Computing Sacker–Sell spectra in Discrete Time Dynamical Systems

Abstract

In this paper we develop boundary value methods for detecting Sacker–Sell spectra in discrete time dynamical systems. The algorithms are advancements of earlier methods for computing projectors of exponential dichotomies. The first method is based on the projector residual $P^2-P$. If this residual is large, then the difference equation has no exponential dichotomy. Further criterions for detecting Sacker–Sell spectral intervals are the norm of end points and midpoints of the solution of a specific boundary value problem. Refined error estimates for the underlying approximation process are given, and the resulting algorithms are applied to an example with known continuous Sacker–Sell spectrum, as well as to the variational equation along orbits of Hénon's map.