View source for Fractals/Iterations of real numbers/r iterations

Dynamics:
* real analytic unimodal dynamics<ref>[https://www.math.stonybrook.edu/~mlyubich/papers/Brin-prize.pdf FORTY YEARS OF UNIMODAL DYNAMICS: ON THE OCCASION OF ARTUR AVILA WINNING THE BRIN PRIZE by  MIKHAIL LYUBICH]</ref>


=Diagram types=
==2D diagrams==
* parameter is a variable on horizontal axis
** bifurcation diagram : P-curves ( = periodic points) versus parameter<ref>[http://abacus.bates.edu/~sross/bifurandorbit01.html Bifurcation and Orbit Diagrams by  Chip Ross ]</ref>
** orbit diagram : points of critical orbit versus parameter 
** skeleton diagram ( [[Fractals/Iterations in the complex plane/def cqp#Critical curves|critical curves = q-curves]])<ref>[https://math.stackexchange.com/questions/2390324/what-are-the-lines-on-a-bifurcation-diagram math.stackexchange question: what-are-the-lines-on-a-bifurcation-diagram ?]</ref>
** Lyapunow diagram : Lyapunov exponent versus parameter<ref>[https://www.researchgate.net/publication/222490223_A_revision_of_the_Lyapunov_exponent_in_1D_quadratic_maps A revision of the Lyapunov exponent in 1D quadratic maps by Gerardo Pastor, Miguel Romera, Fausto Montoya Vitini. PHYSICA D NONLINEAR PHENOMENA 107(1):17 · AUGUST 1997]</ref>
** multiplier diagrams : multiplier of periodic orbit versus parameter
* constant parameter diagrams
** cobweb diagram or a Verhulst diagram <ref>[[w:Cobweb plot|wiki : Cobweb plot]]</ref> = Graphical iteration
** Iterates versus time diagram<ref>[http://www.geom.uiuc.edu/~math5337/ds/part3/part3_trap.html One-Dimensional Dynamical Systems  Part 3: Iteration by Hinke Osinga ]</ref> = connected scatter graph = time series
** invariant density diagram , histogram,<ref>[[w:histogram|wikipedia : histogram]]</ref> the distribution of the orbit,<ref>[http://theworldismysterious.wordpress.com/2013/09/30/chaos-theory-dependence-on-parameter-r-continued/ CHAOS THEORY: DEPENDENCE ON PARAMETER R  ]</ref> frequency distribution <ref>[http://helmingstay.blogspot.com/2011/10/bat-country.html bat-country by xian]</ref> = power spectrum <ref>[http://www.youtube.com/watch?v=84lq0Eh5ZX0 Power Spectrum of the Logistic Map from wolframmathematica ]</ref>
** Poincare plot<ref>[[:w:Poincaré_plot|Poincare plot in wikipedia]]</ref><ref>[https://physics.stackexchange.com/questions/444067/poincar%c3%a9-plane-and-logistic-map?rq=1 physics.stackexchange question : Poincaré plane and Logistic Map]</ref> 
<gallery>
File:LogisticMap BifurcationDiagram.png|orbit diagram
File:Critical curves of real quadratic map.svg|critical curves
File:Bifurcation diagram for real quadratic map. Periodic points for periods 1,2,4,and 8 are shown.png|Bifurcation diagram for real quadratic map. Periodic points for periods 1,2,4,and 8 are shown
File:Lyapunov exponent of real quadratic map.png|Orbit and Lyapunov diagrams
File:Multiplier4 f.png|Multiplier diagram
File:Logistic map cobweb and time evolution a=3.2.png|Logistic map cobweb and time evolution a=3.2
</gallery>


===Transformation===

[[Fractals/Computer_graphic_techniques/2D/plane#Exponential_map|Exponential transformation]] of the parameter axis


