Tor: a toolbox for the continuation of two-dimensional tori in autonomous systems and non-autonomous systems with periodic forcing

The paper’s derivations of the PDE-to-BVP reformulation with the all-to-all coupling v(ϕ+2πρ,0)=v(ϕ,T), the Poincaré phase conditions (one in the time-periodic case and two in the autonomous case), the dimension-deficit count leading to the need to release four parameters, and the Neimark–Sacker initializer all match the model’s solution step-for-step. Specifically, the characteristic reparametrization and boundary condition are derived in the paper’s Sec. 2.1 (eqs. (5)–(11) ), the phase conditions in Sec. 2.2 (eqs. (12)–(13) ), the deficit accounting and Ω2−ω2 coupling in Sec. 4.1 (), and the NS-based torus initializer (u(t), û(θ1,t), rotation by α per period, and x̂(θ1,t)) in Sec. 4.2 (eqs. (33)–(39) ). The model adds standard implicit-function/uniqueness remarks and the converse mapping u from v, consistent with the paper’s framework.