In dynamical systems, connecting orbits are solutions that converge to invariant sets like steady states and connecting orbits as time goes to $-\infty$ and $+\infty$. Such solutions are important for understanding the global dynamics of a system: for instance, their existence can sometimes imply chaotic dynamics. However, actually proving the existence of connecting orbits, let alone describing them precisely, can already be difficult for ODEs, and is extremely challenging for PDEs.

In this talk, I will show how this problem can been approached for parabolic PDEs, using computer-assisted proofs. That is, we first compute an approximate connecting orbit numerically, and then try to prove the existence of a true connecting orbit nearby. This requires several ingredients, among which a rigorous integrator allowing to get approximate solutions of initial value problems for parabolic PDEs, with guaranteed error bounds that prove that the solution actually exists over the entire time interval considered. A large part of the talk we be devoted to the presentation of such rigorous integrator for semilinear parabolic PDEs.