Editing Logistic map (section)

====Solution when {{math|1=''r'' = 4}}====
The special case of {{math|1=''r'' = 4}} can in fact be solved exactly, as can the case with {{math|1=''r'' = 2}};<ref name="schr" /> however, the general case can only be predicted statistically.<ref>{{cite journal | last1 = Little | first1 = M. | last2 = Heesch | first2 = D. | year = 2004 | title = Chaotic root-finding for a small class of polynomials | url = http://www.maxlittle.net/publications/GDEA41040.pdf | journal = Journal of Difference Equations and Applications | volume = 10 | issue = 11| pages = 949–953 | doi = 10.1080/10236190412331285351 | arxiv = nlin/0407042 | s2cid = 122705492 }}</ref> 
The solution when {{math|1=''r'' = 4}} is:<ref name="schr">{{cite journal |last=Schröder |first=Ernst |author-link=Ernst Schröder (mathematician) |year=1870 |title=Ueber iterirte Functionen|journal=Mathematische Annalen |volume=3 |issue= 2|pages=296–322 | doi=10.1007/BF01443992 |s2cid=116998358 }}</ref><ref name=":2">{{cite journal|last=Lorenz |first=Edward |date=1964 |title=The problem of deducing the climate from the governing equations |journal=Tellus |volume=16 |issue=February |pages=1–11|doi=10.3402/tellusa.v16i1.8893 |bibcode=1964Tell...16....1L |doi-access=free }}</ref>

<math display="block">x_{n}=\sin^{2}\left(2^{n} \theta \pi\right),</math>

where the initial condition parameter {{mvar|θ}} is given by

<math display="block">\theta = \tfrac{1}{\pi}\sin^{-1}\left(\sqrt{x_0}\right).</math>

For rational {{mvar|θ}}, after a finite number of iterations {{mvar|x<sub>n</sub>}} maps into a periodic sequence. But almost all {{mvar|θ}} are irrational, and, for irrational {{mvar|θ}}, {{mvar|x<sub>n</sub>}} never repeats itself – it is non-periodic. This solution equation clearly demonstrates the two key features of chaos – stretching and folding: the factor {{math|2<sup>''n''</sup>}} shows the exponential growth of stretching, which results in [[sensitive dependence on initial conditions]], while the squared sine function keeps {{mvar|x<sub>n</sub>}} folded within the range {{math|[0,1]}}.

For {{math|1=''r'' = 4}} an equivalent solution in terms of [[complex number]]s instead of trigonometric functions is<ref name="schr" />

<math display="block">x_n=\frac{-\alpha^{2^n} -\alpha^{-2^n} +2}{4}</math>

where {{mvar|α}} is either of the complex numbers

<math display="block">\alpha = 1 - 2x_0 \pm \sqrt{\left(1 - 2x_0\right)^2 - 1}</math>

with [[absolute value#Complex numbers|modulus]] equal to 1. Just as the squared sine function in the trigonometric solution leads to neither shrinkage nor expansion of the set of points visited, in the latter solution this effect is accomplished by the unit modulus of {{mvar|α}}.

By contrast, the solution when {{math|1=''r'' = 2}} is<ref name="schr" />

<math display="block">x_n = \tfrac{1}{2} - \tfrac{1}{2}\left(1-2x_0\right)^{2^n}</math>

for {{math|''x''<sub>0</sub> ∈ [0,1)}}. Since {{math|(1 − 2''x''<sub>0</sub>) ∈ (−1,1)}} for any value of {{math|''x''<sub>0</sub>}} other than the unstable fixed point 0, the term {{math|(1 − 2''x''<sub>0</sub>)<sup>2<sup>''n''</sup></sup>}} goes to 0 as {{mvar|n}} goes to infinity, so {{mvar|x<sub>n</sub>}} goes to the stable fixed point {{sfrac|1|2}}.