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The '''logistic map''' is a [[polynomial]] [[map (mathematics)|mapping]] (equivalently, [[recurrence relation]]) of [[Quadratic function|degree 2]], often cited as an archetypal example of how complex, [[chaos theory|chaotic]] behaviour can arise from very simple [[non-linear]] dynamical equations.<ref name="Boeing2016Systems">{{cite journal|url=http://geoffboeing.com/publications/nonlinear-chaos-fractals-prediction/|author=Boeing, G.|title=Visual Analysis of Nonlinear Dynamical Systems: Chaos, Fractals, Self-Similarity and the Limits of Prediction|journal=Systems|date=2016|volume=4|issue=4|pages=37|accessdate=2016-12-02|doi=10.3390/systems4040037|arxiv=1608.04416}}</ref> The map was popularized in a 1976 paper by the biologist [[Robert May, Baron May of Oxford|Robert May]],<ref name="May, Robert M 1976">{{cite journal |last=May |first=Robert M. |year=1976 |title=Simple mathematical models with very complicated dynamics |journal=[[Nature (journal)|Nature]] |volume=261 |issue=5560 |pages=459–467 |doi=10.1038/261459a0 |bibcode=1976Natur.261..459M |pmid=934280 }}</ref> in part as a discrete-time demographic model analogous to the [[Logistic function#Logistic differential equation|logistic equation]] first created by [[Pierre François Verhulst]].<ref>"{{MathWorld | urlname=LogisticEquation | title= Logistic Equation}}</ref>
Mathematically, the logistic map is written
{{NumBlk|:|<math>x_{n+1} = r x_n \left(1-x_n\right)</math>|{{EquationRef|1}}}}
where {{mvar|x<sub>n</sub>}} is a number between zero and one that represents the ratio of existing population to the maximum possible population. The values of interest for the parameter {{mvar|r}}  (sometimes also denoted {{mvar|μ}}) are those in the interval {{math|[0,4]}}.
This nonlinear difference equation is intended to capture two effects:
* ''reproduction'' where the population will increase at a rate [[Proportionality (mathematics)|proportional]] to the current population when the population size is small.
* ''starvation'' (density-dependent mortality) where the growth rate will decrease at a rate proportional to the value obtained by taking the theoretical "carrying capacity" of the environment less the current population.

However, as a [[demography|demographic model]] the logistic map has the pathological problem  that some initial conditions and parameter values (for example, if {{math|''r'' > 4}}) lead to negative population sizes.  This problem does not appear in the older [[Ricker model]], which also exhibits chaotic dynamics.

The {{math|''r'' {{=}} 4}} case of the logistic map is a nonlinear transformation of both the [[bit-shift map]] and the {{math|''μ'' {{=}} 2}} case of the [[tent map]].

== Characteristics of the map ==

===Behavior dependent on {{mvar|''r''}}===
The image below shows the amplitude and frequency content of some logistic map iterates for parameter values ranging from 2&nbsp;to&nbsp;4.

[[Image:Logistic map animation.gif]]

