  The contributions of this technical note are twofold. Firstly, we formulate
an optimization problem to obtain a linear representation of a nonlinear vector
field based on a system's trajectory. We also prove that its cost function is
strictly convex, given the trajectory is persistently exciting. Under certain
observability conditions, we provide results that guarantee the Hurwitz
stability of the global minimizer. Secondly, we present a novel algorithm based
on point-wise geometric flows to estimate the boundary of the region of
attraction. We show that the algorithm converges to the exact boundary of the
region of attraction under certain assumptions on the system dynamics. Finally,
we validate the results using simulations on various nonlinear autonomous
systems.
