On statistical inference for non-linear dynamical systems evolving in their global attractor

arXiv:2606.06018v2 Announce Type: replace Abstract: We consider a periodic scalar reaction-diffusion system over a two-dimensional domain under natural conditions on the reaction function and with initial condition $\theta$. We show that on the global attractor $\mathcal A$ of the resulting dynamical system $(u_\theta(t):t>0)$, a reverse Poincar\'e inequality holds true, and that as a consequence the map $\theta \mapsto u_\theta(t)$ satisfies an $L^2(\Om)$-Lipschitz stability estimate on $\mathcal A$ for any $t>0$ fixed. We then show that statistical recovery of an initial condition $\theta$ in the attractor $\mathcal A$, as well as prediction of the states $u_\theta$, is possible from discrete measurements of the system at parametric $1/\sqrt N$ convergence rates.