## 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 ∂_{1}v + Av = B(t) v_{1} 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 λ_{0}∈Λ, then there is an exponential dichotomy, or exponential trichotomy, for all λ near λ_{0}. We present several illustrations indicating the significance of this robustness property.

Original language | English (US) |
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Pages (from-to) | 471-513 |

Number of pages | 43 |

Journal | Journal of Dynamics and Differential Equations |

Volume | 11 |

Issue number | 3 |

DOIs | |

State | Published - 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