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

Covariant 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) [36] and Wolfe and Samelson (2007) [37]. 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. ► We review the definition and numerical approximation of covariant/Lyapunov/Oseledets vectors. ► We describe two new, and review two recent approaches for computing these vectors numerically. ► Using three case studies we compare the efficacy of each approach. ► The three case studies differ in terms of symmetry, spectral separation, dimension and amount of data available. ► The case studies represent a selection from the broad range of possible applications of Oseledets vectors.