  Motivated by the mathematics literature on the algebraic properties of
so-called polynomial vector flows, we propose a technique for approximating
nonlinear differential equations by linear differential equations. Although the
idea of approximating nonlinear differential equations with linear ones is not
new, we propose a new approximation scheme that captures both local as well as
global properties. This is achieved via a hierarchy of approximations, where
the Nth degree of the hierarchy is a linear differential equation obtained by
globally approximating the Nth Lie derivatives of the trajectories.
  We show how the proposed approximation scheme has good approximating
capabilities both with theoretical results and empirical observations. In
particular, we show that our approximation has convergence range at least as
large as a Taylor approximation while, at the same time, being able to account
for asymptotic stability (a nonlocal behavior). We also compare the proposed
approach with recent and classical work in the literature.
