Computing covariant vectors, Lyapunov vectors, Oseledets vectors, and dichotomy projectors: a comparative numerical study

TL;DR

This paper compares four computational methods for approximating covariant, Lyapunov, and Oseledets vectors, introducing two new approaches and evaluating their performance across diverse dynamical systems and data availability scenarios.

Contribution

The paper introduces two novel methods based on singular value decomposition and exponential dichotomies, and provides a comparative analysis of four approaches for computing these vectors.

Findings

New methods perform well with limited data.

Performance varies across different dynamical system types.

Existing methods are improved and benchmarked against new approaches.

Abstract

Covariant vectors, Lyapunov vectors, or Oseledets vectors are increasingly being used for a variety of model analyses in areas such as partial differential equations, nonautonomous differentiable dynamical systems, and random dynamical systems. These vectors identify spatially varying directions of specific asymptotic growth rates and obey equivariance principles. In recent years new computational methods for approximating Oseledets vectors have been developed, motivated by increasing model complexity and greater demands for accuracy. In this numerical study we introduce two new approaches based on singular value decomposition and exponential dichotomies and comparatively review and improve two recent popular approaches of Ginelli et al. (2007) and Wolfe and Samelson (2007). We compare the performance of the four approaches via three case studies with very different dynamics in terms of symmetry, spectral separation, and dimension. We also investigate which methods perform well with limited data.