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

Abstract

We consider a two-dimensional periodic reaction-diffusion system 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 a $L^2$-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 `fast' near parametric convergence rates.