<gallery>
Lyapunov exponent of real quadratic map.png|normal horizontal axis
Lyapunov exponent of real quadratic map with exponential transformation on the real axis.png|transformed horizontal axis to show period doubling cascade
Bifurcation diagram of complex quadratic map.png| 3 windows with non-linear, i.e. logarithmic x-axis scale:
</gallery>

==3D diagrams ==
<gallery>
LogisticCobwebChaos.gif|An animated cobweb diagram
Logistic map examples.gif|An animated connected scatter  diagram
Logistic Map.webm
</gallery>

=Maps=
* Due to [[Fractals/Mathematics/Numerical|numerical errors]] different implementations of the same equation can give different trajectories. For example:  <code>r*x*(1-x)</code> and  <code>r*x - r*x*x</code>
* Sensitivity to initial conditions: "small difference in the initial condition will produce large differences in the long-term behaviour of the system. This property is sometimes called the 'butterfly effect'."<ref>[http://homepages.ulb.ac.be/~dgonze/TEACHING/logistic.pdf The logistic equation by Didier Gonze]</ref>


Map types
* [[Fractals/Mathematics/doubling|doubling map]]


==Tent map ==

Orbits of tent map: 
<gallery>
Time series of the Tent map for the parameter m=2.0 which shows numerical error.svg|double precision computation and numerical error
Time series of the Tent map for the parameter m=2.0 made with increased precision.svg|increased precision with no error
</gallery>

==Logistic map==

names : 
* logistic map <ref name="May, Robert M 1976">{{cite journal |last=May |first=Robert M. |year=1976 |title=Simple mathematical models with very complicated dynamics |journal=[[:w:Nature (journal)|Nature]] |volume=261 |issue=5560 |pages=459–467 |doi=10.1038/261459a0 |bibcode=1976Natur.261..459M |pmid=934280 |hdl=10338.dmlcz/104555 |hdl-access=free }}</ref>: <math>f(x) = r x (1 - x),</math><ref>[http://abel.harvard.edu/archive/118r_spring_05/docs/may.pdf Simple mathematical models with very complicated dynamics by R May]</ref>
* logistic equation   <math>x_{n+1} = f(x_n) ,</math>
* logistic difference equation   <math>x_{n+1} = r x_n (1 - x_n) ,</math>
* discrete dynamical system

The logistic map<ref>[http://www.egwald.ca/nonlineardynamics/logisticsmapchaos.php The Logistic Map and Chaos   by   Elmer G. Wiens]</ref><ref>[http://www.springer.com/physics/complexity/book/978-3-540-28366-9 Ausloos, Marcel, Dirickx, Michel (Eds.) : The Logistic Map and the Route to Chaos]</ref> is defined by a recurrence relation (  difference equation) :

:<math>x_{n+1} = r x_n (1 - x_n),</math>

where :
* <math>r</math> is a given constant parameter
* <math>x_0</math> is given the initial term  
* <math>x_n</math> is subsequent term determined by this relation

Do not confuse it with : 
* differential equation ( which gives continous version )

Bash code <ref>[http://tcode.tcs.auckland.ac.nz/~corpus/logistic.html Logistic map  by M.R. Titchener ]</ref>
Javascript code from Khan Academy <ref>[https://www.khanacademy.org/cs/logistic-map/1060730277 khanacademy    ; logistic-map]</ref>
[[MATLAB]] code:

<syntaxhighlight lang="matlab">
r_values = (2:0.0002:4)';
iterations_per_value = 10;
y = zeros(length(r_values), iterations_per_value);
y0 = 0.5;
y(:,1) = r_values.*y0*(1-y0);
for i = 1:iterations_per_value-1
    y(:,i+1) = r_values.*y(:,i).*(1-y(:,i));
end
plot(r_values, y, '.', 'MarkerSize', 1);
grid on;
</syntaxhighlight>

See also Lasin <ref>[http://lab.fs.uni-lj.si/lasin/?page_id=269&lang=en Lasin - Matlab code]</ref>