By varying the parameter {{mvar|r}}, the following behavior is observed:
[[File:Logistic map.gif|300px|thumb|right|Evolution of different initial conditions as a funcrion of {{mvar|r}}]]
* With {{mvar|r}} between 0 and 1, the population will eventually die, independent of the initial population.
* With {{mvar|r}} between 1 and 2, the population will quickly approach the value {{math|{{sfrac|''r'' − 1|''r''}}}}, independent of the initial population.
* With {{mvar|r}} between 2 and 3, the population will also eventually approach the same value {{math|{{sfrac|''r'' − 1|''r''}}}}, but first will fluctuate around that value for some time. The [[rate of convergence]] is linear, except for {{math|''r'' {{=}} 3}}, when it is dramatically slow, less than linear (see [[Bifurcation memory]]).
* With {{mvar|r}} between 3 and 1&nbsp;+&nbsp;{{sqrt|6}} ≈ 3.44949, from [[almost all]] initial conditions the population will approach  permanent oscillations between two values. These two values are dependent on {{mvar|r}}.
* With {{mvar|r}} between 3.44949 and 3.54409 (approximately), from almost all initial conditions the population will approach permanent oscillations among four values. The latter number is a root of a 12th degree polynomial {{OEIS|id=A086181}}.
* With {{mvar|r}} increasing beyond 3.54409, from almost all initial conditions the population will approach oscillations among 8 values, then 16, 32, etc. The lengths of the parameter intervals that yield oscillations of a given length decrease rapidly;  the ratio between the lengths of two successive bifurcation intervals approaches the [[Feigenbaum constant]] {{math|''δ'' ≈ 4.66920}}. This behavior is an example of a [[period-doubling bifurcation|period-doubling cascade]].
* At {{math|''r'' ≈ 3.56995}} {{OEIS|id=A098587}} is the onset of chaos, at the end of the period-doubling cascade. From almost all initial conditions, we no longer see oscillations of finite period.  Slight variations in the initial population yield dramatically different results over time, a prime characteristic of chaos.
* Most values of {{mvar|r}} beyond 3.56995 exhibit chaotic behaviour, but there are still certain isolated ranges of {{mvar|r}} that show non-chaotic behavior; these are sometimes called ''islands of stability''. For instance, beginning at 1&nbsp;+&nbsp;{{sqrt|8}}<ref>{{cite journal|last=Zhang |first=Cheng |title=Period three begins |journal=[[Mathematics Magazine]] |volume=83 |date=October 2010 |pages=295–297 |doi=10.4169/002557010x521859}}</ref> (approximately 3.82843) there is a range of parameters {{mvar|r}} that show oscillation among three values, and for slightly higher values of {{mvar|r}} oscillation among 6 values, then 12 etc.
* The development of the chaotic behavior of the logistic sequence as the parameter {{mvar|r}} varies from approximately 3.56995 to approximately 3.82843 is sometimes called the [[Pomeau–Manneville scenario]], characterized by a periodic (laminar) phase interrupted by bursts of aperiodic behavior. Such a scenario has an application in semiconductor devices.<ref name="carson82">{{cite journal|first1=Carson |last1=Jeffries|first2=José |last2=Pérez |journal=[[Physical Review A]]|year=1982|title=Observation of a Pomeau–Manneville intermittent route to chaos in a nonlinear oscillator|volume=26 |issue=4 |pages=2117–2122|doi=10.1103/PhysRevA.26.2117|bibcode = 1982PhRvA..26.2117J |url=http://www.escholarship.org/uc/item/2dm2k8mm}}</ref> There are other ranges that yield oscillation among 5 values etc.; all oscillation periods occur for some values of {{mvar|r}}.  A ''period-doubling window'' with parameter {{mvar|c}} is a range of {{mvar|r}}-values consisting of a succession of subranges.  The {{mvar|k}}th subrange contains the values of {{mvar|r}} for which there is a stable cycle (a cycle that attracts a set of initial points of unit measure) of period {{math|2<sup>''k''</sup>''c''}}.  This sequence of sub-ranges is called a ''cascade of harmonics''.<ref name="May">{{cite journal | first = R. M. |last=May | title = Simple mathematical models with very complicated dynamics | journal = Nature | year = 1976 | volume = 261 | issue = 5560 | pages = 459–67 | doi = 10.1038/261459a0 | pmid = 934280|bibcode = 1976Natur.261..459M }}</ref> In a sub-range with a stable cycle of period {{math|2<sup>''k''*</sup>''c''}}, there are unstable cycles of period {{math|2<sup>''k''</sup>''c''}} for all {{math|''k'' < ''k''*}}.  The {{mvar|r}} value at the end of the infinite sequence of sub-ranges is called the ''point of accumulation'' of the cascade of harmonics.  As {{mvar|r}} rises there is a succession of new windows with different {{mvar|c}} values.  The first one is for {{math|''c'' {{=}} 1}}; all subsequent windows involving odd {{mvar|c}} occur in decreasing order of {{mvar|c}} starting with arbitrarily large {{mvar|c}}.<ref name="May" /><ref>{{cite journal |last1=Baumol |first1=William J. |authorlink=William Baumol |last2=Benhabib |first2=Jess |authorlink2=Jess Benhabib |title=Chaos: Significance, Mechanism, and Economic Applications |journal=[[Journal of Economic Perspectives]] |date=February 1989 |volume=3 |issue=1 |pages=77–105 |doi=10.1257/jep.3.1.77 }}</ref>  
* Beyond {{math|''r'' {{=}} 4}}, almost all initial values eventually leave the interval {{math|[0,1]}} and diverge.

