Deformation of invariant tori under perturbation

The model reproduces the paper’s two “complete forms” for the first-order functional variation of P_k (eqs. (21) and (27) in the paper) and uses the same chain-rule logic; it also arrives at the normal equation (DP_m−I) δ⊥x_cyc = −δP_m (eq. (7)) and the tangency of δP_m on the torus when the returning map remains well-defined (eq. (9)). Minor issue: the model states that for k = m one may set δ(k*Δθ)=0 because m*Δθ≡2π·1; in the paper this vanishing is only asserted when Δθ is P-independent. For k=m, if m is allowed to depend on P (as the paper later does via δm), then δ(m*Δθ)=0 holds only after including δm terms (cf. eqs. (8)–(9)); nevertheless, this extra assumption is unnecessary for the model’s final tangency conclusion because the δ(k*Δθ)-term in eq. (27) is tangent and vanishes under normal projection. Overall, the derivations align closely with the paper’s arguments and results.