Systems of differential delay equations: Floquet multipliers and discrete Lyapunov functions
type: Item
instance of: scholarly article
title: Systems of differential delay equations: Floquet multipliers and discrete Lyapunov functions
zbMATH Open document ID: 0849.34055
author: John Mallet-Paret
author: George R. Sell
published in: Journal of Differential Equations
publication date: 1996-11-14T00:00:00Z
review text: This paper deals with systems of differential delay equations with a cyclic feedback structure of the form \(\dot x^0 (t) = f^0 (t,x^0 (t), x^1 (t))\), \(\dot x^i (t) = f^i (t,x^{i - 1} (t), x^i (t), x^{i + 1} (t))\) for \(1 \leq i \leq N - 1\) and \(\dot x^N (t) = f^N (t,x^{N - 1} (t)\), \(x^N(t)\), \(x^0(t - 1))\) which is a nearest neighbor system: the derivative of each \(x^i\) depends on \(x^i\) and on its neighbors \(x^{i \pm 1}\). The functions \(f^0\), \(f^1, \dots, f^N\) satisfy some monotonicity conditions. The purpose of the paper is twofold. First the authors define and investigate a Lyapunov-like integer-valued function \(V^\delta\), where \(\delta =\pm 1\), as follows: Let \(\mathbb{K}\) be the closed interval \([-1,0]\) joined with the positive integers and let \(\varphi : \mathbb{K} \to \mathbb{R}\) be a continuous function not identically zero. Let \(sc (\varphi)\) denote the number of sign changes of \(\varphi (\cdot)\) if this is finite, otherwise it is \(+ \infty\). Then define \(V^+ (\varphi) = sc (\varphi)\), if \(sc (\varphi)\) is even or \(+ \infty\), \(V^+ (\varphi) = sc (\varphi) + 1\), if \(sc (\varphi)\) is odd and \(V^- (\varphi) = sc (\varphi)\), if \(sc (\varphi)\) is odd or \(+ \infty\), \(V^- (\varphi) = sc (\varphi) + 1\), if \(sc (\varphi)\) is even. If \((x^0, x^1, \dots,x^N)\) is a solution and \(t\) is fixed, define \(x_t(\vartheta)=x^0(t+\vartheta)\), if \(-1\leq\vartheta\leq 0\) and \(x_t (\vartheta) = x^\vartheta (t)\), if \(\vartheta = 1, 2, \dots, N\). After these denotations some sufficient conditions are provided such that \(V^\pm (x_t)\) is nonincreasing in \(t\) (Theorem 2.10) or finite (Theorem 2.4). The second purpose of the paper is to apply the above general results to nonautonomous linear nearest neighbor systems in order to provide information about the relation between the Floquet multipliers and the values of \(V^\pm (x_t)\) when the \(x_t\) belong to the Floquet subspaces. Some nice results are given for the au…
Mathematics Subject Classification ID: 34K99
Mathematics Subject Classification ID: 34K20
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 877802
zbMATH Keywords: systems of differential delay equations with a cyclic feedback structure
zbMATH Keywords: nonautonomous linear nearest neighbor systems
zbMATH Keywords: Floquet multipliers
reviewed by: George Karakostas
MaRDI profile type: Publication
OpenAlex ID: W2065450947
DOI: 10.1006/JDEQ.1996.0036
description: scientific article; zbMATH DE number 877802