Custom software was developed to calculate the Lyapunov exponent of a chaotic dynamical system over a range of its four parameters. The raw data from this 4-dimensional parameter space was fed to another program to generate thousands of individual images, where X and Y represent two of the four coefficients. Every single pixel in these images gets its value from the calculation of the Lyapunov exponent of the strange attractor generated by the four coefficients it represents. The second program outputs four 16-bit grayscale images, which represent the different “components” of the spectrum of Lyapunov exponents. These images were combined in Photoshop using a pseudo-color technique to bring out subtle coloration in the final images.
The following resources were very helpful and inspirational in my exploration of lyapunov space and in the development of my software and artwork.
Extracting beauty from chaos by Andy Burbanks
Lyapunov Exponents by Jan Thor
Lyapunov fractal on Wikipedia
Z-Lyapunov by Georges Rosset
C code for numerical calculation of the Lyapunov exponent by Paul Bourke
Complexity and ‘art’: Lyapunov graphs by Diethard Janßen
Markus-Lyapunov Fractals by Charles Vassallo
Lyapunov Exponents Lab Report by Earl F. Glynn II
Lyapunov Space – A Software Developer’s Odyssey by Bill Storage