SVD algorithms to approximate spectra of dynamical systems
Mathematics and Computers in Simulation
On the error in approximating stability spectra for discrete dynamical systems
Mathematics and Computers in Simulation
On the Error in the Product QR Decomposition
SIAM Journal on Matrix Analysis and Applications
Advances in Computational Mathematics
Computing Sacker-Sell spectra in Discrete Time Dynamical Systems
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
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Different definitions of spectra have been proposed over the years to characterize the asymptotic behavior of nonautonomous linear systems. Here, we consider the spectrum based on exponential dichotomy of Sacker and Sell [J. Differential Equations, 7 (1978), pp. 320--358] and the spectrum defined in terms of upper and lower Lyapunov exponents. A main goal of ours is to understand to what extent these spectra are computable. By using an orthogonal change of variables transforming the system to upper triangular form, and the assumption of integral separation for the diagonal of the new triangular system, we justify how popular numerical methods, the so-called continuous QR and SVD approaches, can be used to approximate these spectra. We further discuss how to verify the property of integral separation, and hence how to a posteriori infer stability of the attained spectral information. Finally, we discuss the algorithms we have used to approximate the Lyapunov and Sacker--Sell spectra and present some numerical results.