In this talk, I explore how internal deformations in spinless extended bodies, such as periodic pulsations or oscillations of the body’s shape, can trigger chaotic motion even when the background spacetime is fixed and the underlying geodesic dynamics is integrable. Using Dixon’s multipolar framework, I show how time-periodic finite-size effects can split separatrices and generate homoclinic chaos, diagnosed via the Melnikov method. I discuss Schwarzschild black holes for nearly spherical bodies with oscillating oblateness, as well as spherically symmetric pulsations in non-vacuum spacetimes, including electrovac/charged black holes and black holes embedded in dark-matter halos. Finally, I highlight a genuinely relativistic boundary of the point-particle idealization: in Ricci-flat vacuum, spinless spherical bodies move geodesically through quadrupole order, yet finite-size deviations can arise at hexadecapole order, where even vacuum Schwarzschild admits nontrivial corrections and chaos under pulsations.

On a background Minkowski spacetime, the Euler equations (both relativistic and not) are known to admit unstable homogeneous solutions with finite-time shock formation. Such shock formation can be suppressed on cosmological spacetimes whose spatial slices expand at an accelerated rate. However, situations with decelerated expansion, which are relevant in our early universe, are not as well understood. I will present some recent joint work in this direction, based on collaborations with David Fajman, Maciej Maliborski, Todd Oliynyk and Max Ofner.

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 variety of possible asymptotic states which in some cases are described by partially hyperbolic lines of equilibria, bands of periodic orbits or generalised Liénard systems.

Asymptotically Euclidean initial data sets (IDS) in General Relativity model instants in time for isolated systems. In this talk, we show that an IDS is asymptotically Euclidean if it admits a cover by closed hypersurfaces of constant spacetime mean curvature (STCMC), provided these hypersurfaces satisfy certain geometric estimates, some weak foliation properties, and each surface exhibits generalized stability. Building on the work of Cederbaum and Sakovich (2021), which established that every asymptotically Euclidean IDS has a unique STCMC foliation, we conclude that the existence of such a foliation characterizes asymptotically Euclidean IDS. Furthermore, we explore the connections to the center of mass and show why these coordinates seem well-adapted to describe this concept. This is joint work with O. Vičánek Martínez.

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.