Editing Logistic map (section)

===== Qualitative Summary =====
[[File:Feigenbaum Tree.gif|class=skin-invert-image|thumb|284x284px|Evolution of different initial conditions as a function of {{mvar|r}} (The parameter k from the figure corresponds to the parameter r from the definition in the article.)]]
[[File:Feigenbaum tree with bias.gif|class=skin-invert-image|thumb|Evolution of different initial conditions as a function of ''{{mvar|r}}'' with bias (The parameter k from the figure corresponds to the parameter r from the definition in the article.)|283x283px]]

* 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.
* This number shall be compared and understood as the equivalent of the [[Reynolds number]] for the onset of other chaotic phenomena such as [[turbulence]] and similar to the [[critical temperature]] of a [[phase transition]]. In essence the [[phase space]] contains a full subspace of cases with extra dynamical variables to characterize the microscopic state of the system, these can be understood as [[Eddies]] in the case of turbulence and [[order parameters]] in the case of [[phase transitions]].
* 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 |issue=4 |pages=295–297 |doi=10.4169/002557010x521859|s2cid=123124113 }}</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.
* At <math>r = 1 + \sqrt 8 = 3.8284...</math>, the stable period-3 cycle emerges.<ref>{{Cite journal |last=Bechhoefer |first=John |date=1996-04-01 |title=The Birth of Period 3, Revisited |url=https://doi.org/10.1080/0025570X.1996.11996402 |journal=Mathematics Magazine |volume=69 |issue=2 |pages=115–118 |doi=10.1080/0025570X.1996.11996402 |issn=0025-570X}}</ref>
* 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 |s2cid=119466337 |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=":2" group="May, Robert M. (1976)"/> 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=":2" group="May, Robert M. (1976)"/><ref>{{cite journal |last1=Baumol |first1=William J. |author-link=William Baumol |last2=Benhabib |first2=Jess |author-link2=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 |doi-access=free }}</ref>  
* At <math>r = 3.678..., x = 0.728...</math>, two chaotic bands of the bifurcation diagram intersect in the first [[Misiurewicz point]] for the logistic map. It satisfies the equations <math>r^3 - 2r^2 - 4r -8 = 0, x = 1-1/r</math>.<ref>{{Cite web |title=Misiurewicz Point of the Logistic Map |url=https://sprott.physics.wisc.edu/chaos/mispoint.htm |access-date=2023-05-08 |website=sprott.physics.wisc.edu}}</ref>  
* Beyond {{math|''r'' {{=}} 4}}, almost all initial values eventually leave the interval {{math|[0,1]}} and diverge. The set of initial conditions which remain within {{math|[0,1]}} form a [[Cantor set]] and the dynamics restricted to this Cantor set is chaotic.<ref>{{cite book |last1=Teschl |first1=Gerald | author-link1=Gerald Teschl |title=Ordinary Differential Equations and Dynamical Systems |url=https://www.mat.univie.ac.at/~gerald/ftp/book-ode/index.html |url-access=registration |publisher=Amer. Math Soc. |year=2012 |isbn=978-0-8218-8328-0 }}</ref>

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 |last1=Collet |first1=Pierre |first2=Jean-Pierre |last2=Eckmann |author-link2=Jean-Pierre Eckmann |title=Iterated Maps on the Interval as Dynamical Systems |url=https://archive.org/details/iteratedmapsonin0000coll |url-access=registration |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.

[[File:Logistic Bifurcation map High Resolution.png|class=skin-invert-image|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]] 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.

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.


<div class=skin-invert-image>{{multiple image
| align             = center
| direction         = horizontal
| total_width       = 900
| image1            = Subsection Bifurcation Diagram Logistic Map.png
| caption1          = Magnification of the chaotic region of the map
| image2            = Logistic map bifurcation diagram magnifications.png
| caption2          = Stable regions within the chaotic region, where a tangent bifurcation occurs at the boundary between the chaotic and periodic attractor, giving intermittent trajectories as described in the [[Pomeau–Manneville scenario]]
}}</div>

We can also consider negative values of {{mvar|r}}:
* For {{mvar|r}} between -2 and -1 the logistic sequence also features chaotic behavior.<ref name="Takashi Tsuchiya, Daisuke Yamagishi, 1997">{{cite journal|last1=Tsuchiya|first1=Takashi|last2=Yamagishi|first2=Daisuke|date=February 11, 1997|title=The Complete Bifurcation Diagram for the Logistic Map|journal=Z. Naturforsch.|volume=52a|issue=6–7 |pages=513–516|doi=10.1515/zna-1997-6-708 |bibcode=1997ZNatA..52..513T |s2cid=101491730 |doi-access=free}}</ref>
* With {{mvar|r}} between -1 and 1&nbsp;-&nbsp;{{sqrt|6}} and for {{mvar|x}}<sub>0</sub> between 1/{{mvar|r}} and 1-1/{{mvar|r}}, the population will approach permanent oscillations between two values, as with the case of {{mvar|r}} between 3 and 1&nbsp;+&nbsp;{{sqrt|6}}, and given by the same formula.<ref name="Takashi Tsuchiya, Daisuke Yamagishi, 1997"/>