Dichotomy spectra and Morse decompositions of linear nonautonomous differential equations

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Abstract

Recently, the existence of Morse decompositions for nonautonomous dynamical systems was shown for three different time domains: the past, the future and—in the linear case—the entire time. In this article, notions of exponential dichotomy are discussed with respect to the three time domains. It is shown that an exponential dichotomy gives rise to an attractor–repeller pair in the projective space, which is a building block of a Morse decomposition. Moreover, based on the notions of exponential dichotomy, dichotomy spectra are introduced, and it is proved that the corresponding spectral manifolds lead to Morse decompositions in the projective space.

MSC

34D05
34D09
34D45
37B25
37B55
37C70

Keywords

Attractor
Attractor–repeller pair
Dichotomy spectrum
Exponential dichotomy
Finest Morse decomposition
Morse decomposition
Nonautonomous dynamical system
Repeller

Cited by (0)

Research supported by the Bayerisches Eliteförderungsgesetz of the State of Bavaria and the Graduiertenkolleg Nichtlineare Probleme in Analysis, Geometrie und Physik (GK 283) financed by the DFG.