Editing Logistic map (section)

== Universality ==
=== A class of mappings that exhibit homogeneous behavior ===
[[File:正弦関数による力学系のグラフ.png|class=skin-invert-image|thumb|Graph of the sine map ( 4-1 )]]
[[File:正弦関数による力学系の軌道図.png|class=skin-invert-image|thumb|Orbit diagram of the sine map ( 4-1 )]]

The bifurcation pattern shown above for the logistic map is not limited to the logistic map <!--[ 239 ]-->. It appears in a number of maps that satisfy certain conditions . The following dynamical system using sine functions is one example <!--[ 251 ]-->:

{{NumBlk|:|<math>{\displaystyle x_{n+1}=b\sin \pi x_{n}}</math>|{{EquationRef|4-1}}}}

Here, the domain is 0 ≤ b ≤ 1 and 0 ≤ x ≤ 1 <!--[ 251 ]-->. The sine map ( 4-1 ) exhibits qualitatively identical behavior to the logistic map ( 1-2 ) <!--[ 251 ]-->: like the logistic map, it also becomes chaotic via a period doubling route as the parameter b increases, and moreover, like the logistic map, it also exhibits a window in the chaotic region <!--[ 251 ]-->.

Both the logistic map and the sine map are one-dimensional maps that map the interval [0, 1] to [0, 1] and satisfy the following property, called unimodal <!--[ 252 ]-->.

<math>f(0)=f(1)= 0</math>.
The map is differentiable and there exists a unique critical point c in [0, 1] such that <math>f'( c ) = 0</math>.
In general, if a one-dimensional map with one parameter and one variable is unimodal and the vertex can be approximated by a second-order polynomial, then, regardless of the specific form of the map, an infinite period-doubling cascade of bifurcations will occur for the parameter range 3 ≤ r ≤ 3.56994... , and the ratio δ defined by equation ( 3-13 ) is equal to the Feigenbaum constant, 4.669... <!--[ 253 ]-->.

The pattern of stable periodic orbits that emerge from the logistic map is also universal <!--[ 254 ]--> . For a unimodal map, <math>x_{n +1} = cf ( x_n )</math> , with parameter c, stable periodic orbits with various periods continue to emerge in a parameter interval where the two fixed points are unstable, and the pattern of their emergence (the number of stable periodic orbits with a certain period and the order of their appearance) is known to be common <!--[ 255 ]--><!--[ 256 ]-->. In other words, for this type of map, the sequence of stable periodic orbits is the same regardless of the specific form of the map <!--[ 257 ]--> . For the logistic map, the parameter interval is 3 < a < 4, but for the sine map ( 4-1 ), the parameter interval for the common sequence of stable periodic orbits is 0.71... < b < 1 <!--[ 256 ]-->. This universal sequence of stable periodic orbits is called the U sequence <!--[ 254 ]-->.

In addition, the logistic map has the property that its Schwarzian derivative is always negative on the interval [0, 1] . The Schwarzian derivative of a map f (of class C3 ) is 

{{NumBlk|:|<math>{\displaystyle Sf(x)={\frac {f'''(x)}{f'(x)}}-{\frac {3}{2}}\left({\frac {f''(x)}{f'(x)}}\right)^{2}}	</math>|{{EquationRef|4-2}}}}

<!--[ 258 ]--> 
In fact, when we calculate the Schwarzian derivative of the logistic map, we get

{{NumBlk|:|<math>{\displaystyle S(ax(1-x))={\frac {-6}{(1-2x)^{2}}}<0}</math>|{{EquationRef|4-3}}}}

where the Schwarzian derivative is negative regardless of the values of a and x . <!--[ 259 ]--> It is known that if a one-dimensional mapping from [0, 1] to [0, 1] is unimodal and has a negative Schwarzian derivative, then there is at most one stable periodic orbit . <!--[ 260 ]-->

=== Topological conjugate mapping ===

Let the symbol ∘ denote the composition of maps . In general, for  a topological space X, Y, two maps f  : X → X and g  : Y → Y are composed by a homeomorphism h : X → Y.