[[Maxima |Maxima CAS]] code <ref>[http://www.telefonica.net/web2/biomates/maxima/gpdraw/multiplots/index.html Logistic diagram by Mario Rodriguez Riotorto using Maxima CAS draw package]</ref>
<pre>
/* Logistic diagram by Mario Rodriguez Riotorto using Maxima CAS draw packag  */
pts:[];
for r:2.5 while r <= 4.0 step 0.001 do /* min r = 1 */
		(x: 0.25,
		for k:1 thru 1000 do x: r * x * (1-x), /* to remove points from image compute and do not draw it */
		for k:1 thru 500  do (x: r * x * (1-x), /* compute and draw it */
        	 	 pts: cons([r,x], pts))); /* save points to draw it later, re=r, im=x */
load(draw);
draw2d(	terminal   = 'png,
	file_name = "v",
        dimensions = [1900,1300],
	title      = "Bifurcation diagram, x[i+1] = r*x[i]*(1 - x[i])",
	point_type = filled_circle,
	point_size = 0.2,
	color = black,
	points(pts));
</pre>

===explicit solutions===

The system with r=4 has the explicit solution for the nth iteration :<ref>[https://www.uv.es/oteo/pubs/PRE_OR_mpfun_Logistic_Map_2007.pdf Double precision errors in the logistic map: Statistical study and dynamical interpretation J. A. Oteo J. Ros]</ref>

  <math> x_n = sin^2(2^n arcsin \sqrt{x_0} ) </math>

===Precision===
Numerical Precision in the Chaotic Regime : " the number of digits of precision which must be specified is about 0.6 of the number of iterations. Hence, to determine is x10 000, we need about 6000 digits."

"Hence it is not possible to predict the value of xn for very large n in the chaotic regime." <ref>[http://physics.ucsc.edu/~peter/242/logistic.pdf The Logistic Map by A. Peter Young]</ref>

===Better image ===

[[Image:LogisticMap BifurcationDiagram.png|thumb|right|Bifurcation diagram of the logistic map]]

To show more detaile use [[:File:LogisticMap_BifurcationDiagram.png|tips]] by [[:commons:User:PAR:|User:PAR]]:

"The horizontal axis is the r parameter, the vertical axis is the x variable. The image was created by forming a 1601 x 1001 array representing increments of 0.001 in r and x. A starting value of x=0.25 was used, and the map was iterated 1000 times in order to stabilize the values of x. 100,000 x -values were then calculated for each value of r and for each x value, the corresponding (x,r) pixel in the image was incremented by one. All values in a column (corresponding to a particular value of r) were then multiplied by the number of non-zero pixels in that column, in order to even out the intensities. Values above 250,000 were set to 250,000, and then the entire image was normalized to 0-255. Finally, pixels for values of r below 3.57 were darkened to increase visibility."

See also :
* tips from learner.org <ref>[http://www.learner.org/courses/mathilluminated/units/13/textbook/06.php earner.org  textbook]</ref>



    "The "problem" with pretty much all fractal-type systems, is, that, what happens at the beginning of an actually infinite iterational scheme, is often not descriptive of the long-time limit behaviour. And that's happening with the perturbed Lyapunov exponent from Mario Markus' algorithm as well. If I recall correctly, he states in his 90's article something like "let the x-value settle in". It's similar to the statement "for sufficiently large N" in math papers or in general with convergent series.
    The actual value of how many skipping iterations one performs, is, however, (at least afaik) just a guess till you're satisfied with the quality of the image. In my images, I could not get rid of those artifacts in full, one example being the UFO image, where vasyan did a great job removing those spots:
    vasyan's: https://fractalforums.org/index.php?action=gallery;sa=view;id=2388
    my version (with spots): https://fractalforums.org/index.php?action=gallery;sa=view;id=1960" marcm200<ref>[https://fractalforums.org/noobs-corner/76/what-could-these-artifacts-be-on-my-lyapunov-fractal-renders/4352/msg29782#msg29782 fractalforums.org : what-could-these-artifacts-be-on-my-lyapunov-fractal-renders]</ref>

