This paper deals with the coupled Hamiltonian 1:2 resonance, i.e. the Hamiltonian 1:2:1:2 resonance. This resonance is of the first order. We isolate several integrable cases. Our main focus is on two models. In the first part of the paper, we present a discrete symmetric normal form truncated to order three and we compute the relative equilibria for its corresponding system. In the second part, the paper is devoted to the study of the Hamiltonian describing the four-wave mixing (FWM) model. In addition to the Hamiltonian, the corresponding system possesses three more independent integrals. We use these integrals to obtain estimates for the phase space and total energy. Further, we compute the relative equilibria of the FWM system for the 1:2:1:2 resonance. Finally, we carry out some numerical experiments for the detuned system.
This paper deals with Lagrangian fibrations in coupled resonant oscillators containing swinging spring oscillators of four degrees of freedom. We will consider the cases with zero angular momenta with 1 : 1 and 1 : 2 (Fermi) resonances. Moreover, we also consider a coupled spatial swinging spring oscillator with a harmonic oscillator. Actually, they make different four degrees of freedom Hamiltonian resonances, i.e. 1 : 2 : 1 : 1 and 1 : 2 : 1 : 2 resonances. We will obtain the third-order normalization of these Hamiltonian resonances. We consider the linear Casimir energy and we approximate models by normal forms. Then our models are Liouville integrable. For one case with zero angular momentum and coupled swinging spring, we will identify Hopf fibration. The relative equilibria and normal modes are identified in the space of orbits. Our analysis is based on affecting energy levels in oscillators. These energy levels have been marked with distinguished parameters which show the constants of motions or integrals.
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