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Abstract:

The problem of numerical computation of a few Lyapunov exponents of nite-dimensional dynamical systems is considered from the viewpoint of the dierential geometry of Stiefel manifolds. Whether one computes one, many or all Lyapunov exponents of a continuous dynamical system by time integration, discrete or continuous orthonormalization is essential for stable numerical integration. A dierential geometric view of continuous orthogonalization suggests that one restrict the linearized vectoreld to a Stiefel manifold. However, 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 which use the bre-bundle characterization of these manifolds. This leads to a new class of systems for continuous orthogonalization which have strong numerical invariance properties and the strong skew-symmetry property. Numerical integration of these new systems with geometric integrators leads to a new class of numerical methods for computing a few Lyapunov exponents which preserve orthonormality to machine accuracy. This idea is also taken a step further by making the Stiefel manifold an attracting invariant manifold in which case standard explicit Runge-Kutta algorithms can be used. This leads to an algorithm which requires only marginally more computation than a standard integration without orthogonalization. This class of methods should be particularly eective for computing a few Lyapunov exponents for large-dimension dynamical systems. The new

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