Differential systems, the mapping over period for which is represented by a product of three exponential matrices
type: Item
instance of: scholarly article
title: Differential systems, the mapping over period for which is represented by a product of three exponential matrices
zbMATH Open document ID: 1136.34003
published in: Journal of Mathematical Analysis and Applications
publication date: 2007-02-26T00:00:00Z
review text: Let \(\varphi(t;t_0,x_0)\) denote the solution to the system \[ x^{\prime} =X(t,x),\quad t\in\mathbb R,\quad x \in\mathbb R^{n} x(t_0)=x_{0} \] where \(X\) is a continuously differentiable function. The function \(F(t,x)=\varphi(-t;t,x)\) is called the reflecting function for this system. For a linear system \[ x^{\prime}=P(t)x,\quad t\in\mathbb R,\quad x\in\mathbb R^{n} \] its reflecting function is linear and given by \(F(t,x)=F(t)x\), where \(F(t):=\Phi(-t)\Phi^{-1}(t)\), and \(\Phi (t)\) is the fundamental matrix of the linear system. The matrix \(F(t)\) is called the reflecting matrix of this linear system. The author studies the case when the reflecting matrix \(F(t)\) can be written as the product of three exponential matrices in the form \[ F(t)=e^{\alpha(t)A}e^{\beta(t)B}e^{-\alpha(-t)A}, \] where \(\alpha(t)\) and \(\beta(t)\) are arbitrary scalar functions, and, \(A\) and \(B\) are constant \(n\times n\) matrices. He deduces sufficient conditions for existence and stability of periodic solutions to a periodic ordinary differential system.
reviewed by: Behzad Djafari Rouhani
Mathematics Subject Classification ID: 34A30
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34D20
zbMATH DE Number: 5129673
zbMATH Keywords: Reflecting function
zbMATH Keywords: reflecting matrix
zbMATH Keywords: periodic solution
zbMATH Keywords: asymptotic orbital stability
author: Èduard V. Musafirov
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.06.085
OpenAlex ID: W2000377327
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DOI: 10.1016/J.JMAA.2006.06.085
description: scientific article; zbMATH DE number 5129673