  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.
