Analytic Phase-Space and Spectral Modeling of Amplitude-Dependent Resonance Island Dynamics Near the Third-Integer Resonance

Abstract We present an analytic phase-space and spectral framework for single-particle dynamics near the third-integer resonance in circular accelerators, emphasizing amplitude-dependent nonlinear effects. Starting from a resonant Hamiltonian in action-angle variables, we obtain elliptic-function solutions that give closed-form relations between island action, separatrix geometry, and amplitude-dependent tune for both the primary orbit and the resonance islands. Mapping this parameterization into the frequency domain yields analytic predictions for the main spectral line and higher-order sidebands, providing a quantitative interpretation of structures such as the devil’s staircase. Multi-turn tracking with ELEGANT shows good agreement in both phase space and spectra, establishing a rigorous, interpretable basis for analyzing nonlinear resonance effects in Transverse Resonance Island Buckets (TRIBs), RF knock-out (RF-KO), and other resonance-island-based beam operations.