For any value of {{mvar|r}} there is at most one stable cycle.  If a stable cycle exists, it is globally stable,  attracting almost all points.<ref>{{cite book |last=Collet |first=Pierre |first2=Jean-Pierre |last2=Eckmann |authorlink2=Jean-Pierre Eckmann |title=Iterated Maps on the Interval as Dynamical Systems |url=https://archive.org/details/iteratedmapsonin0000coll |url-access=registration |location= |publisher=Birkhauser |year=1980 |isbn=978-3-7643-3026-2 }}</ref>{{rp|13}}  Some values of {{mvar|r}} with a stable cycle of some period have infinitely many unstable cycles of various periods.

The  [[bifurcation diagram]] at right summarizes this. The horizontal axis shows the possible values of the parameter {{mvar|r}} while the vertical axis shows the set of values of {{mvar|x}} visited asymptotically from almost all initial conditions by the iterates of the logistic equation with that {{mvar|r}} value.

[[Image:Logistic Bifurcation map High Resolution.png|thumb|right|[[Bifurcation diagram]] for the logistic map.
The [[attractor]] for any value of the parameter {{mvar|r}} is shown on the vertical line at that {{mvar|r}}.]]
The bifurcation diagram is a [[self-similar]]: if we zoom in on the above-mentioned value {{math|''r'' ≈ 3.82843}} and focus on one arm of the three, the situation nearby looks like a shrunk and slightly distorted version of the whole diagram. The same is true for all other non-chaotic points. This is an example of the deep and ubiquitous connection between [[chaos (mathematics)|chaos]] and [[fractal]]s.
[[File:Subsection Bifurcation Diagram Logistic Map.png|thumb|Magnification of the chaotic region of the map.]]
[[File:Logistic map bifurcation diagram magnifications.png|thumb|Stable regions within the chaotic region.]]

===Chaos and the logistic map===
[[Image:Logistic map scatterplots large.png|thumb|right|Two- and three-dimensional [[Poincaré plot]]s show the stretching-and-folding structure of the logistic map]]
[[Image:LogisticCobwebChaos.gif|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|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.<ref name="Boeing2016Systems" />  A rough description of chaos is that chaotic systems exhibit a great sensitivity to initial conditions—a property of the logistic map for most values of {{mvar|r}} between about 3.57 and 4 (as noted above).<ref name="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), right, 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.

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 [[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" />

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 }}</ref> the invariant measure is

:<math>\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.

== 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> \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="Schröder" /> 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 | format = PDF | journal = Journal of Difference Equations and Applications | volume = 10 | issue = 11| pages = 949–953 | doi = 10.1080/10236190412331285351 | arxiv = nlin/0407042 }}</ref>  
The solution when {{math|1=''r'' = 4}} is,<ref name="Schröder">
{{cite journal |doi=10.1007/BF01443992 |last=Schröder |first=Ernst |authorlink=Ernst Schröder |year=1870 |title=Über iterierte Funktionen |journal=Math. Ann. |volume=3 |issue= 2|pages=296–322 |id= |url= |accessdate= |quote= }}</ref><ref>{{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}}</ref>

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

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

:<math>\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 numbers instead of trigonometric functions is<ref name="schr">{{cite journal |last=Schröder |first=Ernst |authorlink=Ernst Schröder |year=1870 |title=Ueber iterirte Functionen|journal=Mathematische Annalen |volume=3 |issue= 2|pages=296–322 | doi=10.1007/BF01443992 }}</ref>

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

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

:<math>\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>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'' ≥ 1}}.  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>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>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.

== Related concepts ==

===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}}</ref> is well visible with map proposed as a toy 
model for discrete laser dynamics: 
<math> x \rightarrow G x (1 - \tanh (x))</math>,
where <math>x</math> stands for electric field amplitude, <math>G</math><ref name="Okulov, A Yu 1986">{{cite journal|last1=Okulov|first1=A Yu|last2=Oraevskiĭ|first2=A N|year=1986|title=Space–temporal behavior of a light pulse propagating in a nonlinear nondispersive medium|journal=J. Opt. Soc. Am. B|volume=3|issue=5|pages=741–746|bibcode=1986OSAJB...3..741O|doi=10.1364/JOSAB.3.000741}}</ref> is laser gain as bifurcation parameter. 
[[Image:Bifurcation_diagr_Tanh_Map_jpg_wiki.jpg|thumb|right|[[Bifurcation diagram]] for the hyperbolic tangent map.
It is a [[self-similar]] in a wider range of bifurcation parameter G. This is the else ubiquitous connection between [[chaos (mathematics)|chaos]] and [[fractal]]s.]]
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.

