CAPPS

Studying the global dynamics of nonlinear differential equations is a very challenging task. Even proving the existence of non trivial equilibria or periodic orbits is often difficult, at least if one moves away from perturbative or asymptotic parameter regimes, to say nothing of establishing connecting orbits between these types of solutions, nor of proving the existence of more complex, e.g. chaotic, dynamics. In this context, there is often a large gap between what can be observed and conjectured from numerical simulations, and what can actually be proven mathematically. The goal of this project is to bridge this gap by developing new computer-assisted techniques, essentially allowing to turn numerical simulations into mathematical theorems. Computer-assisted proofs have recently risen in popularity in the fields of differential equations an dynamical systems, and can now be used to study ODEs in a rather systematic way. Our main objective is to develop such tools for studying some classes of PDEs and SDEs, allowing us to obtain new results that are currently out of reach of purely analytical techniques. Here are the principal directions we want to investigate: - Existence and precise description of patterns (i.e. non homogeneous steady states) in quasilinear PDEs; - Stability properties, of the above-mentioned patterns, but more generally of solutions obtained using computer-assisted techniques; - Large deviation estimates for finite time Lyapunov exponents of SDEs, allowing to predict a sign change of the first Lyapunov exponent; - A unified framework to describe computer-assisted proofs based on finite elements methods, finite volume methods and on spectral techniques. We plan to focus on some specific problems, like chemotaxis models in population dynamics, but the new computer-assisted techniques produced by this project will by nature be rather robust, and will therefore provide a new set of tools for studying a wide array of nonlinear differential equations.