Linearization of Hyperbolic Finite-Time Processes

Details Linearization of Hyperbolic Finite-Time Processes Linearization of Hyperbolic Finite-Time Processes

2011-09-06

arxiv.org/abs/1109.1162v2

Mathematics

Dynamical Systems

33 pages

Scientific paper

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.

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