Biased Nonlinear Ship Roll as a Z2-Equivariant Oscillator: Symmetry Breaking, Closed-Form Capsize Boundaries, and a Second-Generation Intact-Stability Perspective

The roll equation of a port–starboard symmetric ship is a clean physical realization of an order-two reflection (Z2)-equivariant nonlinear oscillator: odd restoring and damping make the upright state a symmetric equilibrium whose two angles of vanishing stability form a heteroclinic-connected pair. A steady heeling action—beam wind, off-center load or list—enters as a single symmetry-breaking parameter c. Although the qualitative effect of such a bias is known, we show that this one parameter organizes the whole capsize problem in closed form. Equivariant singularity theory identifies c as the imperfection that unfolds the symmetric pitchfork of equilibria into a cusp, turning the heteroclinic pair into a homoclinic loop. The biased Melnikov boundary then yields two directional capsize thresholds and a damping-independent asymmetry index Δfcr=2cIc/A(Ω), exactly linear in the bias: for a lightly damped hull a heel below 0.1∘ already halves the port/starboard split. For parametric roll, Floquet analysis gives the bias-corrected stability boundary, recovering ΔGM/GM>4ζ in the symmetric limit and translating the principal tongue; for the dead-ship condition, the stationary Fokker–Planck solution—exact for the adopted one-degree-of-freedom (1-DOF) model—shows that, in the light-damping energy-diffusion regime, lightly damped capsize occurs over the lowered barrier with probability approaching one. A small-bias expansion reveals a sensitivity hierarchy—beam-sea capsize and dead-ship survival are first-order in the heel, and parametric detuning is only second-order—and a two-parameter cusp links the bias to pure loss of stability. All results are validated against safe-basin erosion, Floquet multipliers and Monte-Carlo simulation for a real vessel, offering the International Maritime Organization (IMO) second-generation intact-stability criteria (SGISC) a transparent, analytically based correction for the port–starboard asymmetry their symmetric assumption omits.