A projector-based convergence proof of the Ginelli algorithm for covariant Lyapunov vectors

TL;DR

This paper provides a rigorous mathematical proof of the convergence of the Ginelli algorithm for computing covariant Lyapunov vectors, extending previous approaches to a more general setting.

Contribution

It introduces a projector-based convergence proof for the Ginelli algorithm, accommodating degenerate Lyapunov spectra and utilizing the Multiplicative Ergodic Theorem.

Findings

Established a rigorous convergence proof for Ginelli's algorithm.

Extended the proof to handle degenerate Lyapunov spectra.

Utilized the Multiplicative Ergodic Theorem for asymptotic characterization.

Abstract

Linear perturbations of solutions of dynamical systems exhibit different asymptotic growth rates, which are naturally characterized by so-called covariant Lyapunov vectors (CLVs). Due to an increased interest of CLVs in applications, several algorithms were developed to compute them. The Ginelli algorithm is among the most commonly used. Although several properties of the algorithm have been analyzed, there exists no mathematically rigorous convergence proof yet. In this article we extend existing approaches in order to construct a projector-based convergence proof of Ginelli's algorithm. One of the main ingredients will be an asymptotic characterization of CLVs via the Multiplicative Ergodic Theorem. In the proof, we keep a rather general setting allowing even for degenerate Lyapunov spectra.