Ikeda Map - Chaos from Laser Light

From 20,000 starting points iterating to chaos

Controls

Parameters

u (dissipation) 0.90

κ (coupling) 0.40

Presets

The Map

t = κ − 6/(1 + x² + y²)
x' = 1 + u(x·cos(t) − y·sin(t))
y' = u(x·sin(t) + y·cos(t))

Rotation by angle t, then scaling by u, plus offset.

Statistics

Points 0

Iterations 0

Behavior -

Kensuke Ikeda (1979)

Japanese physicist Kensuke Ikeda created this map to model light bouncing inside a ring cavity with a nonlinear optical medium. The light's electric field evolves chaotically—intensity unpredictably varies despite deterministic equations! This was one of the first demonstrations of optical chaos.

Stretch and Fold

Like all chaotic maps, the Ikeda map "stretches" (by rotation and scaling) and "folds" (through the nonlinear term). Nearby points separate exponentially—sensitivity to initial conditions. The folding keeps the attractor bounded despite stretching!

Parameter u Controls Chaos

When u < 0.6: stable fixed point. As u increases: period-doubling bifurcations. Near u = 0.9: full chaos with fractal strange attractor. The parameter u represents energy dissipation in the optical resonator—more dissipation = less chaos!

Shrimp Structures

Recent research (2024) reveals "shrimp-shaped" regions in parameter space—islands of stability within chaos. These self-similar structures appear at multiple scales, showing the intricate boundary between order and chaos in this seemingly simple system!

The Ikeda Map