Perturbation Theory for Approximation of Lyapunov Exponents by Qr Methods

Motivated by a recently developed backward error analysis for QR methods, we consider the error in the Lyapunov exponents of perturbed triangular systems. We consider the case of stable and distinct Lyapunov exponents as well as the case of stable but not necessarily distinct exponents. We illustrate our analytical results with a numerical example. 1. Introduction. Lyapunov exponents are often employed in the numerical study of nonlinear dynamical systems and are probably the most widely used quantities for detecting chaos, estimating dimensions of attractors, entropy; e.g., see [7, 21, 2, 3]. However, there is little error analysis of the techniques used to approximate Lyapunov exponents; the works [8, 11, 17, 19] are the only works of which we know dealing precisely with error analysis for approximation of Lyapunov exponents. In this paper we provide quantifiable error bounds for Lyapunov exponents approximated by QR techniques. We recall that, to approximate Lyapunov exponents for a linear non-autonomous system ˙ x = A(t)x, the basic idea of QR methods consists in first triangularizing (via the QR factorization) an underlying fundamental matrix solution X: X = QR, and then extracting the Lyapunov exponents from the diagonal of the transformed triangular system: ˙ R = BR, with B triangular. This basic approach is common in analytical works on the subject (see [12, 15, 20]) and it is also the most common approach for numerical methods to approximate Lyapunov exponents (see [5, 8, 9] and references therein). In the recent work [11], we gave a backward error analysis for QR methods used to approximate Lyapunov exponents. Our analysis showed that, by QR methods (i.e., by a numerical realization of QR methods), one obtains an exact triangularization of a fundamental matrix solution of a perturbed triangular problem with coefficient function B + E, instead of B. We were also able to give quantitative bounds on the perturbation E and showed that in principle this perturbation can be made arbitrarily small by controlling the accuracy of the computation. Now, for systems with stable Lyapunov exponents (a necessary condition for trying to approximate them), small perturbations reflect in small errors in the Lyapunov exponents; e.g., see [1] for necessary and sufficient conditions for the stability of Lyapunov exponents. In this paper, we clarify, and quantify, the error induced by a small perturbation E on the Lyapunov exponents. We consider both the case of stable distinct and stable but not …