Stationary pattern interfaces

Symmetric superpatterns near a Turing instability. Choose a direction (p, q with cot θ = √3·p/q, or η), pick which inscribed hexagons and isolated rolls to combine, and steer μ₀, β₂ directly on the existence map.

Direction

p
q

Existence & parameters

K₀
K₂
drag the point
full only a none current
Axes μ₀ (horizontal), β₂ (vertical). Green edge on the roll side is b²=0 — lower K₂ to open the region for hexagon+roll combinations.

Central set Σc

mode selected hexagon active roll pole (Jordan-4)

Pattern u(x)

scroll = zoom · drag = pan

Inscribed hexagons

Isolated rolls  Σciso

Central modes on no hexagon; each carries a roll at amplitude b.

Solution

Leading-order (ε=0) amplitudes of the reduced equations; symmetric ansatz (all hexagons share amplitude a, all rolls share b). Companion to the quadratic-resonances explorer.