Smooth SVD on the Lorentz group with applications to computations of Lyapunov exponents

In this work we give a constructive argument to establish existence of a smooth singular value decomposition (SVD) for a $C^k$ function X in the Lorentz group. We rely on the explicit structure of the polar factorization of X in order to justify the form of the SVD. Our construction gives a simple algorithm to 5nd the SVD of X , which we have used to approximate the Lyapunov exponents of a di6erential system whose fundamental matrix solution evolves on the Lorentz group. Algorithmic details and examples are given.