An efficient method for recovering Lyapunov vectors from singular vectors

Lyapunov vectors are natural generalizations of normal modes for linear disturbances to aperiodic deterministic flows and offer insights into the physical mechanisms of aperiodic flow and the maintenance of chaos. Most standard techniques for computing Lyapunov vectors produce results which are norm-dependent and lack invariance under the linearized flow (except for the leading Lyapunov vector) and these features can make computation and physical interpretation problematic. An efficient, norm-independent method for constructing the n most rapidly growing Lyapunov vectors from n − 1 leading forward and n leading backward asymptotic singular vectors is proposed. The Lyapunov vectors so constructed are invariant under the linearized flow in the sense that, once computed at one time, they are defined, in principle, for all time through the tangent linear propagator. An analogous method allows the construction of the n most rapidly decaying Lyapunov vectors from n decaying forward and n − 1 decaying backward singular vectors. This method is demonstrated using two low-order geophysical models.