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Zbl 1169.34040
Berger, A.; Doan, T.S.; Siegmund, S.
A definition of spectrum for differential equations on finite time.
(English)
[J] J. Differ. Equations 246, No. 3, 1098-1118 (2009). ISSN 0022-0396

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 $\Phi(t,s)$ is the evolution operator of a linear system on an interval $I$, then $$|\Phi(t,s)\xi|\leq\exp(-\alpha(t-s))|\xi|$$ for $t,s\in I$ with $t\geq s$ and for vectors $\xi$ from the stable subspace" at time $s$ (and a similar estimate holds for the unstable subspace" and $t\leq s$). \par 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.
[Sergei Yu. Pilyugin (St. Petersburg)]
MSC 2000:
*34D09 Dichotomy, trichotomy
34A30 Linear ODE and systems

Keywords: linear differential equations; finite-time dynamics; exponential dichotomy; hyperbolicity; spectral theorem

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