Periodic Klein-Gordon Chains with Three Particles in 1:2:2 Resonance

In this paper, we deal with the three degrees of freedom Hamiltonian systems describing the Klein-Gordon chains with three particles of equal masses and periodic boundary conditions. Specially, we focus on the case that the frequencies of the linearization are in 1 : 2 : 2 resonance. After second normalization the truncated normal form gives rise to an integrable system. Also, we calculate the coefficients of the terms that remain in normal form. Considering perturbation in frequencies, we analyze the dynamical features of this one degree of freedom system on the reduced phase space by calculating its equilibria and bifurcations. Specifically supercritical and subcritical Hamiltonian pitchfork bifurcations are found in different scenarios of parameters.