{{NumBlk|:|<math>{\displaystyle h\circ f=g\circ h}</math>|{{EquationRef|4-4}}}}

f and g are said to be phase conjugates if they satisfy the relation <!--[ 261 ]-->. The concept of phase conjugation plays an important role in the study of dynamical systems <!--[ 262 ]-->. Phase conjugate f and g exhibit essentially identical behavior, and if the behavior of f is periodic, then g is also periodic, and if the behavior of f is chaotic, then g is also chaotic <!--[ 262 ]-->.

In particular, if a homeomorphism h is linear, then f and g are said to be linearly conjugate . <!--[ 263 ]--> Every quadratic function is linearly conjugate with every other quadratic function . <!--[ 264 ]--> Hence,

{{NumBlk|:|<math>{\displaystyle x_{n+1}=x_{n}^{2}+b}</math>|{{EquationRef|4-5}}}}

{{NumBlk|:|<math>{\displaystyle x_{n+1}=1-cx_{n}^{2}}</math>|{{EquationRef|4-6}}}}

{{NumBlk|:|<math>{\displaystyle x_{n+1}=d-x_{n}^{2}}</math>|{{EquationRef|4-7}}}}

are linearly conjugates of the logistic map for any parameter a <!--[ 265 ]-->. Equations ( 4-6 ) and ( 4-7 ) are also called logistic maps <!--[ 266 ]-->. In particular, the form ( 4-7 ) is suitable for time-consuming numerical calculations, since it requires less computational effort <!--[ 134 ]-->.

[[File:Tent map cobweb diagram, example of parameter 2.png|class=skin-invert-image|thumb|Orbital view of the tent map ( 4-8 ). It has a topological conjugate relationship with the a = 4 logistic map.]]

Moreover, the logistic map <math>f_{a=4}</math> for <math>r = 4</math> is topologically conjugate to the following tent map T  ( x ) and Bernoulli shift map B  ( x ) <!--[ 267 ]-->.

{{NumBlk|:|<math>{\displaystyle T(x_{n})={\begin{cases}2x_{n}&\left(0\leq x_{n}\leq {\frac {1}{2}}\right)\\2(1-x_{n})&\left({\frac {1}{2}}\leq x_{n}\leq 1\right)\end{cases}}}</math>|{{EquationRef|4-8}}}}

{{NumBlk|:|<math>{\displaystyle B(x_{n})={\begin{cases}2x_{n}&\left(0\leq x_{n}<{\frac {1}{2}}\right)\\2x_{n}-1&\left({\frac {1}{2}}\leq x_{n}\leq 1\right)\end{cases}}}</math>|{{EquationRef|4-9}}}}

These phase conjugate relations can be used to prove that the logistic map <math>f_{a=4}</math> is strictly chaotic and to derive the exact solution ( 3-19 ) of <math>f_{r=4}</math> <!--[ 268 ]-->.

Alternatively, introducing the concept of symbolic dynamical systems, consider the following shift map σ defined on the symbolic string space consisting of strings of 0s and 1s as introduced above <!--[ 269 ]-->:

{{NumBlk|:|<math>{\displaystyle \sigma (s_{0}s_{1}s_{2}\cdots )=(s_{1}s_{2}\cdots )}</math>|{{EquationRef|4-10}}}}

Here, <math>s_i</math> is 0 or 1. On the set <math>\Lambda</math> introduced in equation ( 3-18 ), the logistic map <math>f_{r>4}</math> is [[topologically conjugate]] to the shift map, so we can use this to derive that <math>f_{r>4}</math> on <math>\Lambda</math> is chaotic <!--[ 270 ]-->.

=== Period-doubling route to chaos ===

In the logistic map, we have a function <math>f_r (x) = rx(1-x)</math>, and we want to study what happens when we iterate the map many times. The map might fall into a fixed point, a fixed cycle, or chaos. When the map falls into a stable fixed cycle of length <math>n</math>, we would find that the graph of <math>f_r^n</math> and the graph of <math>x\mapsto x</math> intersects at <math>n</math> points, and the slope of the graph of <math>f_r^n</math> is bounded in <math>(-1, +1)</math> at those intersections. 

