Existence of periodic solutions of differential systems with small parameters
type: Item
instance of: scholarly article
title: Existence of periodic solutions of differential systems with small parameters
zbMATH Open document ID: 1160.34040
author: Zhengxin Zhou
published in: Journal of Applied Mathematics and Computing
publication date: 2008-10-21T00:00:00Z
review text: The method of reflecting functions is used to study periodic solutions of periodic systems with small parameters. Given the differential equation \noindent \((1)\)\hfill \(x'=X(t,x),\quad t\in \mathbb{R},\,\,x\in D\subset \mathbb{R}^n\,,\) \hfill {} \noindent then the continuously differentiable vector function \(F(t,x)\) on \(\mathbb{R}\times \mathbb{R}^n\) is called a reflective function [\textit{V. I. Mironenko}, Differ. Equations 28, No.6, 789-794 (1992); translation from Differ. Uravn. 28, No.6, 984-991 (1992; Zbl 0834.34048)] if it is solution of the Cauchy problem \[ F_t(t,x)+F_x(t,x) X(t,x)+X(-t,F(t,x))=0,\quad F(0,x)=x\,. \] In this paper the reflective functions are applied to study the existence of periodic solutions of periodic systems with small parameters. The following result is a typical one (the paper establishes several results of this type): Theorem. Suppose that \(F(t,x)\) is the reflective function of the \(n\)-dimensional \(2\omega\)-periodic system \[ x'=X(t,x,0)\,, \] and there is a \(\bar x\) satisfying \(F(-\omega,\bar x)=\bar x\) and \(\text{ det\,}(F_x(-\omega,\bar x)-E)\neq 0\). Then, for \(| \varepsilon| \) sufficiently small, the periodic system \[ x'=X(t,x,\varepsilon)+F_x^{-1}(t,x) P(t,x)-P(-t,F(t,x)) \] has a \(2\omega\)-periodic solution.
reviewed by: Ivan Ginchev
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 5354629
zbMATH Keywords: reflecting function
zbMATH Keywords: periodic system
zbMATH Keywords: periodic solutions
MaRDI profile type: Publication
full work available at URL: s12190-008-0064-1
OpenAlex ID: W1973328504
cites work: Q3746142
cites work: Q4856815
cites work: Reflection function and the small parameter method
cites work: On the reflective function of polynomial differential system.
cites work: The nonlinear reflective function of differential system
DOI: 10.1007/S12190-008-0064-1
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description: scientific article; zbMATH DE number 5354629