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Balloting

Simple trig

Logistic map

The Attractor of Henon

Number doubling

Barnsley's Fern

The Sierpinski Triangle

Logistic Map

Also called the logistic difference equation
or the quadratic difference equation.

X = rX(1 - X)

(Complete program code at bottom of page)

This was Mitchell Feigenbaum's break-through
equation on his road to discovering universality
across different chaotic systems.

Iterating this equation produces regions of
distinct values, involving period doubling,
as well as regions of chaos.  It is the rate of
convergence of period doubling that is universal.

Explanation:

A distinct set of values occur at a certain value
of r (r1).  The number of distinct values suddenly 
doubles at a higher value of r (r2).  Take the 
difference between r2 and r1.  Divide that difference 
by the difference between r3 (where the next doubling
occurs) and r2.  That number (the convergence ratio)
approaches 4.6692016... as r increases.  That 
convergence ratio appears in many other chaotic 
systems involving very different equations.

There are four ways to graph this iterated equation 
in a static manner:

  1. In a bifurcation diagram, the values of X are 
     plotted against the values of the parameter r.
     One such diagram is shown at the top of this page.

  2. Plotting X against the previous value of X for 
     a given value of r produces a parabolic curve, 
     always in segments corresponding to the range
     of values that X acquires for that value of r.

     Of special interest to investigators is the 
     question of whether a point lands to the left or 
     to the right of the previous point.  Various 
     LRLRRRL etc patterns are found to repeat.

     Below the diagrams section, there is a table
     showing some LRRR patterning and a table
     showing L shift : R shift ratios for various
     values of the parameter r.

  3. Plotting each value of X against each iteration 
     produces a straight line graph which reveals how 
     many iterations are needed for convergence to 
     distinct values for X for a given value of r.  
     (Anywhere from just a few iterations to four 
     million iterations prove to be sufficient, 
     depending on the value of r.)

  4. Graphing in a circular manner:  Plotting the 
     sine and cosine of an incrementally increasing 
     angle times the ever evolving X value reveals 
     hidden patterns amidst the chaotic zones.


All four types of graphs are displayed further down on
this page.  (Types 3 and 4 are my own creations.)

The initial value (0 - 1) for X makes no difference
in the appearance of any of the graphs.  For some values 
of r, the equation is sensitive to the initial value of 
X, but the resulting values are not significantly
different from one initial value of X to another.

The range of values for r is 0 - 4.

Here are some period doubling events for approximate
values of r:


distinct values      r  (range 0 - 4)
_______________     _________________

     1               1
     2               3
     4               3.4494899
     8               3.5440905
    16               3.5644074
    32               3.5687595
    64               3.5696917
   128               3.5698914
   256               3.5699341
   512               3.5699433
  1024               3.5699453
  2048               3.5699456
     .
     .
   2^n               3.5699468376


and so on, without end as r approaches 3.56994692, 
at which point chaos begins to take over.  

The highest value of r below 3.56994692 for which
I discovered distinct values in the form 2^n was 
3.5699468376.  There must have been millions of 
distinct values there.

Distinct values also appear when r > 3.56994692,
in between periods of chaos.  For instance, one such 
set is found at r = 3.57004996211.

Distinct values also occur in groups of
6, 12, 24...  and  5, 10...  in this higher region. 
I even found a group of 18 values.


(See notes below the diagrams section for general 
information about generating a bifurcation diagram.
Program code for all four diagram types follows that.)






================================================================================


This algorithm is infinitely deep.  Similar, yet never 
identical, patterns repeat as one looks infinitely deeper.

After displaying the following four bifurcation
diagrams exploring the infinite depth of this algorithm, 
we'll look at some graphs for single values of the
parameter r.








=======================================================================================


To appreciate just how abruptly chaos ends
and distinct value lines begin, see the
following diagram.




Shown below are the final two X values out of twenty
thousand iterations for the middle distinct value 
line for each value of r.  See the graphs immediately 
above.

