Nondegenerate Hamiltonian Hopf Bifurcations in $\omega: 3: 6$ Resonance $(\omega=1 \, \text{or}\, 2)$

This paper deals with the analysis of Hamiltonian Hopf bifurcations in three-degree-of-freedom systems, for which the frequencies of the linearization of the corresponding Hamiltonians are in $\omega: 3: 6$ resonance $(\omega=1\, \text{or}\, 2)$. We obtain the truncated second-order normal form that is not integrable and expressed in terms of the invariants of the reduced phase space. The truncated first-order normal form gives rise to an integrable system that is analyzed using a reduction to a one-degree-of-freedom system. In this paper, some detuning parameters are considered and nondegenerate Hamiltonian Hopf bifurcations are found. To study Hamiltonian Hopf bifurcations, we transform the reduced Hamiltonian into standard form.