关于单频系统的平均原理
On the average principle for one-frequency systems
摘要：
我们考虑一个具有一个频率的扰动可积系统，以及通过在角上平均给出的作用量的近似动力学。经典理论认为，对于一个阶数为1的扰动，这种近似的误差在时间尺度为O（1/1）时为O（1/1），其中1/2-> 0。我们用一个完全定量的估计来代替这个定性的陈述;在某些情况下，我们的方法在大于1/2的时间尺度上也给出了可靠的误差估计。一些例子，在许多情况下，我们的估计几乎一致的包络线之间的扰动和平均系统的行动迅速振荡的距离。在某些“共振”情况下，也得到了相当好的结果，其中角频率沿系统的轨迹是小的。尽管我们的估计在理论上得到了证明，但在特定应用中的计算通常需要微分方程系统的数值解。然而，这个系统的时间尺度比扰动系统的时间尺度小一个因子。由于这个原因，我们的估计的计算速度比直接数值解的扰动系统，估计也迅速发现的情况下，当时间尺度不可能（在合理的CPU时间）或不可靠的直接解决方案的扰动系统。
英文摘要：
We consider a perturbed integrable system with one frequency, and the approximate dynamics for the actions given by averaging over the angle. The classical theory grants that, for a perturbation of order epsilon, the error of this approximation is O(epsilon) on a time scale O(1/epsilon), for epsilon -> 0. We replace this qualitative statement with a fully quantitative estimate; in certain cases, our approach also gives a reliable error estimate on time scales larger than 1/epsilon. A number of examples are presented; in many cases our estimator practically coincides with the envelope of the rapidly oscillating distance between the actions of the perturbed and of the averaged systems. Fairly good results are also obtained in some "resonant" cases, where the angular frequency is small along the trajectory of the system. Even though our estimates are proved theoretically, their computation in specific applications typically requires the numerical solution of a system of differential equations. However, the time scale for this system is smaller by a factor epsilon than the time scale for the perturbed system. For this reason, computation of our estimator is faster than the direct numerical solution of the perturbed system; the estimator is rapidly found also in cases when the time scale makes impossible (within reasonable CPU times) or unreliable the direct solution of the perturbed system.
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