Waves at sub-harmonic resonance condition (D.3) for synchronised systems, as a function of (dimensionless) forcing frequency f and N₀/Ω_s. The other notations are identical to the ones introduced in the main text. White regions: no compatible waves satisfying (D.3). Stars (yellow area): hyperbolic waves ℋ₁. Right slash (purple area): hyperbolic waves ℋ₂. Dots (green area): elliptic waves ℰ₁. Back slash (blue area): elliptic waves ℰ₂. The classical allowable region of the instability is 0 ≤ f ≤ 4 in neutral fluids.