Editing Logistic map (section)

==Chaos and the logistic map==
[[Image:LogisticCobwebChaos.gif|class=skin-invert-image|thumb|right|A [[cobweb diagram]] of the logistic map, showing chaotic behaviour for most values of {{math|''r'' > 3.57}}]]
[[File:Iterated logistic functions.svg|class=skin-invert-image|right|thumb|[[Logistic function]] {{mvar|f}} (blue) and its iterated versions {{math|''f''{{isup|2}}}}, {{math|''f''{{isup|3}}}}, {{math|''f''{{isup|4}}}} and {{math|''f''{{isup|5}}}} for {{math|''r'' {{=}} 3.5}}. For example, for any initial value on the horizontal axis, {{math|''f''{{isup|4}}}} gives the value of the iterate four iterations later.]]

The relative simplicity of the logistic map makes it a widely used point of entry into a consideration of the concept of chaos. A rough description of chaos is that chaotic systems exhibits:<ref name=":2" group="Devaney 1989">{{harvnb|Devaney|1989|p=50}}</ref> (see [[Chaos theory#Chaotic dynamics|Chaotic dynamics]])
* [[butterfly effect|Great sensitivity on initial conditions]]: i.e. for a small or infinitesimal variation in the initial conditions you may have a large finite effect.
* [[Topologically transitive]]: i.e. the system tends to occupy all available states in a similar sense to fluid mixing.<ref>{{cite web | url=https://encyclopediaofmath.org/wiki/Topological_transitivity | title=Topological transitivity |website=Encyclopedia of Mathematics }}</ref>
* The system exhibits [[Dense set|dense]] [[periodic orbits]]

These are properties of the logistic map for most values of {{mvar|r}} between about 3.57 and 4 (as noted above).<ref name=":2" group="May, Robert M. (1976)"/>  A common source of such sensitivity to initial conditions is that the map represents a repeated folding and stretching of the space on which it is defined. In the case of the logistic map, the [[quadratic polynomial|quadratic]] [[difference equation]] describing it may be thought of as a stretching-and-folding operation on the interval {{math|(0,1)}}.<ref name="Gleick">{{cite book |last=Gleick |first=James |title=Chaos: Making a New Science |year=1987 |publisher=Penguin Books |location=London |isbn=978-0-14-009250-9 }}</ref>

The following figure illustrates the stretching and folding over a sequence of iterates of the map. Figure (a), left, shows a two-dimensional [[Poincaré plot]] of the logistic map's [[state space]] for {{math|''r'' {{=}} 4}}, and clearly shows the quadratic curve of the difference equation ({{EquationNote|1}}).  However, we can [[embedding|embed]] the same sequence in a three-dimensional state space, in order to investigate the deeper structure of the map. Figure (b) demonstrates this, showing how initially nearby points begin to diverge, particularly in those regions of {{mvar|x<sub>t</sub>}} corresponding to the steeper sections of the plot.

[[Image:Logistic map scatterplots large.png|class=skin-invert-image|center|upright=1.8|Two- and three-dimensional [[Poincaré plot]]s show the stretching-and-folding structure of the logistic map]]

This stretching-and-folding does not just produce a gradual divergence of the sequences of iterates, but an exponential divergence (see [[Lyapunov exponent]]s), evidenced also by the [[complexity]] and [[Predictability|unpredictability]] of the chaotic logistic map.  In fact, exponential divergence of sequences of iterates explains the connection between chaos and unpredictability: a small error in the supposed initial state of the system will tend to correspond to a large error later in its evolution.  Hence, predictions about future states become progressively (indeed, [[exponential growth|exponentially]]) worse when there are even very small errors in our knowledge of the initial state. This quality of unpredictability and apparent randomness led the logistic map equation to be used as a [[pseudo-random number generator]] in early computers.<ref name="Gleick" />

At {{mvar|r}} = 2, the function <math>rx(1-x)</math> intersects <math>y = x</math> precisely at the maximum point, so convergence to the equilibrium point is on the order of <math>\delta^{2^n}</math>. Consequently, the equilibrium point is called "superstable". Its Lyapunov exponent is <math>-\infty</math>. A similar argument shows that there is a superstable <math>r</math> value within each interval where the dynamical system has a stable cycle. This can be seen in the Lyapunov exponent plot as sharp dips.<ref name=":0">{{Cite book |last=Strogatz |first=Steven |url=https://www.worldcat.org/oclc/1112373147 |title=Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering |date=2019 |isbn=978-0-367-09206-1 |edition=2nd |location=Boca Raton |chapter=10.1: Fixed Points and Cobwebs |oclc=1112373147}}</ref>

