Revised Hamiltonian near third-integer resonance and implications for an electron storage ring

In electron 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 the 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 resonance islands for a wide range of parameter space and facilitates its optimization.