# zbMATH — the first resource for mathematics

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