:

== See also ==
* [[Logistic function]], solution of the logistic map's continuous counterpart: the [[Logistic differential equation]].
* [[Lyapunov stability#Definition for discrete-time systems]] 
* [[Malthusian growth model]]
* [[Periodic points of complex quadratic mappings]], of which the logistic map is a special case confined to the real line
* [[Radial basis function network]], which illustrates the inverse problem for the logistic map.
* [[Schröder's equation]]
* [[Stiff equation]]

==Notes==
{{reflist|30em}}

==References==
* {{cite journal
 | author1-link=Peter Grassberger|first1=P. |last1=Grassberger |first2=I. |last2=Procaccia
 | title=Measuring the strangeness of strange attractors
 | journal=Physica D
 | year = 1983 | volume = 9
 | issue=1–2 | pages=189–208
 | bibcode = 1983PhyD....9..189G
 | doi = 10.1016/0167-2789(83)90298-1
 }}
* {{cite journal
 | author1-link=Peter Grassberger|first1=P. |last1=Grassberger
 | title = On the Hausdorff dimension of fractal attractors
 | journal=Journal of Statistical Physics
 | year=1981 | volume=26
 | issue = 1 | pages=173–179
 | doi = 10.1007/BF01106792
 |bibcode = 1981JSP....26..173G }}
* {{cite book | last = Sprott | first = Julien Clinton | title = Chaos and Time-Series Analysis | publisher = Oxford University Press | year = 2003 | isbn = 978-0-19-850840-3 }}
* {{cite book | last = Strogatz | first = Steven | title = Nonlinear Dynamics and Chaos | publisher = Perseus Publishing | year = 2000 | isbn = 978-0-7382-0453-6 | url-access = registration | url = https://archive.org/details/nonlineardynamic00stro }}
* {{cite book <!-- Citation bot deny--> | first = Nicholas | last = Tufillaro |first2=Tyler |last2=Abbott |first3=Jeremiah |last3=Reilly  | title = An Experimental Approach to Nonlinear Dynamics and Chaos | url = https://archive.org/details/unset0000unse_q2b7 | url-access = registration | publisher = Addison-Wesley New York | year = 1992 | isbn = 978-0-201-55441-0}}

==External links==
* [https://web.archive.org/web/19991006003907/http://www.geocities.com/CapeCanaveral/Hangar/7959/logisticmap.html  Logistic Map]. Contains an interactive computer simulation of the logistic map.
* [http://hypertextbook.com/chaos/ The Chaos Hypertextbook]. An introductory primer on chaos and fractals.
* [https://web.archive.org/web/20060211001015/http://www.ibiblio.org/e-notes/MSet/Logistic.htm Interactive Logistic map] with iteration and bifurcation diagrams in Java.
* [https://web.archive.org/web/20090821122553/http://www.users.bigpond.com/pmurray/Java/LogisticMap.html Interactive Logistic map] showing fixed points.
* [http://www.egwald.ca/nonlineardynamics/logisticsmapchaos.php The Logistic Map and Chaos] by Elmer G. Wiens
* [https://web.archive.org/web/20070415033246/http://lectures.nsitlounge.in/ Complexity & Chaos (audiobook)] by Roger White. Chapter 5 covers the Logistic Equation.
* "[http://www.wolframscience.com/nksonline/page-918c-text History of iterated maps]," in ''[[A New Kind of Science]]'' by [[Stephen Wolfram]]. Champaign, IL: Wolfram Media, p.&nbsp;918, 2002.
* [http://ChaosBook.org/~predrag/papers/universalFunct.html "A very brief history of universality in period doubling" by P. Cvitanović]
* [https://chaosbook.blogspot.com/1993/05/acceptance-speech-1993-nkt-research.html "A not so short history of Universal Function"  by P. Cvitanović]
* [http://demonstrations.wolfram.com/DiscreteLogisticEquation/ Discrete Logistic Equation] by Marek Bodnar after work by Phil Ramsden, [[Wolfram Demonstrations Project]].
* [http://demonstrations.wolfram.com/OrbitDiagramOfTwoCoupledLogisticMaps/ Multiplicative coupling of 2 logistic maps] by C. Pellicer-Lostao and R. Lopez-Ruiz after work by Ed Pegg Jr, [[Wolfram Demonstrations Project]].
* [http://www.walkingrandomly.com/?p=2006 Using SAGE to investigate the discrete logistic equation]

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[[Category:Chaotic maps]]