The coupled 1:2 resonance in a symmetric case and parametric amplification model

Abstract

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.

Lagrangian fibrations in coupled resonant oscillators in the paired case of four degrees of freedom containing swinging spring oscillator

“…We can find a fourfold perturbed Hamiltonian resonance in [25]. Recently, we classified Hamiltonian resonances with four degrees of freedom and arranged them based on the order of their normal forms [26]. However, we cannot find a systematic study on the Hamiltonian resonances with four degrees of freedom generally.…”