===Lyapunov exponent ===
* code in Matlab<ref>[http://lab.fs.uni-lj.si/lasin/wp/IMIT_files/chaotic/Lyap_logistic/ly_logistic.html calculate lyapunov of the logistic map -LASIN]</ref>
* [http://www.tic.itefi.csic.es/gerardo/publica/Pastor97.pdf G. Pastor, M. Romera and F. Montoya, "A revision of the Lyapunov exponent in 1D quadratic maps", Physica D, 107 (1997), 17-22. Preprint]

===Invariant Measure===

An invariant measure or probability density in state space <ref>[http://pages.physics.cornell.edu/~sethna/StatMech/ComputerExercises/InvariantMeasure/InvariantMeasure.html Invariant Measure Exercise by James P. Sethna, Christopher R. Myers.]</ref>

Video on Youtube<ref>[http://www.youtube.com/watch?v=yiuVFUSUXPQ Invariant measure of logistic map by todo314 ]</ref>

==Real quadratic map==
[[File:Lyapunov exponent of real quadratic map.png|right|thumb|Lyapunov exponent - image and Maxima CAS code]]

Great images by Chip Ross<ref>[http://abacus.bates.edu/~sross/index.html Chip Ross :  Bifurcation and Orbit diagrams]</ref>

For <math>x_{n+1}=x_n^2-c</math>, the code in [[MATLAB]] can be written as:

<syntaxhighlight lang="matlab">
c = (0:0.001:2)';
iterations_per_value = 100;
y = zeros(length(c), iterations_per_value);
y0 = 0;
y(:,1) = y0.^2 - c;
for i = 1:iterations_per_value-1
    y(:,i+1) = y(:,i).^2 - c;
end
plot(c, y, '.', 'MarkerSize', 1, 'MarkerEdgeColor', 'black');
</syntaxhighlight>

Maxima CAS code for drawing real quadratic map : <math>f_c(z) = z^2 + c </math> :

<pre>
/* based on the code by by Mario Rodriguez Riotorto */
pts:[];
for c:-2.0 while c <= 0.25 step 0.001 do 
                (x: 0.0,
                for k:1 thru 1000 do x: x * x+c, /* to remove points from image compute and do not draw it */
                for k:1 thru 500  do (x:  x * x+c, /* compute and draw it */
                         pts: cons([c,x], pts))); /* save points to draw it later, re=r, im=x */
load(draw);
draw2d( terminal   = 'svg,
        file_name = "b",
        dimensions = [1900,1300],
        title      = "Bifurcation diagram, x[i+1] = x[i]*x[i] +c",
        point_type = filled_circle,
        point_size = 0.2,
        color = black,
        points(pts));

</pre>

===Lyapunov exponent===
<syntaxhighlight lang =pascal>
program lapunow;

  { program draws bifurcation diagram y[n+1]=y[n]*y[n]+x,} { blue}
  {  x: -2 < x < 0.25 }
  {  y: -2 < y < 2    }
  {  and Lyapunov exponet for each x { white}

  uses crt,graph,
        { modul niestandardowy }
       bmpM, {screenCopy}
       Grafm;
  var xe,xemax,xe0,yemax,i1,i2:integer;
      yer,y,x,w,dx,lap:real;

  const xmin=-2;         { wspolczynnik   funkcji fx(y) }
        xmax=0.25;
        ymax=2;
        ymin=-2;
        i1max=100;            { liczba iteracji }
        i2max=20;
        lapmax=10;
        lapmin=-10;

   function wielomian2st(y,x:real) :real;
     begin
       wielomian2st:=y*y+x;
     end;  { wielomian2st }

  procedure wstep;
     begin
       opengraf;
       randomize;            { przygotowanie generatora liczb losowych }
       xemax:=getmaxx;              { liczba pixeli }
       yemax:=getmaxy;
       w:=(yemax+1)/(ymax-ymin);
       dx:=(xmax-xmin)/(xemax+1);
     end;

