Continuous time algorithm for Lyapunov exponents. I

Abstract

The Lyapunov exponent algorithm of Benettin et al. [C.R. Acad. Sci. Paris 286A, 431 (1978); Meccanica 15, 9 (1980)] and of Shimada and Nagashima [Prog. Theor. Phys. 61, 1605 (1979)] is converted to a continuous time algorithm. The Gram-Schmidt orthonormalization process is incorporated into the differential equations, making orthogonalization continuous. Differential equations for the k-dimensional fiducial volumes are also derived. It is further shown that this algorithm is a factorization of the fundamental matrix \ensuremath{\Phi} in the form \ensuremath{\Phi}(t)=E(t)W(t)${\mathit{E}}^{\mathrm{\ensuremath{-}}1}$(${\mathit{t}}_{0}$), where E(t) is orthonormal and W(t) is upper triangular. Numerical stability of the algorithm is considered, and it is shown that the standard Gram-Schmidt process can be used to stabilize the (possibly) unstable equations of motion for the orthonormal basis vectors. A numerical example is presented.