Ikeda map: one knob named a — Classical Chaos

The Ikeda map starts at (0, 0). Each step spits out the next (x, y). Discrete map, not a continuous trail like Lorenz. Kensuke Ikeda, 1979; ring-cavity optics in the paper background. On Classical Chaos the canvas is the map, not a full optics simulator.

Site default is a = 1 (range [0, 1]). An example preset sits near a = 0.87719. The band people usually poke for chaos is about [0.6, 1].

The update

A state-dependent angle f turns the plane, then a scales the stretch:

const f = 0.4 - 6 / (1 + x * x + y * y)
const nextX = 1 + a * (x * Math.cos(f) - y * Math.sin(f))
const nextY = a * (x * Math.sin(f) + y * Math.cos(f))
x = nextX
y = nextY

The knob is named a here (not c). Nudge it and the cloud rebuilds from (0, 0). Near 0.87719 the folds often feel looser than at the default a = 1; shove much lower and the structure dulls.

Full canvas

On the full Ikeda stage twist a in [0, 1], try the example near 0.87719, and watch the cloud rebuild from the origin.