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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.

In this paper we consider smooth orthonormal decompositions of smooth time varying matrices. Among others, we consider QR-, Schur-, and singular value decompositions, and their block-analogues. Sufficient conditions for existence of such decompositions are given and differential equations for the factors are derived. Also generic smoothness of these factors is discussed.

Abstract. In this paper the issue of integrating matrix differential systems whose solutions are unitary matrices is addressed. Such systems have skew-Hermitian coecient matrices in the linear case and a related structure in the nonlinear case. These skew systems arise in a number of applications, and interest originates from application to continuous orthogonal decoupling techniques.In this case, the matrix system has a cubic nonlinearity.Numerical integration schemes that compute a unitary approximate solution for all stepsizes are studied. These schemes can be characterized as being of two classes: autoraatic and projected unitary schemes. In the former class, there belong those standard finite difference schemes which give a unitary solution; the only ones are in fact the Gauss-Legendre point Runge-Kutta (Gauss RK) schemes. The second class of schemes is created by projecting approximations computed by an arbitrary scheme into the set of unitary matrices. In the analysis of these unitary schemes, the stability considerations are guided by the skew-Hermitian character of the problem. Various error and implementation issues are considered, and the methods are tested on a number of examples.

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