Spectral Theory of Nonautonomous Differential Equations

The central aim in this chapter is to develop a spectral theory for linear nonautonomous differential equations of the form \(\displaystyle \dot x(t)=A(t)x. \) In Sect. 1.1, we recall two well-developed spectral theories, namely, Lyapunov spectrum and Sacker-Sell spectra, for nonautonomous differential equations on the half real line ( \(I=\mathbb {R}_{0}^{+}\) ) and on the whole real line ( \(I=\mathbb {R}\) ). Section 1.2 is devoted to developing a Bohl spectral theory for nonautonomous differential equations. The content of this section is a part of Doan et al. (J Dyn Differ Equ 29(4):1459–1485, 2017).