  We present a new algorithm for the rigorous integration of the variational
equation (i.e. producing $\mathcal C^1$ estimates) for a class of dissipative
PDEs on the torus. As an application for some parameter value for the
Kuramoto-Sivashinsky PDE on the line with odd and periodic boundary conditions
we prove the existence of infinite number of homo- and heteroclinic orbits to
two periodic orbits. The proof is computer assisted.
