Editing Logistic map (section)

====Finding cycles of any length when {{math|''r'' {{=}} 4}}====
For the {{math|''r'' {{=}} 4}} case, from almost all initial conditions the iterate sequence is chaotic. Nevertheless, there exist an infinite number of initial conditions that lead to cycles, and indeed there exist cycles of length {{mvar|k}} for ''all'' integers {{math|''k'' > 0}}. We can exploit the relationship of the logistic map to the [[dyadic transformation]] (also known as the ''bit-shift map'') to find cycles of any length. If {{mvar|x}} follows the logistic map {{math|''x''<sub>''n'' + 1</sub> {{=}} 4''x<sub>n</sub>''(1 − ''x<sub>n</sub>'')}} and {{mvar|y}} follows the ''dyadic transformation''

<math display="block">y_{n+1}=\begin{cases}2y_n & 0 \le y_n < \tfrac12 \\2y_n -1 & \tfrac12 \le y_n < 1, \end{cases}</math>

then the two are related by a [[homeomorphism]]

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

The reason that the dyadic transformation is also called the bit-shift map is that when {{mvar|y}} is written in binary notation, the map moves the binary point one place to the right (and if the bit to the left of the binary point has become a "1", this "1" is changed to a "0"). A cycle of length 3, for example, occurs if an iterate has a 3-bit repeating sequence in its binary expansion (which is not also a one-bit repeating sequence): 001, 010, 100, 110, 101, or 011. The iterate 001001001... maps into 010010010..., which maps into 100100100..., which in turn maps into the original 001001001...; so this is a 3-cycle of the bit shift map. And the other three binary-expansion repeating sequences give the 3-cycle 110110110... → 101101101... → 011011011... → 110110110.... Either of these 3-cycles can be converted to fraction form: for example, the first-given 3-cycle can be written as {{sfrac|1|7}} → {{sfrac|2|7}} → {{sfrac|4|7}} → {{sfrac|1|7}}. Using the above translation from the bit-shift map to the <math>r = 4</math> logistic map gives the corresponding logistic cycle 0.611260467... → 0.950484434... → 0.188255099... → 0.611260467.... We could similarly translate the other bit-shift 3-cycle into its corresponding logistic cycle. Likewise, cycles of any length {{mvar|k}} can be found in the bit-shift map and then translated into the corresponding logistic cycles.

However, since almost all numbers in {{math|[0,1)}} are irrational, almost all initial conditions of the bit-shift map lead to the non-periodicity of chaos. This is one way to see that the logistic {{math|''r'' {{=}} 4}} map is chaotic for almost all initial conditions.

The number of cycles of (minimal) length {{math|''k'' {{=}} 1, 2, 3,…}} for the logistic map with {{math|''r'' {{=}} 4}} ([[tent map]] with {{math|''μ'' {{=}} 2}}) is a known integer sequence {{OEIS|id=A001037}}: 2, 1, 2, 3, 6, 9, 18, 30, 56, 99, 186, 335, 630, 1161.... This tells us that the logistic map with {{math|''r'' {{=}} 4}} has 2 fixed points, 1 cycle of length 2, 2 cycles of length 3 and so on. This sequence takes a particularly simple form for prime {{mvar|k}}: {{math|2 ⋅ {{sfrac|2<sup>''k'' − 1</sup> − 1|''k''}}}}. For example: 2&nbsp;⋅&nbsp;{{sfrac|2<sup>13 − 1</sup> − 1|13}} = 630 is the number of cycles of length 13. Since this case of the logistic map is chaotic for almost all initial conditions, all of these finite-length cycles are unstable.