Continuity of the Sacker-Sell spectrum on the half line

The Sacker-Sell (also called dichotomy or dynamical) spectrum is an important notion in the stability theory of nonautonomous dynamical systems. For instance, when dealing with variational equations on the (nonnegative) half line, the set Σ + determines uniform asymptotic stability or instability of...

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Bibliographic Details
Published in:Dynamical systems (London, England) England), 2018-01, Vol.33 (1), p.27-53
Main Author: Pötzsche, Christian
Format: Article
Language:English
Subjects:
Dichotomies
Dichotomy spectrum
difference equation
Dynamic stability
exponential dichotomy
Mathematical analysis
nonautonomous hyperbolicity
Operators (mathematics)
Primary 34D09, Secondary 37C60, 37C75, 39A30, 47B37, 93D09
robust stability
Sacker-Sell spectrum
shift operator
Stability
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Summary:The Sacker-Sell (also called dichotomy or dynamical) spectrum is an important notion in the stability theory of nonautonomous dynamical systems. For instance, when dealing with variational equations on the (nonnegative) half line, the set Σ + determines uniform asymptotic stability or instability of a solution and more general, it is crucial to construct invariant manifolds from the stable hierarchy. Compared to the spectrum associated to dichotomies on the entire line, Σ + has stronger and more flexible perturbation features. In this paper, we study continuity properties of the Sacker-Sell spectrum by means of an operator-theoretical approach. We provide an explicit example that the generally upper-semicontinuous set Σ + can suddenly collapse under perturbation, establish continuity on the class of equations with discrete spectrum and identify system classes having a continuous spectrum. These results for instance allow to vindicate numerical approximation techniques.
ISSN:1468-9367
1468-9375
DOI:10.1080/14689367.2017.1293613