Revised Hamiltonian near third-integer resonance and implications for transverse resonance island buckets in storage rings

In storage rings, an accurate description of particle dynamics near third-integer resonance is crucial for various applications. The conventional approach is to extrapolate far-resonance dynamics to near-resonance, but difficulty arises because the nonlinear detuning parameter diverges at this critical point. Here we derive, via a suitable application of the canonical perturbation theory, a revised detuning parameter that is well-behaved near resonance. The resultant theory accurately describes the morphology of transverse resonance island buckets (TRIBs) for a wide range of parameter space. Our results have important implications for advanced applications of storage rings, as well as for the underlying physics of resonant particle dynamics.