an:05490733
Zbl 1169.34040
Berger, A.; Doan, T. S.; Siegmund, S.
A definition of spectrum for differential equations on finite time
EN
J. Differ. Equations 246, No. 3, 1098-1118 (2009).
00243808
2009
j
34D09 34A30
linear differential equations; finite-time dynamics; exponential dichotomy; hyperbolicity; spectral theorem
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.
Sergei Yu. Pilyugin (St. Petersburg)