Stellarator - 2D Poincaré section

Field lines in a toroidal trap

A magnetic field line in a stellarator winds around the torus and also rotates poloidally; the average poloidal turn per toroidal turn is the rotational transform ι. Cutting the torus at one toroidal angle and marking where a line pierces the plane after every turn gives a Poincaré section. With perfect symmetry the points of one line lie on a closed circle (a flux surface). Here the surfaces are circular with ι(r) = ι₀ + ι₂ r², and the field-line map is the area-preserving twist map ψ' = ψ + ε sin(mθ), θ' = θ + 2πι(ψ'), with ψ = r²/2.

Islands and chaos

Where mι is an integer a helical error field resonates with the field line: the surface breaks into a chain of magnetic islands whose half-width is δψ = 2√(ε/(2πmι')), exactly the pendulum separatrix. Neighbouring chains overlap when the Chirikov parameter σ = (δ₁ + δ₂)/gap exceeds about one; then field lines wander chaotically across the plasma and run into the wall, which spoils confinement. Shear (a large ι₂) shrinks the islands and spreads the resonances apart.

Honest simplifications: circular flux surfaces and a single resonant harmonic instead of a real three-dimensional vacuum field computed from coil currents (no Shafranov shift, no pressure-driven currents, no helical coil geometry); the map is a standard-map-style model, not a Biot–Savart integration; the measured island widths use the first-order pendulum approximation and drift from it at large ε; the wall is a plain circle; no particle drifts or collisions, only field lines.

• Start with Nested surfaces: every line stays on its own circle.
• Pick Island chain: three islands open at the yellow circle where ι = 2/3.
• Raise ε with the slider until the islands touch the next resonance and the lines scatter.
• Try Strong shear and see the islands shrink.