Links

General


• older class link (2003) -- good explanations, pictures and code of the attractors we explored in class
• older class link (2006) -- check the links here; many are defunct but some are still ok
• luke_strange_attractors.zip -- the older Max/MSP/jitter patch done by Luke DuBois; a pdf of the book he used with the attractor formulas is included
• ibiblio Chaotic Maps -- good description and info on strange attractors in general
• Paul Bourke's Math pages -- excellent source of formulas + code for a great many things (attractors, fractals, cool stuff!)
  
  
  Logistic Map ('population equation')
  
  
• Logistic Map page at ASU
• MATHEematics Illuminated chapter -- both of these are good math descriptions of the chaos behind the logistic map
  
  
  Lorenz Attractor
  
  
• Lorenz Model Demo from Caltech -- applet exploring the Lorenz attractor (written by Michael Cross)
• basic Lorenz info -- wikipedia entry on the Lorenz attractor
• Exploratorium Lorenz applet -- with this one, you can start a number of orbit-plots simultaneously. Sensitivity to initial conditions, yeah.
• Paul Bourke's Lorenz page -- really snazzy pix
• 3DAttractors.app.zip -- nice OSX app dsiplaying the Lorenz and (related) Roessler attractors (written by Juan G. Restrepo, click on the "Programs" link to get to this)
  
  
  Henon Map
  
  
• basic Henon info -- wikipedia entry on the Henon map
• Henon Map Demo from Caltech -- nice Henon map-generatimg applet (you my want to slow the speed down to watch it fill in), also written by Michael Cross
• Henon Zoom Demo -- this one allows you to select a part of the Henon map and zoom in to see the fractal nature of the attractor (written by Achim Luhn)

Class Downloads

• attractor-patches.zip -- class patches for the Logistic map (population equation), the Lorenz attractor and the Henon map. Audio 'sonification' is added in the last one(s) of each series.

  The basic equation for the Logistic map is:

  y = Rx(1-x)
  
  When R is < 3.0, the iteration of this equation leads to a single value. When R reaches 3.0, the equation starts randomly oscillating between 2 values ("period doubling"), and as R goes larger than 3.0 it begins to enter true chaotic areas. What fun! The code for the iteration of this equation is imbedded in the [rtcmix~] objects in the patches.

  The equations for the Lorenz attractors are:

  dx/dt = a(y-x)
  dy/dt = bx - y - zx
  dz/dt = xy -cz
  
  We used values of a = 10.0, b = 28.0 and c = 8.0/3.0. We also 'moved' this set of differential equations along by adding a "delta" value each time. That value was set to 0.01. The equations (also imbedded in the [rtcmix~] objects in the patches) with the delta value were coded as follows:

  delta = 0.01
  
  dx = 10.0 * (y-x)
  dy = x * (28.0-z) - y
  dz = x*y - (8.0/3.0)*z
  xnew = x + delta*dx
  ynew = y + delta*dy
  znew = z + delta*dz
  
  x = xnew
  y = ynew
  z = znew
  

  The x, y and z variables were initialized to 1.0 (this could have been anything, however).

  The iterated equations for the Henon map are:

  x_n = y_n-1 = ax_n-1² + 1.0
  y_n = bx_n-1
  
  with a = 1.4 and b = 0.3. The code for doing this calculation is straightforward:

  xnew = y - (a*(x*x)) + 1.0
  ynew = b * x
  
  x = xnew
  y = ynew
  

  with x and y initialized to arbitrary values (we used 0.0).