On the Computational Power and Super-turing Capabilities of Dynamical Systems Ecole Normale Supérieure De Lyon on the Computational Power and Super-turing Capabilities of Dynamical Systems

We explore the simulation and computational capabilities of dynamical systems. We rst introduce and compare several notions of simulation between discrete systems. We give a general framework that allows dynamical systems to be considered as computational machines. We introduce a new discrete model of computation: the analog automaton model. We determine the computational power of this model and prove that it does have super-Turing capabilities. We then prove that many very simple dynamical systems from literature are actually able to simulate analog automata. From this result we deduce that many dynamical systems have intrinsically super-Turing capabilities. Nous etudions la puissance de calcul et de simulation des syst emes dy-namiques. Dans un premier temps, nous introduisons et comparons plusieurs notions de simulation entre syst emes discrets. Nous donnons un cadre g en eral permettant de consid erer les syst emes dynamiques comme des mod eles de calcul. Nous d eenissons un nouveau mod ele de calcul: le mod ele des automates analogiques. Nous d eterminons la puissance de calcul de ce mod ele et prouvons qu'il poss ede des capacit es super-Turing. Nous prouvons alors que des syst emes dynamiques tr es simples tir es de la litt erature sont en fait capable de simuler les automates analogiques. De ces r esultats, nous en d eduisons que beaucoup des syt emes dynamiques ont intrins equement des capacit es super-Turing.