In many engineering problems, when studying the existence of periodic solutions to a nonlinear system with a small parameter via the local averaging theorem, it is necessary to verify some properties of the fundamental solution matrix to the corresponding linearized system along the periodic solution of the unperturbed system. But sometimes, it is difficult or it requires a lot of calculations. In this paper, a few simple and effective methods are introduced to investigate the existence of periodic solutions for a kind of small parametric systems. In order to prove the existence of periodic solutions by these ideas, we also introduce a forced autoparametric vibrating system and a generalized model of the tuned mass absorber with pendulum discussed by Brzeski, Perlikowski, and Kapitaniak. Then, we also propose an averaging method to study the existence of periodic solutions.

中文翻译：

---

用平均法求解非线性微分系统的周期解

在许多工程问题中，当通过局部平均定理研究小参数非线性系统周期解的存在性时，需要沿着未扰动的周期解验证相应线性化系统的基本解矩阵的一些性质。系统。但有时，这很困难或需要大量计算。本文介绍了一些简单有效的方法来研究一类小参数系统周期解的存在性。为了通过这些想法证明周期解​​的存在，我们还引入了一个强制自参数振动系统和一个由 Brzeski、Perlikowski 和 Kapitaniak 讨论的带钟摆的调谐质量吸收器的广义模型。然后，