Fractals/Iterations of real numbers/r iterations

Dynamics:

• real analytic unimodal dynamics

Diagram types

2D diagrams

• parameter is a variable on horizontal axis
• bifurcation diagram : P-curves ( = periodic points) versus parameter
• orbit diagram : points of critical orbit versus parameter
• skeleton diagram ( critical curves = q-curves)
• Lyapunow diagram : Lyapunov exponent versus parameter
• multiplier diagrams : multiplier of periodic orbit versus parameter
• constant parameter diagrams
• cobweb diagram or a Verhulst diagram  = Graphical iteration
• Iterates versus time diagram = connected scatter graph = time series
• invariant density diagram , histogram, the distribution of the orbit, frequency distribution  = power spectrum 
• Poincare plot

Transformation

Exponential transformation of the parameter axis

Maps

• Due to numerical errors different implementations of the same equation can give different trajectories. For example: r*x*(1-x) and r*x - r*x*x
• 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'."

Tent map

Orbits of tent map:

Logistic map

names :

• logistic map : $f(x)=rx(1-x),$ • logistic equation $x_{n+1}=f(x_{n}),$ • logistic difference equation $x_{n+1}=rx_{n}(1-x_{n}),$ • discrete dynamical system

The logistic map is defined by a recurrence relation ( difference equation) :

$x_{n+1}=rx_{n}(1-x_{n}),$ where :

• $r$ is a given constant parameter
• $x_{0}$ is given the initial term
• $x_{n}$ is subsequent term determined by this relation

Do not confuse it with :

• differential equation ( which gives continous version )

Bash code  Javascript code from Khan Academy  MATLAB code:

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;

Maxima CAS code 

/* 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 */
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));

explicit solutions

The system with r=4 has the explicit solution for the nth iteration :

$x_{n}=sin^{2}(2^{n}arcsin{\sqrt {x_{0}}})$ 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." 

Better image

"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."

• tips from learner.org 

"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

Invariant Measure

An invariant measure or probability density in state space 

Great images by Chip Ross

For $x_{n+1}=x_{n}^{2}-c$ , the code in MATLAB can be written as:

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');

Maxima CAS code for drawing real quadratic map : $f_{c}(z)=z^{2}+c$ :

/* 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 */
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));

Lyapunov exponent

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.

Turbo Pascal 7.0  Borland
MS-Dos / Microsoft}

Points

• Misiurewicz Point

Properities

self-similarity, scaling and renormalization

• Feigenbaums Scaling Law For TheLogistic Map