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A definition of spectrum for differential equations on finite time. (English) Zbl 1169.34040

The theory of hyperbolicity for linear systems of ordinary differential equations on the line is now well-developed. The authors study hyperbolic linear systems on compact time intervals. In this case, hyperbolicity means that if ${\Phi }\left(t,s\right)$ is the evolution operator of a linear system on an interval $I$, then

$|{\Phi }\left(t,s\right)\xi |\le exp\left(-\alpha \left(t-s\right)\right)|\xi |$

for $t,s\in I$ with $t\ge s$ and for vectors $\xi$ from the “stable subspace” at time $s$ (and a similar estimate holds for the “unstable subspace” and $t\le s$).

They introduce the notion of a finite time spectrum, prove an analog of the Sacker-Sell theorem, and treat the problem of uniqueness for spectral manifolds.

##### MSC:
 34D09 Dichotomy, trichotomy 34A30 Linear ODE and systems, general
##### References:
 [1] Aulbach, B.: Gewöhnliche differentialgleichungen, (2004) [2] Berger, A.; Doan, T. S.; Siegmund, S.: Nonautonomous finite-time dynamics, Discrete contin. Dyn. syst. Ser. B 9, 463-492 (2008) · Zbl 1148.37010 · doi:doi:10.3934/dcdsb.2008.9.463 [3] A. Berger, T.S. Doan, S. Siegmund, Two notions of finite-time hyperbolicity, in: Proceedings of EQUADIFF 2007, in press [4] Berger, M.; Gostiaux, B.: Differential geometry: manifolds, curves, and surfaces, (1988) [5] Coppel, W. A.: Dichotomies in stability theory, Lecture notes in math. 629 (1978) · Zbl 0376.34001 [6] Hahn, W.: Stability of motion, (1967) · Zbl 0189.38503 [7] Sell, G.: The Floquet problem for almost periodic linear differential equations, Lecture notes in math. 415 (1974) · Zbl 0329.34037 [8] Siegmund, S.: Dichotomy spectrum for nonautonomous differential equations, J. dynam. Differential equations 14, 243-258 (2002) · Zbl 0998.34045 · doi:doi:10.1023/A:1012919512399