Field-line islands at the Farey rationals

The standard map as the local model of a field-line Poincaré map: θ is the poloidal angle, ι the rotational transform. Next to it, the Ford circles of the same rationals. The statements below are standard results: the standard map (Chirikov 1979), the critical K of the golden-mean circle (Greene 1979), Hurwitz's theorem (1891) and Ford circles (Ford 1938).

Perturbation K K = 0.60

How to read it

One turn around the torus maps (θ, ι) to (θ + ι′, ι′) with ι′ = ι + (K/2π) sin 2πθ (both mod 1). The step has Jacobian exactly 1: like the flux of a divergence-free magnetic field, area is preserved, so lines never drain anywhere — they circulate.

At K = 0 every line stays on its own surface ι = const. Turning K on breaks the surfaces with rational ι = n/m first: each becomes a chain of m islands, and the low-denominator ones open widest. The magnetic differential equation shows why — its Fourier modes divide by m ι − n, which vanishes exactly at those rationals. The Ford circle at n/m has radius 1/(2m²), the same "simplest first" ordering.

The surfaces that survive longest have rotational transforms as far from every rational as possible. The golden mean (dashed lines, ι = 1/φ and 1/φ²) is the worst approximable number, with Hurwitz's constant 1/√5, and its circle is the last to break, at Greene's K_c ≈ 0.9716. Beyond it no rotational circle is left and a line can wander through every ι.

Scope: a local model, not a tokamak equilibrium. Orbits are seeded on a 12 × 24 grid and iterated 500 times each; the picture is diagnostic. B. V. Chirikov, Phys. Rep. 52, 263 (1979); J. M. Greene, J. Math. Phys. 20, 1183 (1979); A. Hurwitz, Math. Ann. 39, 279 (1891); L. R. Ford, Amer. Math. Monthly 45, 586 (1938).