ANALYSIS OF ADIABATIC TRAPPING PHENOMENA FOR QUASI-INTEGRABLE AREA-PRESERVING MAPS IN THE PRESENCE OF TIME-DEPENDENT EXCITERS

The paper establishes, under an adiabatic two-stage protocol (first ramp the exciter strength at fixed frequency, then sweep frequency), the separatrix-crossing area rule and its consequence for trapping fractions, and validates scaling laws for the areas of Regions I and II near the 1:3 resonance. Specifically: (i) the capture probabilities use the standard Θ-derivative formula and Liouville measure arguments (Eq. (1)-(2)), and for a uniform initial ensemble supported on Regions I ∪ II the trapped fraction equals τ_{I,II→II} = AII/(AI + AII) (their Eq. (28)) ; (ii) the island area scales as AII ∝ ε_m^{1/2} (Fig. 7) ; (iii) the central region area can be fit by AI(ε_m) = a + f(ε_m,b,ε0) ε_m^{-2/3} with f(ε_m,b,ε0) = b/[1 + (ε0/ε_m)^{2/3}] (their Eq. (24)) ; and (iv) the minimal action required to remain trapped during a slow sweep obeys J_min ∝ (ε/ε)^{2/m}, hence J_min ∝ ε^{-2/3} for m=3 (Appendix B, Eqs. (58)-(60)) . The candidate solution reproduces these same pillars using the resonant normal form and the pendulum picture, arriving at the same τ formula and the same exponents 1/2 (AII) and −2/3 (AI and J_min), and it correctly notes the Hamiltonian–map parameter scaling ε_h/ε_m = (ω_{2,0}/Ω_{2,0})^{3/2} (their Eq. (14)) . One overreach in the model is the assertion of universal ε log ε corrections for the static separatrix lobe area AII(ε_m); the paper does not claim such a term and the textbook pendulum area is exactly 16√μ without logarithmic corrections. This does not affect the leading-order scalings or the τ result, so the core conclusions agree.