To understand the global behavior of a nonlinear system, the first step is
to study its invariant set. Indeed, specific solutions like steady states, periodic orbits and connections between them are building blocks that organize the global dynamics. While there are many deep, general and theoretical mathematical results about the existence of such solutions, it is often difficult to apply them to a specific example. Besides, when dealing with a precise application, it is not only the existence of these solutions, but also their qualitative properties that are of interest. In that case, a powerful and widely used tool is numerical simulations, which is well adapted to the study of an explicit system and can provide insights for problems where the nonlinearities hinder the use of purely analytical techniques.
However, one can do even better. Using numerical results as a starting
point, and combining them with a posteriori estimates, one can then get rigorous results and prove the existence of a genuine solution close to the numerical one. In this talk, I will explain how such computer-assisted theorem can be obtained. I will then focus on some examples where these techniques can be useful, namely to study non homogeneous steady states of cross-diffusion systems, and to prove the existence of periodic solutions of the Navier-Stokes equations in a Taylor-Green flow.