  We present a computational method for studying transverse homoclinic orbits
for periodic solutions of delay differential equations, a phenomenon that we
refer to as the \emph{Poincar\'{e} scenario}. The strategy is geometric in
nature, and consists of viewing the connection as the zero of a nonlinear map,
such that the invertibility of its Fr\'{e}chet derivative implies the
transversality of the intersection. The map is defined by a projected boundary
value problem (BVP), with boundary conditions in the (finite dimensional)
unstable and (infinite dimensional) stable manifolds of the periodic orbit. The
parameterization method is used to compute the unstable manifold and the BVP is
solved using a discrete time dynamical system approach (defined via the
\emph{method of steps}) and Chebyshev series expansions. We illustrate this
technique by computing transverse homoclinic orbits in the cubic Ikeda and
Mackey-Glass systems.
