Editing Logistic map (section)

==== 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 ]-->