Periodic Orbits and KAM Tori in the Hénon–Heiles Hamiltonian System: The 1: ω Resonance Case

We prove the existence and determine the linear stability of periodic orbits in the Hénon–Heiles Hamiltonian system under a 1:ω resonance, with ω≥2. The analysis combines Lie–Deprit normalization with regular and singular reduction. We also establish the persistence of two-dimensional KAM tori surrounding the linearly stable periodic orbits and derive periodic asymptotic parametrizations in the original variables.