Nonlinear hydroelastic response of an ice sheet to a moving load: Bifurcation and stability

The nonlinear hydroelastic response of a floating ice sheet to a moving load is investigated in two dimensions through a systematic reduction, spanning from the fully nonlinear Euler equations to a low-dimensional dynamical system. By incorporating linear damping to model dissipation, we demonstrate that the steady-state response exhibits an S-shaped bifurcation structure, giving rise to bistability and hysteretic transitions. By projecting the forced–dissipative Whitham equation onto a two-dimensional manifold, we establish a formal analogy between this hydroelastic wave problem and the forced–damped Duffing oscillator – the prototypical model for such folding bifurcations. This reduction reveals that the spatial lag between the load and the peak ice deflection manifests as a temporal phase shift, which the system adjusts spontaneously to satisfy a global power-balance condition. A critical ratio of damping to load magnitude is identified as the governing threshold for the fold bifurcation, furnishing a predictive criterion for the onset of nonlinear jump phenomena in the ice-sheet response.