I will give an overview talk concerning the possible existence and stability of solutions to non-linear wave equations which are periodic in time. Such solutions can arise in a variety of mathematical models, from fluid dynamics, elasticity, and general relativity, where in particular, there were investigated numerically by Maliborski and Rostworowski (2013) for the Einstein-scalar-field model in spherically symmetry near the Anti-de-Sitter spacetime.

I will start the presentation with some reminder concerning the linear wave equation on Anti-de-Sitter before presenting some results and methods for semi-linear wave equations. In a second part, I will describe a recent construction of special coordinates for 1+1 Lorentzian metric on $\mathbb{R}\times\mathcal{S}^1$ with time-periodic coefficients, which is expected to be an essential step to extend the previous results to quasi linear wave equations.

In this talk I will discuss some results obtained in collaboration with Filipe C. Mena and former PhD student Vítor Bessa on the global dynamics of a minimally coupled scalar field interacting with a perfect-fluid through a friction-like term in spatially flat homogeneous and isotropic spacetimes. In particular, it is shown that the late time dynamics contain a rich varitey of possible asymptotic states which in some cases are described by partially hyperbolic lines of equilibria, bands of periodic orbits or generalised Liénard systems.