  In a smooth flow, the leading-order response of trajectories to infinitesimal
perturbations in their initial conditions is described by the finite-time
Lyapunov exponents and associated characteristic directions of stretching. We
give a description of the second-order response to perturbations in terms of
Lagrangian derivatives of the exponents and characteristic directions. These
derivatives are related to generalised Lyapunov exponents, which describe
deformations of phase-space elements beyond ellipsoidal. When the flow is
chaotic, care must be taken in evaluating the derivatives because of the
exponential discrepancy in scale along the different characteristic directions.
Two matrix decomposition methods are used to isolate the directions of
stretching, the first appropriate in finding the asymptotic behaviour of the
derivatives analytically, the second better suited to numerical evaluation. The
derivatives are shown to satisfy differential constraints that are realised
with exponential accuracy in time. With a suitable reinterpretation, the
results of the paper are shown to apply to the Eulerian framework as well.
