Derivatives and Constraints in Chaotic Flows: Asymptotic Behaviour and a Numerical Method

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.