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 is the evolution operator of a linear system on an interval , then
for with and for vectors from the “stable subspace” at time (and a similar estimate holds for the “unstable subspace” and ).
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.