Slow-fast dynamics of strongly coupled adaptive frequency oscillators

The paper’s Theorem 1 establishes, via Fenichel theory, the existence, stability type, and first-order shape of slow invariant manifolds for the singularly perturbed system after the change of variables, and derives the O(1) reduced slow flow; the candidate solution reproduces the same steps with a slightly more explicit invariance-equation expansion and a standard variation-of-constants bound. Both arguments agree on the critical manifolds Ω+ω=kπ away from F(θ)=0, the stability condition (−1)^{k+1}F(θ), the O(ε) manifold graph ω=(kπ−Ω)[1+ε(−1)^kλ/F(θ)]+O(ε^2), and the slow flow ω(t)=−Ω(0)e^{−λ t}+O(ε), Ω(t)=kπ+Ω(0)e^{−λ t}+O(ε), θ(t)=t on compact subsets; hence they are essentially the same proof.