Preprint of the project: SFB 701: Spectral Structures and Topological Methods in Mathematics - Project A2

Numerical analysis of high-dimensional transfer operators

08-081 Thorsten Hüls.
Numerical computation of dichotomy rates and projectors in discrete time

We introduce a characterization of exponential dichotomies for linear difference equations that can be tested numerically and enables the approximation of dichotomy rates and projectors with high accuracy. The test is based on computing the bounded solutions of a specific inhomogeneous difference equation. For this task a boundary value and a least squares approach is applied. The results are illustrated using Henon's map. We compute approximations of dichotomy rates and projectors of the variational equation, along a homoclinic orbit and an orbit on the attractor. For both approaches, we analyze in detail errors that occur, when restricting the infinite dimensional problem to a finite interval.