Hénon Map

Hénon Map

The Hénon map is a classic 2D discrete dynamical system that exhibits a strange attractor. It was introduced by Michel Hénon in 1976 as a simplified model of the Poincaré section of the Lorenz attractor.

The Map

Given a point (x, y), the next point is:

x_{n+1} = 1 - a·x_n² + y_n
y_{n+1} = b·x_n

With parameters a = 1.4 and b = 0.3, the map produces a strange attractor — a fractal structure with dimension ≈ 1.26.

Properties

• Area contraction: The Jacobian determinant is -b, so volumes shrink by factor |b| = 0.3 each step
• Strange attractor: For a=1.4, b=0.3, almost all initial conditions converge to the attractor
• Fractal structure: The attractor has a self-similar, layered structure

Implement the Hénon Map

Implement three functions:

1. henon_step(x, y, a, b) — compute one iteration
2. henon_iterate(x0, y0, a, b, n) — iterate n times and return final point
3. henon_orbit(x0, y0, a, b, n) — return list of x-coordinates for n iterations