Simply improved averaging for coupled oscillators and weakly nonlinear waves
Simply improved averaging for coupled oscillators and weakly nonlinear waves
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DOI：
10.1016/j.cnsns.2018.11.003
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发表时间：
2018-06
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影响因子：
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通讯作者：
Molei Tao
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作者：
Molei Tao
非线性扰动对线性振荡系统的长时间影响可以用平均法来描述，我们考虑一阶平均法，因为它最简单地适用于高维问题。而不是经典的方法，其中一个使用拉回的线性流隔离慢变量，然后近似的有效动态平均，我们提出了一种替代的坐标变换，更好地近似振荡的平均值。这导致了一个简单的改进的平均系统，这将被证明在理论上和数值上提供一个更准确的近似。然后给出了三个例子：第一个例子是一个由两个耦合振子建模的无线能量传输装置，其结果为该装置的设计和性能量化提供了指导;第二个例子是一个经典的耦合振子问题（费米-帕斯塔-乌拉姆），我们在数值上观察到的精度提高超出了理论上合理的时间尺度;第三个是一个非线性扰动的一阶波动方程，它证明了改进的平均在无限维设置的效果。
The long time effect of nonlinear perturbation to oscillatory linear systems can be characterized by the averaging method, and we consider first-order averaging for its simplest applicability to high-dimensional problems. Instead of the classical approach, in which one uses the pullback of linear flow to isolate slow variables and then approximate the effective dynamics by averaging, we propose an alternative coordinate transform that better approximates the mean of oscillations. This leads to a simple improvement of the averaged system, which will be shown both theoretically and numerically to provide a more accurate approximation. Three examples are then provided: in the first, a new device for wireless energy transfer modeled by two coupled oscillators was analyzed, and the results provide design guidance and performance quantification for the device; the second is a classical coupled oscillator problem (Fermi-Pasta-Ulam), for which we numerically observed improved accuracy beyond the theoretically justified timescale; the third is a nonlinearly perturbed first-order wave equation, which demonstrates the efficacy of improved averaging in an infinite dimensional setting.