A note on linear differential equations with periodic coefficients
type: Item
instance of: scholarly article
title: A note on linear differential equations with periodic coefficients
zbMATH Open document ID: 1180.34048
author: Maite Grau
author: Daniel Peralta-Salas
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2009-07-24T00:00:00Z
review text: The authors consider the periodic linear differential equation \(\dot{x}=A(t)x\) and derive sufficient conditions for the asymptotic stability of the zero solution. They cover a class of systems, where \(A(t)\) is negative semidefinite, i.e. \(x^TA(t)x\leq 0\) for all \(x\) and \(t\), or nearly negative definite, i.e. \(x^TA(t)x\leq \varepsilon_0(t,x)+\varepsilon_1(t)x^2\) with conditions on \(\varepsilon_0\) and \(\varepsilon_1\). The proof uses the construction of a matrix-valued function \(P(t)\) that satisfies the matrix equation \[ Q=\dot{P}+PA+A^TP \] such that \(x^TP(t)x\) is a strict Lyapunov function. Moreover, the authors prove that any configuration of cycles can be realized as hyperbolic and asymptotically stable limit cycles of a polynomial vector field. This solves an open problem stated in [\textit{D. Peralta-Salas}, J. Differ. Equations 217, No. 1, 249--256 (2005; Zbl 1086.34029)].
reviewed by: Peter Giesl
Mathematics Subject Classification ID: 34D08
Mathematics Subject Classification ID: 34A30
Mathematics Subject Classification ID: 37D05
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 34D20
zbMATH DE Number: 5586851
zbMATH Keywords: linear differential equations
zbMATH Keywords: hyperbolicity limit cycles
MaRDI profile type: Publication
full work available at URL: j.na.2009.01.199
OpenAlex ID: W2027653239
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DOI: 10.1016/J.NA.2009.01.199
description: scientific article; zbMATH DE number 5586851