Hénon map: the classic 1976 crescent — Classical Chaos

Michel Hénon wrote this down in 1976 while poking at Lorenz-like chaos. Discrete map: one new point from the last. No continuous ODE, no step size dt. He wanted something like a simplified Poincaré-section sketch of that kind of motion, and ended up with a strange attractor that looks like a folded crescent, a banana if you squint.

Classical Chaos iterates the classic 1976 form. Defaults sit at a = 1.4, b = 0.3, seed (0.03, 0.01).

The classic update

Each tick is one iterate. Point in, point out. The silhouette that piles up is the attractor.

const nextX = 1 - a * x * x + y
const nextY = b * x
x = nextX
y = nextY

Same seed, same two lines, over and over, until the crescent fills in. Knob ranges on the stage sit in about [0, 2] for both a and b.

The Jacobian determinant is −b. With |b| < 1 (here 0.3) patches shrink under the map while stretching and folding still build the strange set. Not area-preserving unless |b| = 1.

Full canvas

On the full Hénon stage twist a and b inside [0, 2] and watch the crescent rebuild from (0.03, 0.01).