Linearization of Hyperbolic Finite-Time Processes

TL;DR

This paper develops a unified spectral framework for analyzing hyperbolic behavior in finite-time linear processes, extending classical concepts to a compact time setting and providing robustness and geometric insights.

Contribution

It introduces exponential monotonicity dichotomy (EMD) for finite-time hyperbolicity, unifies existing approaches, and offers a comprehensive spectral and geometric theory with stability bounds.

Findings

Established a spectral theory based on logarithmic difference quotient.

Proved robustness of EMD under perturbations.

Provided finite-time analogues of classical invariant manifold theorems.

Abstract

We adapt the notion of processes to introduce an abstract framework for dynamics in finite time, i.e.\ on compact time sets. For linear finite-time processes a notion of hyperbolicity namely exponential monotonicity dichotomy (EMD) is introduced, thereby generalizing and unifying several existing approaches. We present a spectral theory for linear processes in a coherent way, based only on a logarithmic difference quotient, prove robustness of EMD with respect to a suitable (semi-)metric and provide exact perturbation bounds. Furthermore, we give a complete description of the local geometry around hyperbolic trajectories, including a direct and intrinsic proof of finite-time analogues of the local (un)stable manifold theorem and theorem of linearized asymptotic stability. As an application, we discuss our results for ordinary differential equations on a compact time-interval.