Hopf and double Hopf bifurcations in a delayed lateral vibration model of footbridges induced by pedestrians

The paper reduces the delay equation near a nonresonant double Hopf point to a 4D center manifold, derives an amplitude–phase normal form with a small parameter, identifies a normally hyperbolic 2-torus for the truncated system, verifies the KAM twist condition det(∂ω1/∂σ)≠0 explicitly, and applies a parameter-dependent KAM theorem to obtain quasi-periodic invariant tori on a large-measure Cantor set; this is then lifted back to the original DDE. The candidate solution follows the same strategy and ingredients (center manifold, normal form, NHIM persistence, KAM with parameters, lifting). Aside from a minor imprecision about the signs of the normal eigenvalues (the torus is a saddle NHIM, not purely contracting), the approaches coincide.