As r approaches 3.8284272, the X values are 
not yet stable (though they are becoming bunched 
quite tightly), and they are not falling downwards 
in an overall sense:


    r               X


3.8284268     .514396041025985
3.8284268     .514396756770264
 
3.8284269     .514516155621660
3.8284269     .514517495144266
 
3.8284270     .514118995580905
3.8284270     .514121155779832
 
3.8284271     .514407341511282
3.8284271     .514407484278966


Chaos ends (for awhile) at r = 3.824271


The .514... values are falling now and they are 
locked in on a single value for each iteration 
of the current parameter r:


3.8284272     .514288397179200
3.8284272     .514288397179200
 
3.8284273     .514253214008774
3.8284273     .514253214008774
 
3.8284274     .514227367177942
3.8284274     .514227367177942
 
3.8284275     .514205928117646
3.8284275     .514205928117646


The same is true for the other two distinct X value
lines at this value of r, as well as for any r value 
splitting event.


================================================================================


The following five images show the straight graphing
(X plotted against iterations in blue) and the parabolic
graphing (X plotted against the previous X in red) 
superimposed onto a bifurcation diagram:


















Here is the only full parabola, that generated at r = 4:








Here is some output showing RRRL patterning:

Note that the plotted point could be on either
the left or right side of the central axis.


    X value           shift

.3186,                "  R"
.7685129016,          "  R"
.6297689087271950,    "  R"
.8253865073602450,    "L"
.5101976177307690,    "  R"
.8846318704178640,    "L"
.3612864678763470,    "  R"
.8168852882604630,    "  R"
.5295263478576550,    "  R"
.881913809528948,     "L"
.368662121002446,     "  R"
.823936279853100,     "  R"
.513531114546066,     "  R"
.884351857644560,     "L"
.362048719319123,     "  R"
.817631832321111,     "  R"
.527849467601022,     "  R"
.882254401326362,     "L"
.367740767239850,     "  R"
.823076533537926,     "  R"
.515500299308996,     "  R"
.884149482153514,     "L"
.362599280778603,     "  R"
.818168489945347,     "  R"
.526641594500876,     "  R"
.882487398066276,      "L"
.367110001734535      "  R"
.822484479197956,     "  R"
.516853312794537,     "  R"
.883994519101387,      "L"

==================================================



The ratio of total L shifts to total R shifts for
a given value of r can take on many values,
such as 

0:1
1:1
1:2
1:2.333...
1:3
1:4
1:4.333...
1:5
1:6
1:7

and, I surmise, other ratios as well.

These ratios remain uniform over specific ranges of r.

These ratios do not correlate with the number of distinct
value lines.  I found examples of the L:R ratio
changing in the midst of a range of a given number
of distinct values.

Between r =  3.56994687 and 3.56994693, the
L:R ratio alternates between two distinct values:

  1:5.04723799963414

  and

  1:5.04724714191982

This is the exact region of r where period
doubling of the 2^n variety ends and chaos begins.

This may be the only region where recurring values
of L:R ratios are found that are not tidy ratios.

Here is a tiny snapshot of the L:R ratios going through
some changes through adjacent ranges of r:



