Robustness of exponential dichotomies in infinite-dimensional dynamical systems

Victor A. Pliss, George R. Sell

Research output: Contribution to journalArticlepeer-review

132 Scopus citations

Abstract

In this paper we examine the issue of the robustness, or stability, of an exponential dichotomy, or an exponential trichotomy, in a dynamical system on an Banach space W. These two hyperbolic structures describe long-time dynamical properties of the associated time-varying linearized equation ∂1v + Av = B(t) v1 where the linear operator B(t) is the evaluation of a suitable Fréchet derivative along a given solution in the set K in W. Our main objective is to show, under reasonable conditions, that if B(t) = B(λ, t) depends continuously on a parameter λ∈Λ and there is an exponential dichotomy, or exponential trichotomy, at a value λ0∈Λ, then there is an exponential dichotomy, or exponential trichotomy, for all λ near λ0. We present several illustrations indicating the significance of this robustness property.

Original languageEnglish (US)
Pages (from-to)471-513
Number of pages43
JournalJournal of Dynamics and Differential Equations
Volume11
Issue number3
DOIs
StatePublished - 1999

Bibliographical note

Funding Information:
This research was supported in part by grants from the Russian Foundation for Fundamental Studies. Both authors express appreciation to the Faculty of Mathematics and Mechanics, in St. Petersburg, and to the Institute for Mathematics and Its Applications and the Minnesota Supercomputer Institute, in Minneapolis, for their help in sponsoring this project.

Copyright:
Copyright 2018 Elsevier B.V., All rights reserved.

Keywords

  • Exponential dichotomy
  • Exponential trichotomy
  • Linear evolutionary equations
  • Navier-Stokes equations
  • Nonlinear wave equation
  • Normal hyperbolicity
  • Ordinary differential equations
  • Partial differential equations
  • Robustness
  • Time-varying coefficients

Fingerprint Dive into the research topics of 'Robustness of exponential dichotomies in infinite-dimensional dynamical systems'. Together they form a unique fingerprint.

Cite this