For example, when <math>r=3.0</math>, we have a single intersection, with slope bounded in <math>(-1, +1)</math>, indicating that it is a stable single fixed point.

As <math>r</math> increases to beyond <math>r=3.0</math>, the intersection point splits to two, which is a period doubling. For example, when <math>r=3.4</math>, there are three intersection points, with the middle one unstable, and the two others stable.

As <math>r</math> approaches <math>r = 3.45</math>, another period-doubling occurs in the same way. The period-doublings occur more and more frequently, until at a certain <math>r\approx 3.56994567</math>, the period doublings become infinite, and the map becomes chaotic. This is the [[Period-doubling bifurcation|period-doubling route to chaos]].
<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          = Relationship between <math>x_{n+2}</math> and <math>x_{n}</math> when <math>a=2.7</math>. Before the period doubling bifurcation occurs. The orbit converges to the 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          = Relationship between <math>x_{n+2}</math> and <math>x_{n}</math> when <math>a=3</math>. 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          = Relationship between <math>x_{n+2}</math> and <math>x_{n}</math> when <math>a=3.3</math>. The fixed point <math>x_{f2}</math> becomes unstable, splitting into a periodic-2 stable cycle.
}}</div>

<div class=skin-invert-image>{{multiple image
| align             = center
| direction         = horizontal
| total_width       = 620
| image1            = Logistic map iterates, r=3.0.svg
| caption1          = When <math>r=3.0</math>, we have a single intersection, with slope exactly <math>+1</math>, indicating that it is about to undergo a period-doubling.
| image2            = Logistic iterates 3.4.svg
| caption2          = When <math>r=3.4</math>, there are three intersection points, with the middle one unstable, and the two others stable.
| image3            = Logistic iterates r=3.45.svg
| caption3          = When <math>r=3.45</math>, there are three intersection points, with the middle one unstable, and the two others having slope exactly <math>+1</math>, indicating that it is about to undergo another period-doubling.
| image4            = Logistic iterates with r=3.56994567.svg
| caption4          = When <math>r\approx 3.56994567</math>, there are infinitely many intersections, and we have arrived at [[Period-doubling bifurcation|chaos via the period-doubling route]].
| perrow            = 2/2
}}</div>

=== Scaling limit ===
{{Main|Feigenbaum function}}
[[File:Logistic_map_approaching_the_scaling_limit.webm|thumb|478x478px|Approach to the scaling limit as <math>r</math> approaches <math>r^* = 3.5699\cdots</math> from below.]][[File:Logistic iterates, together, r=3.56994567.svg|class=skin-invert-image|thumb|489x489px|At the point of chaos <math>r^* = 3.5699\cdots</math>, as we repeat the period-doublings<math>f^{1}_{r^*}, f^{2}_{r^*}, f^{4}_{r^*}, f^{8}_{r^*}, f^{16}_{r^*}, \dots</math>, the graphs seem to resemble each other, except that they are shrunken towards the middle, and rotated by 180 degrees, converging to a fractal.]]
Looking at the images, one can notice that at the point of chaos <math>r^* = 3.5699\cdots</math>, the curve of <math>f^{\infty}_{r^*}</math> looks like a fractal. Furthermore, as we repeat the period-doublings<math>f^{1}_{r^*}, f^{2}_{r^*}, f^{4}_{r^*}, f^{8}_{r^*}, f^{16}_{r^*}, \dots</math>, the graphs seem to resemble each other, except that they are shrunken towards the middle, and rotated by 180 degrees. 

