  We consider constant mean curvature 1 surfaces in $\mathbb{R}^3$ arising via
the DPW method from a holomorphic perturbation of the standard Delaunay
potential on the punctured disk. Kilian, Rossman and Schmitt have proven that
such a surface is asymptotic to a Delaunay surface. We consider families of
such potentials parametrised by the necksize of the model Delaunay surface and
prove the existence of a uniform disk on which the surfaces are close to the
model Delaunay surface and are embedded in the unduloid case.
