Arnold Tongues in Area-Preserving Maps

The paper proves that for the non-exact area-preserving cylinder map, the set of δ admitting p/q-periodic orbits shrinks at least as fast as O(ε^[q/d]) and, via its periodicity and degree arguments, effectively shows the first x-dependent coefficient appears only at order r > [q/d] (hence width O(ε^{ceil(q/d)})), using an implicit-function/iterate–remainder method. The candidate solution uses a Lyapunov–Schmidt reduction of the discrete Euler–Lagrange equation, obtains a scalar bifurcation equation δ = D(ε,x0), and derives the same scaling by a Fourier aliasing rule. Minor differences: the model states the onset order as ceil(q/d) (≥) instead of the paper’s strictly later r > [q/d]; and the model centers δ by subtracting the constant mode, while the paper states the bound around δ = 0. These are presentational/precision differences; the core conclusions agree.