A definition of spectrum for differential equations on finite time

Hyperbolicity of an autonomous rest point is characterised by its linearization not having eigenvalues on the imaginary axis. More generally, hyperbolicity of any solution which exists for all times can be defined by means of Lyapunov exponents or exponential dichotomies. We go one step further and introduce a meaningful notion of hyperbolicity for linear systems which are defined for finite time only, i.e. on a compact time interval. Hyperbolicity now describes the transient dynamics on that interval. In this framework, we provide a definition of finite-time spectrum, study its relations with classical concepts, and prove an analogue of the Sacker–Sell spectral theorem: For a d-dimensional system the spectrum is non-empty and consists of at most d disjoint (and often compact) intervals. An example illustrates that the corresponding spectral manifolds may not be unique, which in turn leads to several challenging questions.