An ergodic averaging method to differentiate covariant Lyapunov vectors

TL;DR

This paper introduces a numerical method to compute derivatives of covariant Lyapunov vectors along their directions, enhancing sensitivity analysis in chaotic systems by leveraging second-order derivatives and Jacobians.

Contribution

The paper presents a novel iterative algorithm for calculating the directional derivatives of CLVs, connecting them with statistical linear response theory and validated on multiple chaotic systems.

Findings

Validated on Smale-Williams attractor

Applied to Arnold Cat maps and Lorenz attractor

Visualized curvature and derivative structures

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 CLVs along their own directions. 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 attractor, obtaining visualizations of the curvature of each attractor. Furthermore, we reveal a fundamental connection of the CLV self-derivatives with a statistical linear response formula.