Invariant manifolds with asymptotic phase for nonautonomous difference equations
type: Item
instance of: scholarly article
title: Invariant manifolds with asymptotic phase for nonautonomous difference equations
zbMATH Open document ID: 1067.39031
DOI: 10.1016/S0898-1221(03)00093-2
publication date: 2004-08-06T00:00:00Z
review text: The concept of asymptotic phase originally occurred in connection with the approach of a solution of an autonomous ordinary differential equation to an arbitrary asymptotically stable periodic solution [cf. \textit{B. Aulbach}, Nonlinear Anal. Theory Methods Appl. 6, 817--827 (1982; Zbl 0509.58034)] for autonomous difference equations with an invariant manifold. Conditions are known which guarantee that a solution approaching this manifold eventually behave like a solution on this manifold [cf. \textit{B. Aulbach}, Continuous and discrete dynamics near manifolds of equilibria. (1984; Zbl 0535.34002); \textit{J. López-Fenner} and \textit{M. Pinto}, Comput. Math. Appl. 33, No.~10, 105-124 (1997; Zbl 0880.39012)]. In this paper the authors generalize the main result of \textit{B. Aulbach} [1984; loc. cit.] for autonomous difference equations to the case of a nonautonomous difference equation \( x(k+1) = f(k,x(k)) \) whose right-hand side is allowed to be invertible and whose invariant manifold doesn't necessarily consist of stationary solutions. The paper consists of 5 sections and an appendix. The main result is given in section 5, that is if \( x(k+1) = f(k,x(k)) \) is a difference equation with invariant compact manifold \(M\), then under 5 conditions \(M\) possesses an asymptotic phase.
Mathematics Subject Classification ID: 39A12
Mathematics Subject Classification ID: 37D10
zbMATH DE Number: 2082241
zbMATH Keywords: invariant manifold
zbMATH Keywords: asymptotic phase
zbMATH Keywords: nonautonomous difference equation
zbMATH Keywords: exponential trichotomy
zbMATH Keywords: reducibility
author: Bernd Aulbach
author: Christian Pötzsche
reviewed by: Ahmed Hegazi
MaRDI profile type: Publication
cites work: Behavior of solutions near manifolds of periodic solutions
cites work: Invariant manifolds with asymptotic phase
cites work: Continuous and discrete dynamics near manifolds of equilibria
cites work: On \((h,k)\) manifolds with asymptotic phase
cites work: \((h,k)\)-trichotomies and asymptotics of nonautonomous difference systems
cites work: Classification of linear time-varying difference equations under kinematic similarity
cites work: A smoothness theorem for invariant fiber bundles
cites work: Manifolds, tensor analysis, and applications.
full work available at URL: s0898-1221(03)00093-2
OpenAlex ID: W2048511227
published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 2082241