An ergodic-averaging method to differentiate covariant Lyapunov vectors

Abstract

Abstract Covariant Lyapunov vectors or CLVs span the expanding and contracting directions of perturbations along trajectories in a chaotic dynamical system. Due to efficient algorithms to compute them that only utilize trajectory information, they have been widely applied across scientific disciplines, principally for sensitivity analysis and predictions under uncertainty. In this paper, we develop a numerical method to compute the directional derivatives of the first CLV along its own direction; the norm of this derivative is also the curvature of one-dimensional unstable manifolds. Similar to the computation of CLVs, the present method for their derivatives is iterative and analogously uses the second-order derivative of the chaotic map along trajectories, in addition to the Jacobian. We validate the new method on a super-contracting Smale–Williams Solenoid attractor. We also demonstrate the algorithm on several other examples including smoothly perturbed Arnold Cat maps, and the Lorenz’63 attractor, obtaining visualizations of the curvature of each attractor. Furthermore, we reveal a fundamental connection of the derivation of the CLV self-derivative computation with an efficient computation of linear response of chaotic systems.