  This article establishes the cutoff phenomenon in the Wasserstein distance
for systems of nonlinear ordinary differential equations with a unique coercive
stable fixed point subject to general additive Markovian noise in the limit of
small noise intensity. This result generalizes the results shown in Barrera,
H\"ogele, Pardo (EJP2021) in a more restrictive setting of Blumenthal-Getoor
index $\alpha>3/2$ to the formulation in Wasserstein distance, which allows to
cover the case of general L\'evy processes with some given moment. The main
proof techniques are based on the close control of the errors in a version of
the Hartman-Grobman theorem and the adaptation of the linear theory established
in Barrera, H\"ogele, Pardo (JSP2021). In particular, they rely on the precise
asymptotics of the nonlinear flow and the nonstandard shift linearity property
of the Wasserstein distance, which is established by the authors in (JSP2021).
Main examples are the Fermi-Pasta-Ulam-Tsingou gradient flow and coercive
nonlinear oscillators subject to small (and possibly degenerate) Brownian or
arbitrary $\alpha$-stable noise.
