Dichotomy spectrum for nonautonomous differential equations. (English) Zbl 0998.34045

Siegmund, Stefan

J. Dyn. Differ. Equations 14, No. 1, 243-258 (2002).

Summary: Here, for nonautonomous linear differential equations \(\dot x=A(t)x\) with locally integrable \(A:\mathbb{R}\to\mathbb{R}^{N\times N}\), the so-called dichotomy spectrum is investigated. As the closely related dichotomy spectrum for skew product flows with compact base (Sacker-Sell spectrum), this dichotomy spectrum for nonautonomous differential equations consists of at most \(N\) closed intervals, which in contrast to the Sacker-Sell spectrum may be unbounded. In the constant coefficients case, these intervals reduce to the real parts of the eigenvalues of \(A\). In any case, the spectral intervals are associated with spectral manifolds comprising solutions with a common exponential growth rate. The main result here is a spectral theorem, which describes all possible forms of the dichotomy spectrum.

MSC:

34D09

Dichotomy, trichotomy of solutions to ordinary differential equations

Keywords:

Sacker-Sell spectrum; nonautonomous linear differential equations; dichotomy spectrum; skew product flows; spectral intervals; spectral manifolds