The problem of numerical computation of a few Lyapunov exponents (LEs) of f
inite-dimensional dynamical systems is considered from the viewpoint of the
differential geometry of Stiefel manifolds. Whether one computes one, many
or all LEs of a continuous dynamical system by time integration, discrete
or continuous orthonormalization is essential for stable numerical integrat
ion. A differential-geometric view of continuous orthogonalization suggests
that one restricts the linearized vectorfield to a Stiefel manifold. Howev
er, the Stiefel manifold is not in general an attracting submanifold of the
ambient Euclidean space: it is a constraint manifold with a weak numerical
invariant. New numerical algorithms for this problem are then designed whi
ch use the fiber-bundle characterization of these manifolds. This framework
leads to a new class of systems for continuous orthogonalization which hav
e strong numerical invariance properties and the strong skew-symmetry prope
rty. Numerical integration of these new systems with geometric integrators
leads to a new class of numerical methods for computing a few LEs which pre
serve orthonormality to machine accuracy. This idea is also taken a step fu
rther by making the Stiefel manifold an attracting invariant manifold in wh
ich case standard explicit Runge-Kutta algorithms can be used. This leads t
o an algorithm which requires only marginally more computation than a stand
ard explicit integration without orthogonalization. These class of methods
should be particularly effective for computing a few LEs for large-dimensio
n dynamical systems. The new schemes are straightforward to implement. A te
st case is presented for illustration, and an example from dynamical system
s is presented where a few LEs are computed for an array of coupled oscilla
tors. (C) 2001 Elsevier Science B.V. All rights reserved.