Continuity of the Sacker–Sell spectrum on the half line

ABSTRACT

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.

Keywords:

• Dichotomy spectrum
• Sacker–Sell spectrum
• exponential dichotomy
• nonautonomous hyperbolicity
• shift operator
• difference equation
• robust stability

MCS/CCS/AMS Classification/CR Category numbers: MSC Classification::

• Primary 34D09, Secondary 37C60, 37C75, 39A30, 47B37, 93D09

Acknowledgments

The author is grateful to a referee for providing the geometric argument proving and preceding Theorem 4.4. The constructive comments of the other referees led to a significant enhancement of the structure and presentation of our work.

Disclosure statement

No potential conflict of interest was reported by the author.