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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 Φ(t,s) is the evolution operator of a linear system on an interval I, then

|Φ(t,s)ξ|exp(-α(t-s))|ξ|

for t,sI with ts and for vectors ξ from the “stable subspace” at time s (and a similar estimate holds for the “unstable subspace” and ts).

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:
34D09Dichotomy, trichotomy
34A30Linear ODE and systems, general
References:
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[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
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[8]Siegmund, S.: Dichotomy spectrum for nonautonomous differential equations, J. dynam. Differential equations 14, 243-258 (2002) · Zbl 0998.34045 · doi:10.1023/A:1012919512399