Description du projet
Dompter l’état sauvage mathématique
Trouver la simplicité dans la complexité, ou comment montrer que le chaos est la clé pour donner un sens au monde qui nous entoure. Les mathématiques peuvent nous y aider. Les modèles s’efforcent de conserver le niveau de complexité nécessaire et suffisant pour décrire un phénomène donné, afin de réduire à la fois le temps de calcul et la consommation d’énergie de ce calcul. En ce qui concerne les processus variables dans le temps en physique et en biologie, la théorie des systèmes dynamiques nous permet de prévoir le comportement du système à un moment ultérieur en suivant son évolution à travers différents états possibles. En ce qui concerne les systèmes chaotiques pour lesquels cela n’est pas possible, les scientifiques utilisent les probabilités, et notamment celle qui veut que le système se trouve dans un état donné à un moment donné. Dans les systèmes chaotiques «sauvages», il existe un nombre infini de possibilités statistiques, ce qui complique encore les choses. Le projet Emergence, financé par l’UE, entreprend la première étude mondiale de ces cas particuliers, dans le but de les simplifier et de les décrire.
Objectif
Many physical or biological systems display time-dependent states which can be mathematically modelled by a differentiable dynamical system. The state of the system consists of a finite number of variables, and the short time evolution is given by a differentiable equation or the iteration of a differentiable map. The evolution of a state is called an orbit of the system. The theory of dynamical systems studies the long time evolution of the orbits.
For some systems, called chaotic, it is impossible to predict the state of an orbit after a long period of time. However, in some cases, one may predict the probability of an orbit to have a certain state. A paradigm is given by the Boltzmann ergodic hypothesis in thermodynamics: over long periods of time, the time spent by a typical orbit in some region of the phase space is proportional to the “measure” of this region. The concept of Ergodicity has been mathematically formalized by Birkhoff. Then it has been successfully applied (in particular) by the schools of Kolmogorov and Anosov in the USSR, and Smale in the USA to describe the statistical behaviours of typical orbits of many differentiable dynamical systems.
For some systems, called wild, infinitely many possible statistical behaviour coexist. Those are spread all over a huge space of different ergodic measures, as initially discovered by Newhouse in the 70's. Such systems are completely misunderstood. In 2016, contrarily to the general belief, it has been discovered that wild systems form a rather typical set of systems (in some categories).
This project proposes the first global, ergodic study of wild dynamics, by focusing on dynamics which are too complex to be well described by means of finitely many statistics, as recently quantified by the notion of Emergence. Paradigmatic examples will be investigated and shown to be typical in many senses and among many categories. They will be used to construct a theory on wild dynamics around the concept of Emergence.
Champ scientifique
Mots‑clés
Programme(s)
Régime de financement
ERC-COG - Consolidator GrantInstitution d’accueil
75794 Paris
France