On the efficient numerical computation of covariant Lyapunov vectors

TL;DR

This paper investigates optimal time windows for computing covariant Lyapunov vectors in Hamiltonian systems, compares two methods, and proposes an adaptation to improve accuracy over long times.

Contribution

It evaluates two methods for determining transient phases in CLV computation and introduces an adaptation to enhance accuracy during long-time integrations.

Findings

Both methods yield similar results for Hamiltonian systems.

Long backward evolution intervals reduce accuracy of center subspace computations.

The proposed adaptation prevents CLV alignment/anti-alignment, improving long-term accuracy.

Abstract

Covariant Lyapunov vectors (CLVs) are useful in multiple applications, but the optimal time windows needed to accurately compute these vectors are yet unclear. To remedy this, we investigate two methods for determining when to safely terminate the forward and backward transient phases of the CLV computation algorithm by Ginelli et al.~\cite{GinelliEtAl2007} when applied to chaotic orbits of conservative Hamiltonian systems. We perform this investigation for two prototypical Hamiltonian systems, namely the well-known H\'enon-Heiles system of two degrees of freedom and a system of three nonlinearly coupled harmonic oscillators having three degrees of freedom, finding very similar results for the two methods and thus recommending the more efficient one. We find that the accuracy of two-dimensional center subspace computations is significantly reduced when the backward evolution stages of the algorithm are performed over long time intervals. We explain this observation by examining the tangent dynamics of the center subspace wherein CLVs tend to align/anti-align, and we propose an adaptation of the algorithm that improves the accuracy of such computations over long times by preventing this alignment/anti-alignment of CLVs in the center subspace.