

{{Image frame|width=620|content=[[File:Bifurcation diagram logistic map (-2 to 4).png|class=skin-invert-image|620px]]
|caption=Orbit diagram for parameter r from −2 to 4. The orbit diverges when the parameter a goes beyond this range, both on the negative and positive sides.|align=center}}

==== Exact solutions for special cases ====
For a logistic map with a specific parameter a, an exact solution that explicitly includes the time n and the initial value x 0 has been obtained as follows.

When r = 4<!--[ 247 ]-->

{{NumBlk|:|<math>{\displaystyle x_{n}={\frac {1-\cos \left[2^{n}\arccos(1-2x_{0})\right]}{2}}}</math>|{{EquationRef|3-19}}}}

When r = 2<!--[ 248 ]-->

{{NumBlk|:|<math>{\displaystyle x_{n}={\frac {1-\exp \left[2^{n}\log(1-2x_{0})\right]}{2}}}</math>|{{EquationRef|3-20}}}}

When r = −2<!--[ 249 ]-->

{{NumBlk|:|<math>{\displaystyle x_{n}={\frac {1}{2}}-\cos \left\{{\frac {1}{3}}\left[\pi -(-2)^{n}\left(\pi -3\arccos({\frac {1}{2}}-x_{0})\right)\right]\right\}}		</math>|{{EquationRef|3-21}}}}

Considering the three exact solutions above, all of them are

{{NumBlk|:|<math>{\displaystyle x_{n}={\frac {1}{2}}\left\{1-f\left[a^{n}f^{-1}(1-2x_{0})\right] \right\}}</math>|{{EquationRef|3-22}}}}