Bifurcation analysis of the problem of a “rubber” ellipsoid of revolution rolling on a plane

The paper reduces the rubber-rolling ellipsoid to a 1-DOF shape equation with energy E = B(ϑ)p_ϑ^2/2 + k^2/(2 sin^2ϑ) + U(ϑ) and, for an axisymmetric ellipsoid, U(ϑ) = α cosϑ + Z(ϑ), Z(ϑ) = √(β^2 sin^2ϑ + cos^2ϑ), B(ϑ) as given explicitly; it also derives ψ̇ = −κ cosϑ/(J(ϑ) sin^2ϑ) (its Eq. (17)) and introduces the rotation number N = −ωψ/ωϑ to classify COM trajectories by resonance, yielding the nine cases listed in the proposition. All of these steps and conclusions match the candidate solution’s derivations and classification, which use an equivalent reconstruction argument (rotated period-increments) rather than the paper’s Fourier expansion, but arrive at the same resonance trichotomy (integer → unbounded drift, rational non-integer → closed, irrational → bounded quasiperiodic) and special cases (equilibria, rotations, separatrices). The only discrepancy is the paper’s displayed formula for the κ=0 threshold εmin, which appears inconsistent (its radicand is negative on the stated regime) and likely contains a typographical error; the candidate’s criterion εmin = sup_{ϑ∈[π/2,π]} U(ϑ) = max{β, 1−α} for the onset of unbounded straight-line motion when κ=0 follows directly from U(ϑ) and the sublevel-set condition and is consistent with the paper’s qualitative discussion of κ=0 (N=0, straight-line motion) and with the proposition’s cases 3–4. See the paper’s reduction and potentials, Eq. (10) and U,B forms for the ellipsoid, the ψ̇ quadrature (17), the ζ̇ reconstruction and rotation-number definition, and the 9-case proposition (classification) in the cited excerpts.