Hamiltonian control to desynchronize Kuramoto oscillators with higher-order interactions

The paper embeds the higher-order Kuramoto model (HOKM) into a Hamiltonian system H = H0 + V and uses Hamiltonian control theory to construct a feedback f(V) via the pseudo-inverse Γ so that the controlled flow is canonically conjugate to the integrable H0; on the invariant torus T1/2 this eliminates synchronization, matching the claimed mechanism e^{-{{ΓV}}}(H0 + V + f(V)) = H0 with f(V) given by the standard series (non-resonant case G=0 implies the simplified A14) . The candidate solution reproduces this construction and its implication for quasi-periodic incommensurable angles. For the pairwise-only control on complete graphs, both derive the same dominant term h̃(N)i ≈ −(1/2) K1^2 R R̃i cos(Ψ − Ψ̃i) and the same qualitative interpretation (stronger near synchrony and for small detuning) . On partial control and when full higher-order terms are needed, the paper provides empirical regimes (h̃ fails when K1 is small relative to K2; full control succeeds broadly) while the candidate adds heuristic inequalities consistent with the paper’s narrative and with the denominators arising in Γ for pairwise and triadic modes . Minor differences are the candidate’s explicit sufficient-condition bounds, which the paper does not prove but are directionally aligned with its results.