具有三次非线性的耦合谐振子中的三比一内共振
Three-to-one internal resonances in coupled harmonic oscillators with cubic nonlinearity
AI总结：
本文研究具有三次非线性的耦合谐振子中的3:1内共振，利用哈密顿微扰论和Birkhoff Normal Form构造新的作用-角变量，得到非线性频率的一阶修正并揭示多达六种拓扑行为，最后应用于超材料蜂窝结构的带隙分析。
摘要：
我们研究了一个具有三次非线性的两个耦合谐振子的一般系统。在没有阻尼的情况下，该系统是哈密顿系统，原点是一个椭圆平衡点，其特征是两个不同的线性频率。为了理解动力学，关键是推导出非线性频率作为所涉及物理参数的函数的显式解析公式。在小振幅区域（微扰情形）下，我们给出了线性频率的一阶非线性修正。虽然该解析表达式已经在非共振情形下被推导出来，但在共振或近共振情形下它是新的。具体来说，我们关注3:1共振，这是唯一参与一阶修正的共振。利用哈密顿结构，我们采用微扰论方法将系统变换为四阶Birkhoff Normal Form。这涉及将系统变换为作用-角变量（辛重标度的极坐标），其中四阶截断哈密顿量依赖于作用量，并且由于共振，还依赖于一个“慢”角。通过构造合适的非线性且不接近恒等变换的坐标变换，我们识别出新的辛作用-角变量集。在这些变量中，所得系统在高阶项意义下是可积的，这意味着它不依赖于角度，频率由能量对作用量的导数得到。这种构造高度依赖于物理参数，因此需要对相图进行详细的分类分析，揭示出多达六种拓扑上不同的行为。作为一个应用，我们研究了具有周期性分布非线性谐振器的超材料蜂窝结构中的波传播，评估了非线性对带隙的影响，特别是在存在共振的情况下。
英文摘要：
We investigate a general system of two coupled harmonic oscillators with cubic nonlinearity. Without damping, the system is Hamiltonian, with the origin as an elliptic equilibrium characterized by two distinct linear frequencies. To understand the dynamics, it is crucial to derive explicit analytic formulae for the nonlinear frequencies as functions of the physical parameters involved. In the small amplitude regime (perturbative case), we provide the first-order nonlinear correction to the linear frequencies. While this analytic expression was already derived for non-resonant cases, it is novel in the context of resonant or nearly resonant scenarios. Specifically, we focus on the 3:1 resonance, the only resonance involved in the first-order correction. Utilizing the Hamiltonian structure, we employ Perturbation Theory methods to transform the system into Birkhoff Normal Form up to order four. This involves converting the system into action-angle variables (symplectically rescaled polar coordinates), where the truncated Hamiltonian at order four depends on the actions and, due to the resonance, on one "slow" angle. By constructing suitable nonlinear and not close-to-the-identity coordinate transformations, we identify new sets of symplectic action-angle variables. In these variables, the resulting system is integrable up to higher-order terms, meaning it does not depend on the angles, and the frequencies are obtained from the derivatives of the energy with respect to the actions. This construction is highly dependent on the physical parameters, necessitating a detailed case analysis of the phase portrait, revealing up to six topologically distinct behaviors. As an application, we examine wave propagation in metamaterial honeycombs with periodically distributed nonlinear resonators, evaluating the nonlinear effects on the bandgap particularly in the presence of resonances.
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