Chirikov Standard Map

Chirikov standard map (B. Chirikov, 1979) — the canonical AREA-PRESERVING map, and the one system here with no attractor at all. A kicked rotor: each iterate kicks the momentum by K·sin θ and lets the angle advance. Because area is preserved nothing contracts onto anything, so it is drawn as an ensemble of orbits rather than one — a single orbit would trace its own invariant curve and tell you nothing about the rest. Turn K up and watch the intact curves break into a chaotic sea with islands of stability stranded in it; near K ≈ 0.9716 the last curve spanning the phase space goes, and orbits can wander in momentum without bound.

p' = p + K·sin θ
θ' = θ + p'   (both mod 2π)

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.