Lyapunov Spectral Intervals: Theory and Computation

Abstract

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.

Lyapunov, Bohl and Sacker-Sell Spectral Intervals for Differential-Algebraic Equations

“…We refer to [26,27,30,44] or [58] for more details on the theory for ODEs. An essential step in the computation of spectral intervals for linear DAEs of the form (1) is to first transform the system to a reduced strangeness-free form (7), which has the same solution set as (1), see [50], and then to consider the spectral results in this framework.…”

“…On the other hand, due to the stability (of the Lyapunov exponents), the inequalities (27) imply that |λ…”

“…The integral separation assumption then ensures that the two sets coincide. We underline that the relation between the Bohl exponents and the Sacker-Sell spectrum of a scalar ODE [58,Lemma 21] has resulted in a more elegant interpretation of SackerSell intervals compared with the approach based on two associated planar systems in [26,27].…”

“…In this section we extend the approaches that were derived for the computation of spectral intervals for ODEs in [26,27,30] to DAEs. We derive numerical methods for computing Lyapunov and Sacker-Sell spectra for DAEs of the form (9) based on smooth QR factorizations.…”

“…Unlike the development of the analytic theory, the development of numerical methods to compute Lyapunov exponents and also other spectral intervals has only recently been studied. In a series of papers, see [23,24,26,27,29,30,31], Dieci and Van Vleck have developed algorithms for the computation of Lyapunov and Bohl exponents as well as Sacker-Sell spectral intervals. These methods have also been analyzed concerning their sensitivity under small perturbations (stability), the relationship between different spectra, the error analysis, and efficient implementation techniques.…”