有限长简谐耦合摆链的稳态大振幅动力学
Stationary Large Amplitude Dynamics of the Finite Chain of Harmonically Coupled Pendulums
摘要：
我们提出了一个解析描述的大振幅平稳振荡的有限离散系统的谐波耦合振荡没有任何限制，其振幅（不包括附近的$π$）。虽然这个模型在物理学的不同领域有许多应用，但它以前只在无限极限中被研究过。所开发的方法允许找到任意振幅的非线性简正模的色散关系。我们强调，由众所周知的sine-Gordon方程描述的长波长近似导致$π/2$的振幅阶的不适当的带结构，即使链足够长。在大振幅下的极其复杂的区域结构对应于非线性简正模之间的大量共振，即使具有强烈不同的波数。由于色散关系的复杂性，波长越短的模式可以具有越小的频率。有限长链的动力学的数值模拟是在一个很好的协议与获得的分析预测。
英文摘要：
We present an analytical description of the large-amplitude stationary oscillations of the finite discrete system of harmonically-coupled pendulums without any restrictions to their amplitudes (excluding a vicinity of $π$). Although this model has numerous applications in different fields of physics it was studied earlier in the infinite limit only. The developed approach allows to find the dispersion relations for arbitrary amplitudes of the nonlinear normal modes. We underline that the long-wavelength approximation, which is described by well- known sine-Gordon equation leads to inadequate zone structure for the amplitude order of $π/2$ even if the chain is long enough. The extremely complex zone structure at the large amplitudes corresponds to lot of resonances between nonlinear normal modes even with strongly different wave numbers. Due to complexity of the dispersion relations the more short wavelength modes can possess the smaller frequencies. The numerical simulation of the dynamics of the finite-length chain is in a good agreement with obtained analytical predictions.
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