We develop some computer-assisted techniques for the analysis of
stationary solutions of dissipative partial differential equations,
their stability, as well as bifurcation diagrams.
As a case study, these methods are applied to the Kuramoto-Sivashinski equation.
This equation has been investigted extensively,
and its bifurcation diagram is well known from a numerical point of view.
Here, we describe rigorously the full graph of solutions
branching off the trivial branch, complete with all secondary bifurcations,
for parameter values between 0 and 80.
We also determine the dimension of the unstable manifold
for the flow at some stationary solution in each branch.