» definitions of some iterative functions which generate chaos:
◊ trigonometric xn+1 = A sin(B yn) + C cos(D xn) yn+1 = E sin(F xn) + G cos(H yn)
where A - H variables determine each attractor;
◊ quadratic xn+1 = A + B xn + C xn2 + D xn yn + E yn + F yn2 yn+1 = G + H xn + I xn2 + J xn yn + K yn + L yn2
where A - L variables determine each attractor;
◊ Hénon xn+1 = 1 - A xn2 + yn yn+1 = B xn
where A, B variables determine each attractor.
» visualizing chaotic attractors:
2-dimensional pictures of attractors are created by analyzing the data received as a result of successive steps of calculation iterative functions which generate the chaos. Hue, saturation and value (HSV color space model) correspond the localized Lyapunov exponent, the number of times the point was visited or distance between value and argument of the function. The result is often beautiful and complex geometric structures.
Presented drawings were generated by own script written in PHP. The number of iterations performed for each of the images - 100 million. More works at http://slide.nethium.pl/.