The Lyapunov fractal available in Saturn & Titan is different to all the other fractals as its algorithm isn’t based on the Mandelbrot or Julia algorithms. I use the name Lyapunov for these fractals as it the name used when I first came across them in the September 1991 edition of Scientific American. The formula used to generate these fractals is named after the Russian mathematician Aleksandr M. Lyapunov, this formula was used by Mario Markus of the Max Plank Institute for Nutrition to model the breakdown of carbohydrates using enzymes and he produces the first pictures, so the fractal should more poperly be called the Markus-Lyapunov fractal. For more information see Introduction to Markus-Lyapunov Fractals.

Unlike all the other fractals implemented in Saturn & Titan colouring is limited to “multi-colour” and “single colour”. The colouring can be be applied separately to positive Lyapunov exponent values and negative Lyapunov exponent values. The best results are obtained by using multi-colour for both positive and negative values with different colour maps (or gradients).

The pictures alter depending on the “sequence string” which in Saturn consists of As and Bs, the simplest sequence is AB, the sequence BA produces the same picture flipped about the diagonal. There are two values which control the quality of the pictures and they are the number of settling cycles and the number of calculating cycles, a cycle is the calculation of the sequence, so if there are 100 settling cycles and 400 calculating cycles and the sequence is AB there are 200 and 800 calculations, the longer the sequence the greater the number of calculations for a give pair of settling and calculating cycles.

There is a distinct shape that crops up frequently in Markus-Lyapunov fractals and that is the “swallow” for a reason that should be obvious, an example is shown below. This example was produced using Mars & Phobos which were fore runners of Saturn & Titan. Red Swallow

There now follow some more Markus-Lyapunov fractals. Lyapumov 25 Lyapunov 34 Lyapunov 35

More examples can be found in my Deviant Art Gallery, use the search gallery box to find the Lyapunov fractals.

Posted 5 March 2012 by in Fractal, Software

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