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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.

34D09Dichotomy, trichotomy
34A30Linear ODE and systems, general
Full Text: DOI
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