论文标题
耦合非线性振荡器的保守系统中的参数共振
Parametric resonance in a conservative system of coupled nonlinear oscillators
论文作者
论文摘要
我们研究了两个非线性耦合振荡器的保守系统。因此，系统的本征模是非线性耦合的，其中一个可以诱导另一个的参数放大，称为系统的自动摄取共振。参数放大意味着两个时间尺度，这是强迫模式的快速尺度，因此慢速扩增，因此多尺度扩展适合获取振幅方程，描述了振荡器的慢速动力学。我们回想起无耗散系统（参数强迫悬挂振荡器）中的参数共振，重点是振荡器与确保参数强迫的源之间的能量传递。当在与参数放大相关的缓慢时间尺度上进行平均时，可以观察到能量保存，这是由振幅方程中运动常数所证明的。然后，我们研究了周期性潜力井中的二聚体，这是一个保守但不可集成的系统。当二聚体能量被困在邻近的电孔中时，我们将得出本征元素振幅的非线性微分方程（质量运动和相对运动的中心）。我们展示了运动的两个常数，这些常数表明振幅方程是可以集成的。我们确定了质量运动中心对相对运动的自动载体扩增的条件，并描述系统的相肖像。在相反的极限中，当二聚体沿外部电势滑动以使质量运动的中心基本上是一种翻译时，我们计算相对运动的振幅方程。在后一种情况下，我们还表现出二聚体颗粒相对运动的自载扩增。在这两种情况下，实际系统的数值整合与渐近分析之间的比较证明了一个很好的一致性。
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.
