Gumowski-Mira

Gumowski-Mira map (I. Gumowski and C. Mira, CERN, 1980) — written to study how a particle beam drifts turn after turn around an accelerator ring, and it produces organic, almost biological figures that look nothing like their origin. The shared nonlinearity g is applied twice per iterate, to x and then to the NEW x. Unlike the others here it is QUASI-PERIODIC at these settings rather than chaotic: the curves it draws are invariant curves the orbit winds around forever, not a fractal dust. Two orbits started a hair apart drift apart slowly instead of separating exponentially, which is why the figure looks woven rather than sprayed.

g(v) = μ·v + 2(1−μ)·v²/(1+v²)
x' = y + a(1 − b·y²)·y + g(x)
y' = −x + g(x')

It opens running, on the model selector, with its own knobs. Rotate it by dragging, zoom with the wheel, and turn the parameters to see what the figure does when the system changes.