Perturbation theory for the approximation of stability spectra by QR methods for sequences of linear operators on a Hilbert space

In this paper, we develop a perturbation analysis for stability spectra (Lyapunov exponents and Sacker–Sell spectrum) for products of operators on a Hilbert space (both real and complex) based upon the discrete QR technique. Error bounds are obtained in both the integrally separated and non-integrally separated cases that correspond to distinct and multiple eigenvalues, respectively, for a single linear operator. We illustrate our results using a linear parabolic partial differential equation in which the strength of the integral separation (the time varying analogue of gaps between eigenvalues) determines the sensitivity of the stability spectra to perturbation.