Animated 3D illustration of the Lorenz Attractor, modeled with five thousand spheres, using the classic parameter set, and color by iteration count. The positions of the spheres represent iterations of the Lorenz equations, calculated using numerical integration. The color transitions from cool blue to warm orange hue based on iteration count.
Another rendition of a frame from this video was contributed to illustrate the article "Chaos at fifty" by Adilson E. Motter and David K. Campbell in "Physics Today", May 2013.
http://scitation.aip.org/content/aip/magazine/physicstoday/66/5
The music - "Hungarian Dance No. 6 In D Major" by Johannes Brahms, Philharmonia Cassovia - is audio swap from YouTube.
Equations:
dx/dt = Sigma (y - x); dy/dt = x (Rho - z) - y; dz/dt = xy - Beta z
Parameters:
Sigma=10, Rho=28, Beta=8/3.
Code:
Integration with step 0.01 (Euler). Coded with Microsoft Visual C++, OpenGL rendering using Cinder (
libcinder.org). Related sample C++ application code can be found here:
https://github.com/stefan-g/LAx.
References:
1. J. Gleick, Chaos: Making a New Science,
http://en.wikipedia.org/wiki/Chaos:_Making_a_New_Science
2. A. Motter and D. Campbell, Chaos at fifty, article in Physics Today,
http://www.physicstoday.org/resource/1/phtoad/v66/i5/p27_s1?bypassSSO=1
3. S. Strogatz, Nonlinear Dynamics and Chaos,
http://books.google.com/books?id=dTvTzBeRn3cC&printsec=frontcover&dq=strogatz+book&hl=en&sa=X&ei=KiyYUdW6N8SSiALqu4GIBQ&ved=0CDAQ6AEwAA
4. Peitgen, Jürgens, Saupe, Chaos and Fractals: New Frontiers of Science,
http://books.google.com/books/about/Chaos_and_Fractals.html?id=jVpS_u0Lg4gC