Numerical Computation of Critical Surfaces for the Breakup of Invariant Tori in Hamiltonian Systems

The paper computes the critical surface for invariant tori numerically (via renormalization and configuration–space conjugation) for precisely the two families in question, derives the same conjugacy equation h (their Eq. (15)), and reports: (i) in d=2, the boundary looks smooth; (ii) in d=3 with ω=(σ,σ^2,1), Ω=(1,1,−1), μ3 fixed at 0.1 (spiral mean), the critical surface exhibits visible cusps, e.g., near (μ1,μ2)≈(0.044,0.23), but offers no rigorous proof of cusp existence . The model solution gives a correct Lyapunov–Schmidt/Crandall–Rabinowitz fold analysis for d=2 and standard A3 cusp conditions for d=3, but explicitly stops short of proving the existence of a cusp for the specific 3D model at μ3=0.1. Hence, the rigorous status of the cusp for that fixed model remains (as of the 2021 arXiv posting and our cutoff) open, while both paper and model align on the qualitative picture.