Since the map is confined to an interval on the real number line, its dimension is less than or equal to unity. Numerical estimates yield a [[correlation dimension]] of {{val|0.500|0.005}} ([[Peter Grassberger|Grassberger]], 1983), a [[Hausdorff dimension]] of about 0.538  ([[Peter Grassberger|Grassberger]] 1981), and an [[information dimension]] of approximately 0.5170976 ([[Peter Grassberger|Grassberger]] 1983) for {{math|''r'' ≈ 3.5699456}} (onset of chaos). Note: It can be shown that the correlation dimension is certainly between 0.4926 and 0.5024.

It is often possible, however, to make precise and accurate statements about the ''[[Frequency#Statistical frequency|likelihood]]'' of a future state in a chaotic system.  If a (possibly chaotic) [[dynamical system]] has an [[attractor]], then there exists a [[probability measure]] that gives the long-run proportion of time spent by the system in the various regions of the attractor. In the case of the logistic map with parameter {{math|''r'' {{=}} 4}} and an initial state in {{math|(0,1)}}, the attractor is also the interval {{math|(0,1)}} and the probability measure corresponds to the [[beta distribution]] with parameters {{math|''a'' {{=}} 0.5}} and {{math|''b'' {{=}} 0.5}}. Specifically,<ref>{{cite journal |last=Jakobson |first=M. |title=Absolutely continuous invariant measures for one-parameter families of one-dimensional maps |journal=Communications in Mathematical Physics |volume=81 |issue=1 |year=1981 |pages=39–88 |doi=10.1007/BF01941800 |bibcode=1981CMaPh..81...39J |s2cid=119956479 |url=http://projecteuclid.org/euclid.cmp/1103920159 }}</ref> the invariant measure is

<math display="block">\frac{1}{\pi\sqrt{x(1-x)}}.</math>

Unpredictability is not randomness, but in some circumstances looks very much like it. Hence, and fortunately, even if we know very little about the initial state of the logistic map (or some other chaotic system), we can still say something about the distribution of states arbitrarily far into the future, and use this knowledge to inform [[decision theory|decision]]s based on the state of the system.
[[File:Logistic map with lyapunov exponent function.png|class=skin-invert-image|thumb|Logistic map with [[Lyapunov exponent]] function]]

=== Graphical representation ===
The [[bifurcation diagram]] for the logistic map can be visualized with the following [[Python (programming language)|Python]] code:
<syntaxhighlight lang="python3" line="1">
import numpy as np
import matplotlib.pyplot as plt

interval = (2.8, 4)  # start, end
accuracy = 0.0001
reps = 600  # number of repetitions
numtoplot = 200
lims = np.zeros(reps)

fig, biax = plt.subplots()
fig.set_size_inches(16, 9)

lims[0] = np.random.rand()
for r in np.arange(interval[0], interval[1], accuracy):
    for i in range(reps - 1):
        lims[i + 1] = r * lims[i] * (1 - lims[i])

    biax.plot([r] * numtoplot, lims[reps - numtoplot :], "b.", markersize=0.02)

biax.set(xlabel="r", ylabel="x", title="logistic map")
plt.show()
</syntaxhighlight>

=== Special cases of the map ===

====Upper bound when {{math|0 ≤ ''r'' ≤ 1}}====
Although exact solutions to the recurrence relation are only available in a small number of cases, a closed-form upper bound on the logistic map is known when {{math|0 ≤ ''r'' ≤ 1}}.<ref>{{cite arXiv|eprint=1710.05053|class=stat.ML|first1=Trevor|last1=Campbell|first2=Tamara|last2=Broderick|title=Automated scalable Bayesian inference via Hilbert coresets|date=2017}}</ref> There are two aspects of the behavior of the logistic map that should be captured by an upper bound in this regime: the asymptotic geometric decay with constant {{mvar|r}}, and the fast initial decay when {{math|''x''<sub>0</sub>}} is close to 1, driven by the {{math|(1 − ''x<sub>n</sub>'')}} term in the recurrence relation. The following bound captures both of these effects:

<math display="block"> \forall n \in \{0, 1, \ldots \} \quad \text{and} \quad x_0, r \in [0, 1], \quad x_n \le \frac{x_0}{r^{-n} + x_0n}. </math>

====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}}.

====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.