This suggests to us a scaling limit: if we repeatedly double the function, then scale it up by <math>\alpha </math> for a certain constant <math>\alpha </math>:<math display="block">f(x) \mapsto - \alpha f( f(-x/\alpha ) ) 
 </math>then at the limit, we would end up with a function <math>g </math> that satisfies <math> g(x) = - \alpha g( g(-x/\alpha ) ) </math>. This is a [[Feigenbaum function]], which appears in most period-doubling routes to chaos (thus it is an instance of '''universality'''). Further, as the period-doubling intervals become shorter and shorter, the ratio between two period-doubling intervals converges to a limit, the first Feigenbaum constant <math>\delta = 4.6692016\cdots </math>.[[File:Logistic scaling with varying scaling factor.webm|thumb|480x480px|For the wrong values of scaling factor <math>\alpha </math>, the map does not converge to a limit, but when <math>\alpha = 2.5029\dots </math>, it converges.]]
[[File:Logistic scaling limit, r=3.56994567.svg|class=skin-invert-image|thumb|487x487px|At the point of chaos <math>r^* = 3.5699\cdots</math>, as we repeat the functional equation iteration <math>f(x) \mapsto - \alpha f( f(-x/\alpha ) ) 
 </math> with <math>\alpha = 2.5029\dots </math>, we find that the map does converge to a limit.]]The constant <math>\alpha </math> can be numerically found by trying many possible values. For the wrong values, the map does not converge to a limit, but when it is <math>\alpha = 2.5029\dots </math>, it converges. This is the second Feigenbaum constant.

=== Chaotic regime ===
In the chaotic regime, <math>f^\infty_r</math>, the limit of the iterates of the map, becomes chaotic dark bands interspersed with non-chaotic bright bands.
[[File:Logistic_map_in_the_chaotic_regime.webm|thumb|470x470px|In the chaotic regime, <math>f^\infty_r</math>, the limit of the iterates of the map, becomes chaotic dark bands interspersed with non-chaotic bright bands.]]

=== Other scaling limits ===
When <math>r</math> approaches <math>r \approx 3.8494344</math>, we have another period-doubling approach to chaos, but this time with periods 3, 6, 12, ... This again has the same Feigenbaum constants <math>\delta, \alpha</math>. The limit of <math display="inline">f(x) \mapsto - \alpha f( f(-x/\alpha ) ) 
 </math> is also the same [[Feigenbaum function]]. This is an example of '''universality'''.[[File:Logistic_map_approaching_the_period-3_scaling_limit.webm|thumb|482x482px|Logistic map approaching the period-doubling chaos scaling limit <math>r^* = 3.84943\dots</math> from below. At the limit, this has the same shape as that of <math>r^* = 3.5699\cdots</math>, since all period-doubling routes to chaos are the same (universality).]]

We can also consider period-tripling route to chaos by picking a sequence of <math>r_1, r_2, \dots</math> such that <math>r_n</math> is the lowest value in the period-<math>3^n</math> window of the bifurcation diagram. For example, we have <math>r_1 = 3.8284, r_2 = 3.85361, \dots</math>, with the limit <math>r_\infty = 3.854 077 963\dots</math>. This has a different pair of Feigenbaum constants <math>\delta= 55.26\dots, \alpha = 9.277\dots</math>.<ref name=":1">{{Cite journal |last1=Delbourgo |first1=R. |last2=Hart |first2=W. |last3=Kenny |first3=B. G. |date=1985-01-01 |title=Dependence of universal constants upon multiplication period in nonlinear maps |url=https://link.aps.org/doi/10.1103/PhysRevA.31.514 |journal=Physical Review A |language=en |volume=31 |issue=1 |pages=514–516 |doi=10.1103/PhysRevA.31.514 |pmid=9895509 |bibcode=1985PhRvA..31..514D |issn=0556-2791}}</ref> And <math>f^\infty_r</math>converges to the fixed point to <math display="block">f(x) \mapsto - \alpha f(f( f(-x/\alpha ) )) 
 </math>As another example, period-4-pling has a pair of Feigenbaum constants distinct from that of period-doubling, even though period-4-pling is reached by two period-doublings. In detail, define <math>r_1, r_2, \dots</math> such that <math>r_n</math> is the lowest value in the period-<math>4^n</math> window of the bifurcation diagram. Then we have <math>r_1 =3.960102, r_2 = 3.9615554, \dots</math>, with the limit <math>r_\infty = 3.96155658717\dots</math>. This has a different pair of Feigenbaum constants <math>\delta= 981.6\dots, \alpha = 38.82\dots</math>. 

In general, each period-multiplying route to chaos has its own pair of Feigenbaum constants. In fact, there are typically more than one. For example, for period-7-pling, there are at least 9 different pairs of Feigenbaum constants.<ref name=":1" />

Generally, <math display="inline">3\delta \approx 2\alpha^2 </math>, and the relation becomes exact as both numbers increase to infinity: <math>\lim \delta/\alpha^2 = 2/3</math>.