  begin {cialo}
    wstep;
    for xe:=xemax downTo 0 do
      begin {xe}
        x:=xmin+xe*dx;     { liniowe skalowanie x=a*xe+b }
        i1:=0;
        i2:=0;
        lap:=0;
        y:=random;        { losowy wybor    y0 : 0<y0<1 }

        while (abs(y)<ymax) and (i1<i1max)
        do
          begin {while i1}
            y:=wielomian2st(y,x);
            i1:=i1+1;
            lap:=lap+ln(abs(2*y)+0.1);
            if keypressed then halt;
          end; {while i1}

        while (i2<i2max) and (abs(y)<ymax)
         do
          begin   {while i2}
            y:=wielomian2st(y,x);
            yer:=(y-ymin)*w;         { skalowanie }
            putpixel(xe,yemax-round(yer),blue); { diagram bifurkacyjny }
            i2:=i2+1;
            lap:=lap+ln(abs(2*y)+0.1);
            if keypressed then halt;
          end; {while i2}

          lap:=lap/(i1max+i2max);
          yer:=(lap-lapmin)*(yemax+1)/(lapmax-lapmin);
          putpixel(xe,yemax-round(yer),white);         { wsp Lapunowa }
          putpixel(xe,yemax-round(-ymin*w),red);       { y=0 }
          putpixel(xe,yemax-round((1-ymin)*w),green);  { y=1}

      end; {xe}

    {..... os 0Y .......................................................}
    setcolor(red);
    xe0:=round((0-Xmin)/dx);     {xe0= xe : x=0 }
    line(xe0,0,xe0,yemax);
     SetColor(red);
    OutTextXY(XeMax-50,yemax-round((0-ymin)*w)+10,'y=0 ');
    SetColor(blue);
    OutTextXY(XeMax-50,yemax-round((1-ymin)*w)+10,'y=1');
    {....................................................................}
    screenCopy('screen',640,480);
    {}
    repeat until keypressed;
    closegraph;

 end.

{ Adam Majewski
Turbo Pascal 7.0  Borland
 MS-Dos / Microsoft}
</syntaxhighlight>

=Zoom=
* [https://fractalforums.org/fractal-mathematics-and-new-theories/28/rendering-bifurcation-diagrams/2074/;topicseen fractal forums: rendering-bifurcation-diagrams]
=Cycles=
* [https://logperiod.appspot.com/ n-cycles of polynomial maps by Cheng Zhang] and [https://arxiv.org/abs/1204.0546 paper: Cycles of the logistic map]

=Points=
* Misiurewicz Points 
** [http://sprott.physics.wisc.edu/chaos/mispoint.htm of the Logistic Map] by the J. C. Sprott
** [http://www.tic.itefi.csic.es/gerardo/publica/Romera96b.pdf M. Romera, G. Pastor and F. Montoya, "Misiurewicz points in one-dimensional quadratic maps", Physica A, 232 (1996), 517-535]
** [http://www.tic.itefi.csic.es/gerardo/publica/Pastor01.pdf G. Pastor, M. Romera, G. Álvarez and F. Montoya, "Misiurewicz point patterns generation in one-dimensional quadratic maps", Physica A, 292 (2001), 207-230]
** [http://www.tic.itefi.csic.es/gerardo/publica/Pastor96a.pdf G. Pastor, M. Romera and F. Montoya, "On the calculation of Misiurewicz patterns in one-dimensional quadratic maps" Physica A, 232 (1996), 536-553. Preprint]
* Accumulation points
* bifurcation points

=Constants=
* [[:w:Feigenbaum constants|Feigenbaum constants in wikipedia]]

=Properities=
==self-similarity, scaling and renormalization==
* Feigenbaums Scaling Law For TheLogistic Map  <ref>[http://demonstrations.wolfram.com/FeigenbaumsScalingLawForTheLogisticMap  demonstrations from wolfram : FeigenbaumsScalingLawForTheLogisticMap]</ref>


