Double Hamiltonian Hopf Bifurcation: normalization and normal form non-integrability

L.M. Lerman, R. Mazrooei-Sebdani, N.E. Kulagin

correctmedium confidence

Category: math.DS
Journal tier: Strong Field
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper derives a T^2-equivariant fourth-order normal form for the double Hamiltonian Hopf unfolding and proves that, after reduction, the 2-DOF truncated system is non-integrable for almost all coefficients, with an explicit exceptional set tied to a reflectionless scattering condition. The candidate solution reproduces the same normal form and reduction, and obtains the same non-integrability conclusion via Morales–Ramis/Galois methods (hyperbolic: Pöschl–Teller; elliptic: Lamé), identifying essentially the same exceptional parameters. Minor differences are methodological and in wording (e.g., calling the zero-momentum level “regular”), not substantive.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The manuscript provides a coherent T\^2-equivariant normal form for the double Hamiltonian Hopf and shows generic non-integrability of the quartic truncation with an explicit discrete exceptional set, complementing known HHB results. The arguments rest on standard, robust tools (reduction, scattering, Galois obstructions). Minor clarifications on the geometry at the zero-momentum level, explicit parameter hypotheses for the scattering argument, and a consolidated enumeration of integrable loci would further improve clarity and self-containment.