Ikeda Map

Ikeda Map

The Ikeda map models the dynamics of light in an optical cavity (a ring laser). It was introduced by Kensuke Ikeda in 1979 and was one of the first physical systems shown to exhibit optical chaos.

The Map

t_n   = 0.4 - 6 / (1 + x_n² + y_n²)
x_{n+1} = 1 + u·(x_n·cos(t_n) - y_n·sin(t_n))
y_{n+1} = u·(x_n·sin(t_n) + y_n·cos(t_n))

The parameter u controls the strength of the nonlinearity:

• u < ~0.6: Fixed point attractor
• u ≈ 0.7: Period-2 cycle
• u ≈ 0.9: Strange attractor — chaotic dynamics

Physical Interpretation

• (x, y) represents the electric field amplitude in the cavity
• t_n is a phase shift that depends on light intensity (x² + y²)
• The rotation by angle t_n followed by scaling by u models the round-trip

Your Task

Implement:

1. ikeda_step(x, y, u) — compute one step of the Ikeda map
2. ikeda_iterate(x0, y0, u, n) — iterate n times, return final point