扰动自治系统周期解在扰动为零时的收敛速率
On the rate of convergence of periodic solutions in perturbed autonomous systems as the perturbation vanishes
摘要：
考虑R^n中的自治系统，其极限环x_0的周期T>0，在适当的意义下是非退化的.然后，我们考虑通过向自治系统添加T-周期（不一定是可微的）项而获得的扰动系统，当小参数e>0趋于0时，其振幅趋于0。本文在假设系统存在T-周期解xe，且当e->0时xe收敛于x 0的条件下，证明了当e->0时de->0的存在性，使得对某个M>0和任何e>0充分小的情况，xe（t+ de）-x 0（t）<=eM.
英文摘要：
We consider an autonomous system in R^n having a limit cycle x_0 of period T>0 which is nondegenerate in a suitable sense. We then consider the perturbed system obtained by adding to the autonomous system a T-periodic, not necessarily differentiable, term whose amplitude tends to 0 as a small parameter e>0 tends to 0. Assuming the existence of a T-periodic solution x_e of the perturbed system and its convergence to x_0 as e->0, the paper establishes the existence of d_e->0 as e->0 such that \|x_e(t+d_e)-x_0(t)\|<=eM for some M>0 and any e>0 sufficiently small.
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