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              end chaos web page

home page:
The preprint for my relativity journal article is at: My book on relativity in absolute terms at the U of M, East Bank Campus: Relativity in absolute terms There are only three sane sections in the Wikipedia article on the twin paradox of special relativity. I authored those sections in 2011, and they've been there continuously since that time: Here's the link to the three sections below. 4. A non space-time approach 5. The equivalence of biological aging and clock time-keeping 12. No twin paradox in an absolute frame of reference
On several fundamental levels, Relativity Trail eclipses the works of Poincare, Lorentz and Einstein regarding the underpinnings and kinematics of special relativity: link: The originality and uniqueness of Relativity Trail
The twin paradox of special relativity cannot be resolved without acknowledging a hierarchy of clock rates dependent on a hierarchy of inertial motion. © 2022 Relativity Trail explains the time differential between reunited clocks, eliminates the twin paradox, diagrams Einstein's clock sychronization in absolute terms, and ends all confusion regarding relative frames of reference. It's completely compatible with, and in fact subsumes, Einstein's relativity. It reveals what is transpiring behind the scenes of Einstein's treatment. Spacetime is shown to be dependent on Einstein's clock synchronization method, and is properly relegated to a geometrical construct which comes up short as a physical reality. © 2008 Relativity Trail, with 192 pages, 65 diagrams and 75 illustrations, will provide you with complete detailed algebraic derivations of all the kinematical effects of special relativity. Everything is charted out in absolute terms against a system at rest with respect to the totality of the universe for perfect clarity as well as soundness of theoretical basis. It is the totality of the universe that imparts the inertial properties of clock rates and lengths which generate the effects of relativity. This is explained in detail in Relativity Trail
Introductory document: Relativity in absolute terms (Twin Paradox in Relativity) (can be read with comprehension in twelve minutes) Absolute versions of Einstein's postulates (a snippet from the above document -- can be read with comprehension in three minutes) Twin Paradox Animation on youtube. Light rays and traveling twins are charted in absolute terms, free of the misleading space-time diagram.
home page: (copy and paste into your browser) Updated 12/19/2023