=See also=
* [https://www.geogebra.org/m/FQrUDRcf Geogebra: The Feigenbaum loci Author:a.zampa]
* [https://oeis.org/search?q=logistic+map&sort=&language=&go=Search oeis  search:  logistic map]
* [https://blbadger.github.io/logistic-map.html Logistic Map by B. L. Badger]
* [https://arxiv.org/abs/1201.2514 Analytical properties of horizontal visibility graphs in the Feigenbaum scenario  Bartolo Luque, Lucas Lacasa, Fernando J. Ballesteros, Alberto Robledo]
* [http://mathcs.holycross.edu/~groberts/Courses/MA374/Classroom/Ch7MainThm.pdf The Quadratic Map  is Topologically Conjugate to the Shift Map - Gareth Roberts]
* [http://sprott.physics.wisc.edu/chaos/nelogmap.htm Numerical Errors in Logistic Map Calculation  J. C. Sprott ]
* [http://www.maa.org/press/maa-reviews/numbers-and-computers Numbers and Computers by Ronald T. Kneusel, Reviewed by David S. Mazel , on 01/27/2016]
* [http://izt.ciens.ucv.ve/ecologia/Archivos/References-I-math/books-math/Math/Skiadas_Skiadas-Chaotic_Modeling_and_Simulation.pdf Chaotic Modelling and Simulation Analysis of Chaotic Models, Attractors and Forms Christos H. Skiadas Charilaos Skiadas]
* [http://www.phaser.com/modules/ecology/delayed_logistic/ delayed_logistic]
* [http://www.maplesoft.com/applications/view.aspx?SID=3507&view=html The logistic map revisited  Jerzy Ombach, Cracow, Poland]
* [http://terpconnect.umd.edu/~bhunt/papers/wind.html Structure in the Parameter Dependence of Order and Chaos for the Quadratic Map  Brian R. Hunt and Edward Ott]
* [http://ibiblio.org/e-notes/MSet/windows.htm Windows of periodicity scaling by Evgeny Demidov]
* [http://rreusser.github.io/logistic-map/ in Webgl by  Ricky Reusser]
* [https://juliacomputing.com/blog/2017/11/27/high-precision-feigenbaum-alpha-calc-using-julia.html high-precision-feigenbaum-alpha-calc-using-julia by Stuart Brorson]
* [http://virtualmathmuseum.org/Fractal/feigenbaum_tree/index.html Feigenbaum Tree by  3DXM Consortium]
* [https://www.researchgate.net/publication/275281657_Pascal_Yang_Hui_triangles_and_power_laws_in_the_logistic_map Pascal Yang Hui triangles and power laws_in_the_logistic_map by Carlos Velarde Alberto Robledo]
* [https://demonstrations.wolfram.com/EstimatingTheFeigenbaumConstantFromAOneParameterScalingLaw/  demonstrations.wolfram : EstimatingTheFeigenbaumConstantFromAOneParameterScalingLaw]
* [http://www.egwald.ca/nonlineardynamics/logisticsmapchaos.php Egwald Mathematics: Nonlinear Dynamics:  The Logistic Map and Chaos  by  Elmer G. Wiens]

=software=
* [https://github.com/gboeing/pynamical pynamical - Python package for modeling, simulating, visualizing, and animating discrete nonlinear dynamical systems and chaos]
* [https://github.com/jonnyhyman/Chaos Chaos. Visualizations connecting chaos theory, fractals, and the logistic map! Written by Jonny Hyman, 2020]

=Videos:=
* [https://www.youtube.com/watch?v=n63azjpd7U8 Illustration of Chaotic Systems Using Logistic Map by Zylasable ]
* [https://www.youtube.com/watch?v=I0aDD9gYCTo Bifurcation Diagram for the Logistic Map Using R by Oxlajuj N'oj]


=References=
<references/>


{{BookCat}}