                   total       total                L:R
init X      r      L shifts    R shifts             ratio     iterations
------   ------    --------    --------            -------    ----------
".90","  3.7416,"  ",200000,"  ",800000,"  ",0,"  ",4,"       ",1000000,"  ",0
".90","  3.7417,"  ",200000,"  ",800000,"  ",0,"  ",4,"       ",1000000,"  ",0
".90","  3.7418,"  ",200000,"  ",800000,"  ",0,"  ",4,"       ",1000000,"  ",0
".90","  3.7419,"  ",300000,"  ",700000,"  ",0,"  ",2.3333,"  ",1000000,"  ",0
".90","  3.7420,"  ",300000,"  ",700000,"  ",0,"  ",2.3333,"  ",1000000,"  ",0
".90","  3.7421,"  ",300000,"  ",700000,"  ",0,"  ",2.3333,"  ",1000000,"  ",0
".90","  3.7422,"  ",300000,"  ",700000,"  ",0,"  ",2.3333,"  ",1000000,"  ",0
".90","  3.7423,"  ",300000,"  ",700000,"  ",0,"  ",2.3333,"  ",1000000,"  ",0
".90","  3.7424,"  ",300000,"  ",700000,"  ",0,"  ",2.3333,"  ",1000000,"  ",0
".90","  3.7425,"  ",300000,"  ",700000,"  ",0,"  ",2.3333,"  ",1000000,"  ",0
".90","  3.7426,"  ",300000,"  ",700000,"  ",0,"  ",2.3333,"  ",1000000,"  ",0
".90","  3.7427,"  ",300000,"  ",700000,"  ",0,"  ",2.3333,"  ",1000000,"  ",0
".90","  3.7428,"  ",250000,"  ",750000,"  ",0,"  ",3,"       ",1000000,"  ",0
".90","  3.7429,"  ",250000,"  ",750000,"  ",0,"  ",3,"       ",1000000,"  ",0





Interesting patterns appear when graphing chaotic zones 
in a circular manner, running sufficient iterations to 
complete hundreds of orbits.

(These same patterns are generated by at least two
other very different algorithms.  One is the Balloting
algorithm seen on another page of this website.  Another
is a repeating line drawing routine used to color a 
plane in a CAD program.  That image is also displayed on
the Balloting page.  These patterns cannot be generated
by simply orbiting random numbers -- an algorithm is
required.)


Here is a graph generated at r = 3.59





Shrinking the image to 1/2 size, previously unseen
strong patterns appear:




At 1/3 size:





Another one at r = 3.64


1/2 size:




1/3 size:





Here is r = 3.64 again, superimposed on another
pattern which we discuss on another page:





Finally, at r = 3.9 we see another chaos zone:






Notes about generating bifurcation diagrams:


The correct way to generate this diagram is to use a nested
loop.  The outside loop iterates values of incremented r
and resets the value of X to its initial value.  The inside 
loop iterates the equation. 

Amazingly, one can also generate a bifurcation diagram by 
using just a single loop, whereby r is incremented for
each iteration of the equation.  I would have thought that 
this would not generate anything but garbage.  

However, the resulting diagram is nearly indistinguishable 
from the one obtained using the nested loop algorithm.  Looking
closely, one can detect a slight difference in the dots 
comprising the chaotic zones, and examining the data read out
reveals that period doubling events are slightly off.  For
example, period two occured at about 3.004, as opposed to 3.




The code:

(X in some of the program code is not the same X that appears 
in the equation appearing on this page or in the discussion
on this page.  Rather, it is the horizontal axis plot value.)

============================================================

The VisualBASIC program code for the bifurcation diagram:


Private Sub bifurbtn_Click()

viewport.ScaleMode = pixel

viewport.DrawWidth = 1

'viewport.Line (80, 60)-(780, 60), RGB(148, 234, 252)   'top
'viewport.Line (80, 760)-(780, 760), RGB(148, 234, 252) 'bottom
'viewport.Line (80, 60)-(80, 760), RGB(148, 234, 252)   'left
'viewport.Line (780, 60)-(780, 760), RGB(148, 234, 252) 'right

'viewport.Line (430, 60)-(430, 760), RGB(148, 234, 252) 'center

yshift = Val(yshiftbox.Text)
'xshift = val(xshiftbox.text)

amp = Val(dsamp.Text)
If amp = 0 Then amp = 1

ITER = Val(itera.Text)
If ITER = 0 Or ITER > 1000 Then ITER = 1000

reset# = Val(numb.Text)
If reset# = 0 Then reset# = 0.9

expand = Val(expandbox.Text)
If expand = 0 Then expand = 1

ri# = Val(parameter.Text)
If ri# = 0 Then ri# = 2.8

rf# = Val(rfbox.Text)
If rf# = 0 Or rf# > 3.999 Then rf# = 3.999

range = CInt((rf# - ri#) * 583.333 * expand)

