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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 \(\Phi(t,s)\) 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 \(t\leq s\)).
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 of solutions to ordinary differential equations
34A30 Linear ordinary differential equations and systems, general
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