  This work develops a functional analytic framework for making computer
assisted arguments involving transverse heteroclinic connecting orbits between
hyperbolic periodic solutions of ordinary differential equations. We exploit a
Fourier-Taylor approximation of the local stable/unstable manifold of the
periodic orbit, combined with a numerical method for solving two point boundary
value problems via Chebyshev series approximations. The a-posteriori analysis
developed provides mathematically rigorous bounds on all approximation errors,
providing both abstract existence results and quantitative information about
the true heteroclinic solution. Example calculations are given for both the
dissipative Lorenz system and the Hamiltonian Hill Restricted Four Body
Problem.
