A GENERAL AVERAGING METHOD FOR AFFINE PERIODIC SOLUTIONS

doi:10.1007/s10473-024-0609-x

数学物理学报(英文版) ›› 2024, Vol. 44 ›› Issue (6): 2207-2224.doi: 10.1007/s10473-024-0609-x

A GENERAL AVERAGING METHOD FOR AFFINE PERIODIC SOLUTIONS

Xue YANG¹, Jiamin XING^2,†, Yong LI^1,2

1. College of Mathematics, Jilin University, Changchun 130012, China;
2. School of Mathematics and Statistics, and Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun 130024, China

收稿日期:2023-08-17 修回日期:2024-07-11 发布日期:2024-12-06
通讯作者: † Jiamin XING, E-mail: xingjiamin1028@126.com
作者简介:Xue YANG, E-mail: xueyang@jlu.edu.cn; Yong LI, E-mail: liyong@jlu.edu.cn
基金资助:
ang's research was supported by the National Natural Science Foundation of China (12371191; 12071175). Xing's research was supported by the NSFC (12071175; 11901080) and the Fundamental Research Funds For the Central Universities (2412023YQ003). Li's research was supported by the NSFC (12071175) and the Natural Science Foundation of Jilin Province (20200201253JC).

A GENERAL AVERAGING METHOD FOR AFFINE PERIODIC SOLUTIONS

Xue YANG¹, Jiamin XING^2,†, Yong LI^1,2

1. College of Mathematics, Jilin University, Changchun 130012, China;
2. School of Mathematics and Statistics, and Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun 130024, China

Received:2023-08-17 Revised:2024-07-11 Published:2024-12-06
Contact: † Jiamin XING, E-mail: xingjiamin1028@126.com
About author:Xue YANG, E-mail: xueyang@jlu.edu.cn; Yong LI, E-mail: liyong@jlu.edu.cn
Supported by:
ang's research was supported by the National Natural Science Foundation of China (12371191; 12071175). Xing's research was supported by the NSFC (12071175; 11901080) and the Fundamental Research Funds For the Central Universities (2412023YQ003). Li's research was supported by the NSFC (12071175) and the Natural Science Foundation of Jilin Province (20200201253JC).

摘要/Abstract

摘要： We consider the persistence of affine periodic solutions for perturbed affine periodic systems. Such $(Q,T)$-affine periodic solutions have the form $x(t+T)=Qx(t)$ for all $t\in\mathbf{R}$, where $T>0$ is fixed and $Q$ is a nonsingular matrix. These are a kind of spatiotemporal symmetric solutions, e.g. spiral waves. We give the averaging method for the existence of affine periodic solutions in two situations: one in which the initial values of the affine periodic solutions of the unperturbed system form a manifold, and another that does not rely on the structure of the initial values of the unperturbed system's affine periodic solutions. The transversal condition is determined using the Brouwer degree. We also present a higher order averaging method for general degenerate systems by means of the Brouwer degree and a Lyapunov-Schmidt reduction.

关键词: affine-periodic solution, perturbed system, Brouwer degree, averaging method

Abstract: We consider the persistence of affine periodic solutions for perturbed affine periodic systems. Such $(Q,T)$-affine periodic solutions have the form $x(t+T)=Qx(t)$ for all $t\in\mathbf{R}$, where $T>0$ is fixed and $Q$ is a nonsingular matrix. These are a kind of spatiotemporal symmetric solutions, e.g. spiral waves. We give the averaging method for the existence of affine periodic solutions in two situations: one in which the initial values of the affine periodic solutions of the unperturbed system form a manifold, and another that does not rely on the structure of the initial values of the unperturbed system's affine periodic solutions. The transversal condition is determined using the Brouwer degree. We also present a higher order averaging method for general degenerate systems by means of the Brouwer degree and a Lyapunov-Schmidt reduction.

Key words: affine-periodic solution, perturbed system, Brouwer degree, averaging method

中图分类号:

34C29

Xue YANG, Jiamin XING, Yong LI. A GENERAL AVERAGING METHOD FOR AFFINE PERIODIC SOLUTIONS[J]. 数学物理学报(英文版), 2024, 44(6): 2207-2224.

Xue YANG, Jiamin XING, Yong LI. A GENERAL AVERAGING METHOD FOR AFFINE PERIODIC SOLUTIONS[J]. Acta mathematica scientia,Series B, 2024, 44(6): 2207-2224.

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A GENERAL AVERAGING METHOD FOR AFFINE PERIODIC SOLUTIONS

A GENERAL AVERAGING METHOD FOR AFFINE PERIODIC SOLUTIONS

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