X = (583.33 * (ri# - 2.8)) + 80
rinc# = 0.0017143

If expand > 1 Then X = 80: rinc# = rinc# / expand

r# = ri#

For g = 1 To range

  num# = reset#
  r# = r# + rinc#
  X = X + 1
      
  For a = 1 To ITER
     
    num# = r# * num# * (1 - num#)

    Y# = (-num# * 700 * amp) + 760 + yshift
                
    viewport.PSet (X, Y#), RGB(0, 0, 0)

100:
   Next a

Next g

totalpoints = g * a

xplotbox.Text = "          "
xplotbox.Text = Str$(g)
totalptsbox.Text = "          "
totalptsbox.Text = Str$(totalpoints)
rincbox.Text = "                        "
rincbox.Text = Str$(rinc#)
finalrbox.Text = "                        "
finalrbox.Text = Str$(r#)
finalnum.Text = "                  "
finalnum.Text = Str$(num#)

End Sub


============================================================

The VisualBASIC program code for bifurcation listing values:


Private Sub quick2loopbtn_Click()

ITER = Val(itera.Text)

reset# = Val(numb.Text)
If reset# = 0 Then reset# = 0.9

rinc# = 0.0000001                   ' or whatever

r# = 3.8284151                      ' or whatever

For g = 1 To 80

  num# = reset#
  r# = r# + 0.0000001
      
  For a = 1 To 4000000              ' or whatever
     
    num# = r# * num# * (1 - num#)
    
    If a <= 999990 Then GoTo 100
    
    Open "ds\" + "list2" + ".txt" For Append As #1
    Write #1, r#; "  "; num#
    Close #1

100:
   Next a

   Open "ds\" + "list2" + ".txt" For Append As #1
   Write #1, "  "
   Close #1

Next g
    
g = g - 1: a = a - 1
totalpoints = g * a

xplotbox.Text = "          "
xplotbox.Text = Str$(g)
totalptsbox.Text = "          "
totalptsbox.Text = Str$(totalpoints)
rincbox.Text = "                        "
rincbox.Text = Str$(rinc)
finalrbox.Text = "                        "
finalrbox.Text = Str$(r)
finalnum.Text = "                  "
finalnum.Text = Str$(num#)

End Sub



============================================================

The VisualBASIC program code for the parbola version:


Private Sub parabtn_Click()

viewport.ScaleMode = pixel
viewport.DrawWidth = 1

viewport.Line (80, 60)-(780, 60), RGB(148, 234, 252)   'top
viewport.Line (80, 760)-(780, 760), RGB(148, 234, 252) 'bottom
viewport.Line (80, 60)-(80, 760), RGB(148, 234, 252)   'left
viewport.Line (780, 60)-(780, 760), RGB(148, 234, 252) 'right

'viewport.Line (430, 60)-(430, 760), RGB(148, 234, 252) 'center

dw = Val(drawwidthbox.Text)
If dw = 0 Then dw = 1
viewport.DrawWidth = dw

ITER = Val(itera.Text)
If ITER = 0 Then ITER = 100000

r = Val(parameter.Text)
para$ = parameter.Text

X# = Val(numb.Text)
If X# = 0 Then X# = 0.9

RR = 0
LL = 0
pt5 = 0

For a = 1 To ITER

   Y# = r * X# * (1 - X#)

   If a < 50000 Then GoTo 50

   TEMPX# = (700 * X#) + 80
   TEMPY# = (-700 * Y#) + 760

   viewport.PSet (TEMPX#, TEMPY#), RGB(255, 0, 0)  
   viewport.PSet (TEMPX#, 770), RGB(255, 0, 0)     ' shows previous X values
   viewport.PSet (70, TEMPY#), RGB(255, 0, 0)      ' shows X values

50:
   X# = Y#
   
Next a

statusbox.Text = "done"

End Sub


============================================================

The VisualBASIC program code for the parbola L:R listing version:

Private Sub quickparabtn_Click()


