The Lyapunov Fractal is a complex structure that appears in the study of dynamical systems, particularly in the context of chaos theory. It's named after Aleksandr Lyapunov, a Russian mathematician known for his work on stability theory. These fractal-like patterns are created by examining the growth rate of sequences of functions, typically logistic maps, which are simple mathematical representations of how a population grows under certain constraints.
Lyapunov Fractal viewed from different positions in space
This is a GLSL (OpenGL Shading Language) implementation designed to render a 3D approximation of a Lyapunov Fractal.
The parameters rA, rB, and rC are computed from the current position and the slice value. These represent different rates in the underlying logistic maps, which are fundamental to the fractal's structure. The core of the fractal generation occurs in a loop where the logistic map is iterated. This loop calculates the next value of x (representing a state in the system) based on rA, rB, or rC, depending on the current iteration. This iterative process is key to revealing the chaotic dynamics that create the fractal pattern.
The sum of the logarithm of the absolute value of the derivative of the map at each step is calculated. This sum, averaged over the number of iterations, gives the Lyapunov exponent (lambda), a measure of the rate of separation of infinitesimally close trajectories.
The final step involves mapping the value of the Lyapunov exponent to a color. Positive values of lambda are mapped to shades of blue, indicating chaotic behavior, while negative values are represented in shades of red and green, indicating stability.
Style transfer experiments with GEN-1
65mm Technicolor film
35mm color film