Dichotomy spectra and Morse decompositions of linear nonautonomous differential equations (Q1011469)

This paper investigates the relationships between exponential dichotomies, dichotomy (Sacker-Sell) spectra and Morse decompositions. In the context of linear nonautonomous ordinary differential equations it is shown that an exponential dichotomy yields an attractor-repeller pair in the projective space representing a building block of a Morse decomposition. Indeed, the corresponding spectral manifolds form a Morse decomposition in the projective space. These results are derived for equations defined on intervals unbounded above or below, as well as on the whole real axes. The required nonautonomous attractor and repeller notions are based on the concept of pullback convergence thoroughly introduced in the author's previous work [Attractivity and bifurcation for nonautonomous dynamical systems. Lecture Notes in Mathematics 1907. Berlin: Springer (2007; Zbl 1131.37001)].