A definition of spectrum for differential equations on finite time

Abstract

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.

Barriers to transport in aperiodically time-dependent two-dimensional velocity fields: Nekhoroshev's theorem and "Nearly Invariant" tori

“…Another approach to the notion of hyperbolicity for finite times would be through a proper generalization of the idea of the spectrum associated with the linearization about a finite time trajectory. This is discussed in Berger et al (2009) and Doan et al (2011).…”

A Remark on Finite‐Time Hyperbolicity

“…With these quantities, recall two notions of hyperbolicity (see [1][2][3] for details, as well as [5,6] for more general background information on finite-time dynamics and its applications). Definition 1.1 (D-Hyperbolicity) System (1) is called D-hyperbolic on I if, for all t ∈ I, S(t) is indefinite and non-degenerate, and ξ, M (t)ξ > 0 for all ξ ∈ Z(t).…”

“…With these quantities, recall two notions of hyperbolicity (see [1][2][3] for details, as well as [5,6] for more general background information on finite-time dynamics and its applications). (1) is called M-hyperbolic on I if there exist constants α < 0 < β and an invariant projector P , i.e., the mapping P : I → R d×d is projection-valued with P (t)Φ(t, s) = Φ(t, s)P (s) for all t, s ∈ I, such that…”

Transient spectral theory, stable and unstable cones and Gershgorinʼs theorem for finite-time differential equations

“…Berger et al [2,4] and the references therein) with applications e.g. in fluid dynamics or satellite imaging of ocean currents.…”

“…Motivated by this work, a new notion of hyperbolicity, namely M-hyperbolicity, is introduced in Berger et al [4]. M-hyperbolicity is based on monotonic growth and decay of solutions.…”