CHAOS   equations - graphs - program code


   chaos  -  complexity  -  order

   Eight equations generating exotic behavior.

   Algorithms, program code and graphical output
   by Roger Luebeck  © 2000, 2017


   The mathematical intrigues on these pages don't rival 
   the sophistication of Mandelboxes and other 3D fractal 
   endeavors which are popular today.  If you like the 
   things you see here, and have yet to discover the world 
   of fractals, just do a YouTube search on Mandelboxes.

   The first two sections below are my creations.  The rest 
   of the sections I merely added my investigations to, 
   especially the logistic map.

   Sections 1, 2, 3 and 4 are by far the most interesting.

1.    Balloting  Florida 2000 Gore vs. Bush

      Iteration of a balloting algorithm, using random 
      number generator.  It grew out of my analysis of 
      the 2000 Gore - Bush Florida balloting.  Surprising 
      patterns amidst chaos.  Includes commentary on the 
      media, the Supreme Court and lawyers.

2.    Simple trig

      I played a hunch and came up with a pair of repeating
      trig equations, with offsetting scalar multipliers 
      for each equation.  Unbelievable patterns. 
      (See side panel and click "Simple trig" to see many more.)

3.    Logistic map (LM)

      X = r * X * (1 - X)

      Mathematician Paul Stein called the complexity of 
      this iterated equation "frightening".

      Iterating this equation produces regions of distinct 
      values, involving period doubling, as well as regions 
      of chaos.  

      On the LM page, you'll find bifurcation diagrams
      at various scales showing the depth of this equation.

      You'll also find the equation graphed parabolically,
      superimposed onto a straight line graph and the 
      bifurcation diagram.  The LRLRR patterning and 
      L:R ratios are examined.

      Finally, you'll find circular plots of the equation,
      revealing patterns amidst the chaos.      

4.    Sine and pi  (no link called for)

      X = r * Sin (pi * X)

      Here's what my investigations turned up:

      Profoundly -

      Iterating this equation generates a bifurcation diagram
      visually indistinguishable from the one generated by the
      above logistic map (LM), though the actual values are much 
      different.  With LM, the valid values for r range from 
      0 to 4 and valid values for initial X range from 0 to 1.
      With the sine pi equation, r can range from 0 to 
      infinity, though there is only chaos beyond r = 57.29578.
      Initial X can range from 0 to infinity, though the graph
      flips upside down when initial X exceeds 57.29578.  It 
      flips right side up again when initial X is double that
      value, with the flipping occuring at each multiple of 
      57.29578, which equals 360/(2 pi).  And just as with 
      the LM, one can generate segments of a parabola by 
      plotting X vs previous X.

5.    The Attractor of Henon

      X = ((previousY + 1) - (1.4 * previousX ^ 2))
      Y = (0.3 * previousX)  

      "Iterating this pair of equations produces a strange
      simple set of curved lines that is poorly understood 
      by mathematicians.  As thousands, then millions of
      points appear, more and more detail emerges.  What 
      appear to be single lines prove, on magnification, 
      to be pairs, then pairs of pairs, and so on to
      infinity.  Whether any two successive points appear 
      nearby or far apart is unpredictable." - James Gleick
      Called a strange attractor.

6.    Number doubling

      Choose an initial value between 0 and 1.  
      Double it.  Drop the integer part.  Repeat.  
      Values occur in discreet bunches.  Bunches 
      occur in multiples of five, depending on the 
      number of decimal places used.  Repeating 
      patterns occur within the bunches.

7.    Barnsley's Fern

      Randomly selected parts of a four part algorithm 
      produces a realistic looking fern.  Invented by 
      Michael Barnsley.

8.    The Sierpinski Triangle

      Randomly selected parts of a three part algorithm 
      produces a triangular lattice of triangles.  I tried 
      doing this with an orderly, alternating selection of 
      the parts of the algorithm, and got a partial gasket.
      Also called The Sierpinski Gasket.


 Roger Luebeck

 Updated 03/24/2024

 More science and technology from this author:

 Special relativity in absolute terms

 A simple 3D CAD system

 Author's home page is