===Feigenbaum universality of 1-D maps===
Universality of one-dimensional maps with parabolic maxima and [[Feigenbaum constants]] <math>\delta=4.669201...</math>, <math>\alpha=2.502907...</math>.<ref>[http://chaosbook.org/extras/mjf/LA-6816-PR.pdf Feigenbaum, M. J. (1976) "Universality in complex discrete dynamics", Los Alamos Theoretical Division Annual Report 1975-1976]</ref><ref>{{cite journal|last=Feigenbaum|first=Mitchell|date=1978|title=Quantitative universality for a class of nonlinear transformations|journal=Journal of Statistical Physics|volume=19|issue=1|pages=25–52|bibcode=1978JSP....19...25F|citeseerx=10.1.1.418.9339|doi=10.1007/BF01020332|s2cid=124498882}}</ref>

The gradual increase of <math>G</math> at interval <math>[0, \infty)</math> changes dynamics from regular to chaotic one <ref name="Okulov, A Yu 1984">{{cite journal|last1=Okulov|first1=A Yu|last2=Oraevskiĭ|first2=A N|year=1984|title=Regular and stochastic self-modulation in a ring laser with nonlinear element|journal=Soviet Journal of Quantum Electronics|volume=14|issue=2|pages=1235–1237|bibcode=1984QuEle..14.1235O|doi=10.1070/QE1984v014n09ABEH006171}}</ref> with qualitatively the same [[bifurcation diagram]] as those for logistic map.

=== Renormalization estimate ===
The Feigenbaum constants can be estimated by a renormalization argument. (Section 10.7,<ref name=":0" />).

By universality, we can use another family of functions that also undergoes repeated period-doubling on its route to chaos, and even though it is not exactly the logistic map, it would still yield the same Feigenbaum constants.

Define the family <math display="block">f_r(x) = -(1+r)x + x^2</math>The family has an equilibrium point at zero, and as <math>r</math> increases, it undergoes period-doubling bifurcation at <math>r = r_0, r_1, r_2, ...</math>.

The first bifurcation occurs at <math>r = r_0 = 0</math>. After the period-doubling bifurcation, we can solve for the period-2 stable orbit by <math>f_r(p) = q, f_r(q) = p</math>, which yields <math display="block">\begin{cases}
p = \frac 12 (r + \sqrt{r(r+4)}) \\
q = \frac 12 (r - \sqrt{r(r+4)})
\end{cases}</math>At some point <math>r = r_1</math>, the period-2 stable orbit undergoes period-doubling bifurcation again, yielding a period-4 stable orbit. In order to find out what the stable orbit is like, we "zoom in" around the region of <math>x = p</math>, using the affine transform <math>T(x) = x/c + p</math>. Now, by routine algebra, we have<math display="block">(T^{-1}\circ f_r^2 \circ T)(x) = -(1+S(r)) x + x^2 + O(x^3)</math>where <math>S(r) = r^2 + 4r - 2, c = r^2 + 4r - 3\sqrt{r(r+4)}</math>. At approximately <math>S(r) = 0</math>, the second bifurcation occurs, thus <math>S(r_1) \approx 0</math>.

By self-similarity, the third bifurcation when <math>S(r) \approx r_1</math>, and so on. Thus we have <math>r_n \approx S(r_{n+1})</math>, or <math>r_{n+1} \approx \sqrt{r_{n}+6}-2</math>. Iterating this map, we find <math>r_\infty = \lim_n r_n \approx \lim_n S^{-n}(0) = \frac 12(\sqrt{17}-3)</math>, and <math>\lim_n \frac{r_\infty - r_n}{r_\infty - r_{n+1}} \approx S'(r_\infty) \approx 1 + \sqrt{17}</math>.

Thus, we have the estimates <math>\delta \approx 1+\sqrt{17} = 5.12...</math>, and <math>\alpha \approx r_\infty^2 +4r_\infty- 3 \sqrt{r_\infty^2+4r_\infty} \approx -2.24...</math>. These are within 10% of the true values.