Lyapunov functions and bounded solutions of linear systems of differential equations
type: Item
instance of: scholarly article
title: Lyapunov functions and bounded solutions of linear systems of differential equations
zbMATH Open document ID: 0612.34048
DOI: 10.1007/BF01056754
author: V. L. Kulik
author: Yu. A. Mitropol'skij
published in: Ukrainian Mathematical Journal
publication date: 1986-01-01T00:00:00Z
review text: In problems involving the existence of invariant manifolds of dynamical systems one has to consider families of equations \(dh/dt=(H(t)\cdot h)\in {\mathbb{R}}^ n.\) The present article continues the research concerning such systems in Ukr. Math. J. 36, 720-729 (1984; Zbl 0555.34026). The system is imbedded in the 2n-dimensional system \(\dot h=H\cdot h\), \(\dot y=-h- H^ Ty\). The main theorem concerns the implications following from the existence of a Lyapunov function V (of a certain type) such that \(\dot V\geq -\| h\|^ 2\). In particular, given an arbitrary continuous vector function f the authors obtain a representation \(h={\mathcal M}f\), \({\mathcal M}={\mathcal M}^ 2\) and show that it represents all invariant manifolds of a 2n-dimensional system of linear differential equations with bounded coefficient matrices.
Mathematics Subject Classification ID: 34D20
Mathematics Subject Classification ID: 34C11
Mathematics Subject Classification ID: 34A30
zbMATH DE Number: 3989837
zbMATH Keywords: invariant manifolds of dynamical systems
zbMATH Keywords: Lyapunov function
reviewed by: Vadim Komkov
MaRDI profile type: Publication
cites work: Q3255589
cites work: Q3218409
cites work: Q5566243
Recommended article: Lyapunov functions and boundedness for differential and delay equations
Recommended article: Q3220931
Recommended article: Q4177901
Recommended article: Asymptotically linear solutions of differential equations via Lyapunov functions
Recommended article: Q3988717
Recommended article: Q3842129
Recommended article: Lyapunov's functions and boundedness of the solutions of impulsive differential equations
Recommended article: Q3322466
Recommended article: Q4027390
Recommended article: Q3983961
description: scientific article; zbMATH DE number 3989837