eureka
experiment 026 · hamiltonian chaos · area-preserving maps

The Standard Map

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. Below the critical coupling Kc = 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.

θ → p ↑

The map

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

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

The Jacobian determinant is identically 1 — the map is area-preserving. It is the stroboscopic Poincaré section of a single mechanical system: a free rotor (angle $\theta$, momentum p) given an instantaneous gravitational kick of strength K once per period. Every Hamiltonian system with a divided phase space looks locally like this, which is why the same picture appears in tokamak field-line diagnostics, particle accelerator dynamic aperture, the kicked top, and the restricted three-body problem.

Three regimes, one slider

At K = 0 the map is integrable: each horizontal line p = const is invariant and the dynamics is a rigid rotation by p. Increase K and the resonant lines (p = 2πq/r for rationals q/r) pinch into chains of resonance islands, each a Poincaré–Birkhoff chain alternating elliptic and hyperbolic fixed points. Between them, irrational-frequency tori survive — the KAM theorem (Kolmogorov 1954, Arnold 1963, Moser 1962) guarantees this for sufficiently irrational rotation numbers and small K.

As K grows the islands fatten; their separatrices acquire a stochastic layer; the tori between them are gradually destroyed. John Greene showed in 1979 that the very last torus to break is the one with rotation number equal to the inverse golden mean, $\gamma$ = (√5 − 1)/2. It vanishes precisely at

$$ K_{c} = 0.971635406… $$

Above Kc the chaotic regions of phase space connect into one global stochastic sea. Momentum is no longer bounded: an orbit started anywhere chaotic can, eventually, reach anywhere. Below Kc KAM tori still wall off the plane into invariant bands, even though most trajectories already look chaotic to the eye.

The largest Lyapunov exponent

Each orbit carries along a tangent vector v = (δp, δθ) advanced by the linearised map

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

After every step the simulation renormalises v and accumulates log|v|; the time-average is the largest Lyapunov exponent λ. On a KAM torus λ → 0 (linear stretching is at most polynomial); in the stochastic sea λ > 0 and grows roughly like ln(K/2) for K ≫ 1 (the quasilinear estimate). The chaotic fraction readout is the proportion of currently active orbits whose running λ exceeds 0.05.

Accelerator modes

Try the Accelerator preset (K ≈ 6.59): around isolated values of K the map admits stable periodic orbits whose momentum drifts ballistically — anomalous diffusion superimposed on the chaotic background. These accelerator-mode islands punch holes in the otherwise quasilinear $D$ ≈ K²/4 momentum-diffusion law and are the cleanest toy model of Lévy flights in deterministic systems.

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