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Periodic solutions of nonlinear differential systems by the method of averaging. (English) Zbl 07250673
Summary: 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 P. Brzeski et al. [Commun. Nonlinear Sci. Numer. Simul. 19, No. 1, 298–310 (2014; Zbl 1344.70009)]. Then, we also propose an averaging method to study the existence of periodic solutions.
MSC:
34C29 Averaging method for ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
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