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Parametric resonance in a conservative system of coupled nonlinear oscillators

We study a conservative system of two nonlinear coupled oscillators. The eigenmodes of the system are thus nonlinearly coupled, and one of them may induce a parametric amplification of the other, called an autoparametric resonance of the system. The parametric amplification implies two time scales, a fast one for the forcing and a slow one for the forced mode, thus a multiscale expansion is suitable to get amplitude equations describing the slow dynamics of the oscillators. We recall the parametric resonance in a dissipationless system, the parametrically forced Duffing oscillator, with emphasis on the energy transfer between the oscillator and the source that ensures the parametric forcing. Energy conservation is observed when averaging is done on the slow time scale relevant to parametric amplification,evidenced by a constant of the motion in the amplitude equation. Then we study a dimer in a periodic potential well, which is a conservative but non integrable system. When the dimer energy is such that it is trapped in neighboring potential wells, we derive coupled nonlinear differential equations for the eigenmodes amplitudes (center of mass motion and relative motion). We exhibit two constants of the motion, which demonstrates that the amplitude equations are integrable. We establish the conditions for autoparametric amplification of the relative motion by the center of mass motion, and describe the phase portraits of the system. In the opposite limit, when the dimer slides along the external potential so that the center of mass motion is basically a translation, we calculate the amplitude equation for the relative motion. In this latter case, we also exhibit autoparametric amplification of the relative motions of the dimer particles. In both cases, the comparison between numerical integration of the actual system and the asymptotic analysis evidences an excellent agreement.