Boris Chirikov's standard map is the simplest object that does something Lorenz cannot: it conserves phase-space volume. No attractors, no friction — just a twist of the torus repeated forever.

The Math

On the cylinder (θ, p) ∈ [0, 2π) × ℝ, iterate:

$$ \begin{aligned} p_{n+1} = p_{n} + K \sin(\theta _{n}) \\ \theta _{n+1} = \theta _{n} + p_{n+1} \end{aligned} $$

Chaos and Order

Below the critical coupling K_c = 0.971635… (Greene, 1979) invariant KAM tori partition the plane into nested rings of quasi-periodic motion. Above it, the last golden-mean torus shatters and chaos becomes globally connected. Drag K across the threshold and watch the geometry of integrability fail in real time.