ITER = Val(itera.Text)
If ITER = 0 Then ITER = 20000

r# = Val(parameter.Text)
para$ = parameter.Text

rinc# = Val(rincbox.Text)

loops = Val(gbox.Text)

resetx# = Val(numb.Text)
init$ = numb.Text

If X# = 0 Then X# = 0.9: init$ = ".90"

RR = 0
LL = 0
pt5 = 0
oops = 0

For g = 1 To loops

X# = resetx#
r# = r# + rinc#

  For a = 1 To ITER

     Y# = r# * X# * (1 - X#)

     'If a < 999000 Then GoTo 50

     'Open "ds\" + "xy-list" + ".txt" For Append As #1
     'Write #1, init$; "  "; r#; "  "; X#; "  "; Y#
     'Close #1


      If X# = 0.5 Then pt5 = pt5 + 1: GoTo 50
      
      If Y# < X# And X# > 0.5 Then sh$ = "  R": RR = RR + 1
      If Y# > X# And X# > 0.5 Then sh$ = "L": LL = LL + 1
      If Y# < X# And X# < 0.5 Then sh$ = "L": LL = LL + 1
      If Y# > X# And X# < 0.5 Then sh$ = "  R": RR = RR + 1

50:
     X# = Y#

  Next a

  If LL = 0 Then LL = 1: oops = oops + 1     ' prevents division by zero
  LRRATIO# = RR / LL

  Open "ds\" + "lr-ratio" + ".txt" For Append As #1
  Write #1, init$; "  "; r; "  "; LL; "  "; RR; "  "; pt5; "  "; LRRATIO#; "  "; ITER; "  "; oops; "  "; X#; "  "; Y#
  Close #1
   
  RR = 0
  LL = 0
  pt5 = 0
  oops = 0
  
Next g

statusbox.Text = "done"

End Sub



============================================================

The VisualBASIC program code for the straight graph version:


Private Sub logdifstrbtn_Click()

viewport.ScaleMode = pixel

viewport.DrawWidth = 1

viewport.Line (80, 60)-(780, 60), RGB(148, 234, 252)   'top
viewport.Line (80, 760)-(780, 760), RGB(148, 234, 252) 'bottom
viewport.Line (80, 60)-(80, 760), RGB(148, 234, 252)   'left
viewport.Line (780, 60)-(780, 760), RGB(148, 234, 252) 'right

'viewport.Line (430, 60)-(430, 760), RGB(148, 234, 252) 'center

r# = Val(parameter.Text)

amp = Val(dsamp.Text)
If amp = 0 Then amp = 1

inc = Val(dsinc.Text)
If inc = 0 Then inc = 0.007

num# = Val(numb.Text)
If num# = 0 Then num# = 0.9

ITER = Val(itera.Text)
If ITER = 0 Then ITER = 100000: inc = 0.007

yshift = Val(yshiftbox.Text)

X = 80

 For a = 1 To ITER
               
    num# = r# * num# * (1 - num#)

    X = X + inc
    
    Y# = (-num# * 700 * amp) + 760 + yshift
        
    viewport.PSet (X, Y#), RGB(0, 0, 255)

 Next a

finalnum.Text = "        "
finalnum.Text = Str$(num#)

End Sub



============================================================

The VisualBASIC program code for the circular graph version:


Private Sub logisticrunbtn_Click()

R = Val(parameter.Text)
num# = Val(numb.Text)
ra = Val(radiusmult.Text) * 450
ITER = Val(itera.Text)
anginc# = Val(angleinc.Text) * (3.14159265358979 / 180)


For a = 1 To ITER
               
    num# = R * num# * (1 - num#)
    
    angle# = angle# + anginc#

    X# = num# * (Cos(angle#) * ra) + 430
    Y# = num# * 1.2284 * (Sin(angle#) * ra) + 530
           
    viewport.PSet (X#, Y#), RGB(0, 0, 0)

Next a

finalnum.Text = "        "
finalnum.Text = Str$(num#)

End Sub

=============================================================

home


Balloting

Simple trig

Logistic map

The Attractor of Henon

Number doubling

Barnsley's Fern

The Sierpinski Triangle