  In this paper we present a computer-assisted procedure for proving the
existence of transverse heteroclinic orbits connecting hyperbolic equilibria of
polynomial vector fields. The idea is to compute high-order Taylor
approximations of local charts on the (un)stable manifolds by using the
Parameterization Method and to use Chebyshev series to parameterize the orbit
in between, which solves a boundary value problem. The existence of a
heteroclinic orbit can then be established by setting up an appropriate
fixed-point problem amenable to computer-assisted analysis. The fixed point
problem simultaneously solves for the local (un)stable manifolds and the orbit
which connects these. We obtain explicit rigorous control on the distance
between the numerical approximation and the heteroclinic orbit. Transversality
of the stable and unstable manifolds is also proven.
