The system seems to give interesting results only when b is close to one. It behaves less chaotically when b > 1 is fixed, so you can actually animate it - click the thumbnail to see.
Continuing the theme of strange attractors, here's the well-known one embedded in the Ikeda map. The thumbnail at left shows the central 'vortex'
of the attractor, and links to a larger viewport.
In these images, I've plotted the real and imaginary components along the x and y axes respectively. But the more popular way to visualize this attractor adds an extra parameter to the system and is expressed in trigonometric functions. Such adaptation of the code below yields these results.
French astronomer Michel Hénon reported on this strange, fractal attractor in 1976. Since then, it has been among the most studied examples of chaotic dynamical systems.
This pattern generator, discovered by Barry Martin, was nicknamed 'Hopalong' when Scientific American introduced it in their September '86 issue.Clicking the window adjusts the viewport position; there is also an alternate version with color and animation.
Also, for a certain Rubyist friend, I wrote another lazily-evaluated, colored and animated implementation in Ruby.
It took me some experimentation to figure out how to color this derivation of the logistic map; I'm still not quite sure how the hues should scale as you zoom. But the bi-tonal method shown below works well enough to produce the image at left - click it for more detail.
What programmer hasn't at some point written an implementation of Benoit Mandelbrot's great discovery, the most famous fractal in the world? Here's my own minimal version, with the simplest possible coloring scheme. To interact with it, just click any two points: the window will zoom in on the rectangle they define.
