Computer-assisted proofs of Hopf bubbles and degenerate Hopf bifurcations

The paper’s Proposition 12 gives verifiable interval conditions (1–8) under which a bubble bifurcation with quadratic fold exists in the projection of Θ_H ∩ W(v1, v*, v2), and its proof proceeds by: (i) using condition 1 to parametrize the Hopf curve by α; (ii) using conditions 2–4 to force an interior constrained extremum of β along the Hopf curve inside the wedge; (iii) eliminating Θ_H'' via the identity a(Θ_H)=0 and showing d^2/dt^2 β(Θ_H) ≠ 0 from condition 7; and (iv) using the local diffeomorphism h(s)=(α(s), a(s)) (condition 6) and uniform definiteness of the Hessian in (α,a) coordinates (condition 8) to conclude a strict local extremum at (α*,0), including ∂_aΓ(α*,0)=0 by amplitude symmetry of the periodic-orbit embedding, as stated in the proof (see the statement of the conditions and proof lines around Proposition 12 and Definition 2). These steps are explicitly laid out in the paper’s text, including the wedge set-up and the exact form of conditions 1–8, the role of Vs, and the proofs of monotonicity and nondegeneracy of the fold (Proposition 11 for the Hopf curve existence; Definition 3 for the wedge; Definition 2; and the proof of Proposition 12) . The candidate solution mirrors the paper’s argument almost step-for-step: it identifies H={a=0} and its tangent space, finds an interior constrained extremum of β on H∩W by conditions (2)–(4), invokes a Lagrange-multiplier relation ∇β=c∇a at the constrained extremum, uses condition (7) to get nonzero second derivative along H, and employs h(s)=(α,a) (condition 6) together with the Hessian definiteness (8) to conclude a quadratic fold. Minor omissions/nuances: the model does not explicitly justify the needed ∂_aΓ(α*,0)=0 (the paper appeals to amplitude symmetry here), and there is a small notational mismatch about whether h’s first coordinate is α or a, which the model acknowledges. These are fixable presentation issues and do not affect correctness of the core argument. Overall, both are correct and substantially the same proof.