Periodic solutions of nonlinear differential systems by the method of averaging

Zhanyong Li; Qihuai Liu; Kelei Zhang

Applications of Mathematics (2020)

  • Volume: 65, Issue: 4, page 511-542
  • ISSN: 0862-7940

Abstract

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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.

How to cite

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Li, Zhanyong, Liu, Qihuai, and Zhang, Kelei. "Periodic solutions of nonlinear differential systems by the method of averaging." Applications of Mathematics 65.4 (2020): 511-542. <http://eudml.org/doc/297048>.

@article{Li2020,
abstract = {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.},
author = {Li, Zhanyong, Liu, Qihuai, Zhang, Kelei},
journal = {Applications of Mathematics},
keywords = {periodic solution; local averaging theorem; forced autoparametric vibrating system; tuned mass absorber},
language = {eng},
number = {4},
pages = {511-542},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Periodic solutions of nonlinear differential systems by the method of averaging},
url = {http://eudml.org/doc/297048},
volume = {65},
year = {2020},
}

TY - JOUR
AU - Li, Zhanyong
AU - Liu, Qihuai
AU - Zhang, Kelei
TI - Periodic solutions of nonlinear differential systems by the method of averaging
JO - Applications of Mathematics
PY - 2020
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 65
IS - 4
SP - 511
EP - 542
AB - 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.
LA - eng
KW - periodic solution; local averaging theorem; forced autoparametric vibrating system; tuned mass absorber
UR - http://eudml.org/doc/297048
ER -

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