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