About the Fractal Note Gallery at the Visual Math Institute

Dedicated to Christian Mira

This site, created in November 2013 by Ralph H. Abraham and the Visual Math Institute, presents images, movies, and sounds from our current research on the chaotic/fractal attractors of simple discrete dynamical systems (iterations) and their bifurcations.

This project extends the work reported in the book Chaos in Discrete Dynamical Systems (aka JPX) by Ralph Abraham, Laura Gardini, and Christian Mira, 1997. The studies reported in this book concern two families of simple mappings of the euclidean plane into itself. Both are defined by quadratic polynomials in two variables. Hence the acronym JPX, for Just Plane Chaos.

Our extension here is based on the extensive computations of Julian C. Sprott reported in his book Strange Attractors (aka SA) of 1993. These attractors:

  • extend quadratic attractors from the simple cases of JPX into the most general cases,
  • extend from quadratic polynomials to polynomials of degrees 3, 4, and 5,
  • extend the work of Fields & Golubitsky on symmetric chaos in 2D, and
  • extend from two dimensions (2D) to three (3D) and beyond.
Each chaotic attractor is presented in this gallery as a static image, colored according to the density of the point cloud defined ay a long trajectory (typically three million points) of the attractor. These images are accompanied by sound files generated by the time sequence of the attractor.

Our primary focus in this project is on the bifurcations of these attractors as one of the polynomial coefficients is varied. These bifurcation sequences are presented as short movies, usually at a rate of ten frames per second. The occurence of windows of periodic (nonchaotic) behavior is of special interest from the point of view of bifurcation theory. Some of our movies have a sound track, each frame accompanied by one tenth of a second of granular audio data (100 samples) encoded at 1000 samples per second.

We are also interested in the application of our images, animations, and sounds in the sphere of digital art.

Revised 30 September 2015 by Ralph Abraham, <abraham@vismath.org>