Robustness of exponential dichotomies in infinite-dimensional dynamical systems

Victor A. Pliss, George R. Sell

Research output: Contribution to journalArticlepeer-review

130 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 - Jan 1 1999

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

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