Lyapunov fractals with Seeeduino Wio Terminal

What is the Lyapunov space?

The formula that generates it, created by the Russian mathematician Aleksandr Lyapunov, uniquely determines a value corresponding to chaos, relating to the behavior of the dynamic system under consideration.

Color shades intercept variable exponents from zero to minus infinity. Between positive and negative values ​​we find the abrupt transition between the brightest colors and black.

The model is based on a variation of the logistic equation. The engine of which consists of an x ​​value and a parameter, linked by the equation in the figure.

As long as the parameter remains below 2, the system is stable. As the value of the parameter in the system increases, it shows an increasing number of attractors, until it shows a self-similar and chaotic fractal behavior for values ​​greater than 3.56.

The program begins with a presentation screen, in which the genesis of the Lyapunov fractal is briefly described.

By pressing the paddle, you go to the next screen, where the graph of the Lyapunov set is presented.

The level of detail (and the speed of execution) depends on the bailout variable, that is how many tests are carried out on each point before establishing its value (and therefore, the color). Notice how the representation of the scene appears almost three-dimensional.

By appropriately selecting the coordinates of the plane, it is possible to isolate the "self-resembling swallows" characteristics (the entire plane is itself a "swallow").

Even the image created on the entire plane, with only 3 iterations, does justice to the fact that we are faced with the description of a chaotic system.

With a threshold of 500 iterations, particularly detailed images are obtained, even with the 320x240 resolution of the Wio Terminal. Unfortunately the photos don't do the images justice: we got a double color palette with 7 bit depth (128 shades)

The implementation on Seeeduino Wio is discussed throughout the video, and did not require any particular work. The source code is available on github.

Note: the video is close-captioned both in English and Italian.