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 $\hat{I}¸$. We show that on the global attractor $\mathcal{A}$ of the resulting dynamical system $(u_{\hat{I}}¸(t):t>0)$, a reverse Poincaré inequality holds true, and that as a consequence the map $\hat{I}¸\mapsto u_{\hat{I}}¸(t)$ 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 $\hat{I}¸$ in the attractor $\mathcal{A}$, as well as prediction of the states $u_{\hat{I}}¸$, is possible from discrete measurements of the system at `fast' near parametric convergence rates.