Projektbeschreibung
Zähmung der (mathematischen) Wildheit
Es ist unerlässlich, Muster in Komplexität oder Chaos zu erkennen, um aus der Welt um uns herum schlau zu werden. Die Mathematik kann uns dabei helfen. Modelle streben nach einem Komplexitätsgrad, der notwendig und hinreichend ist, um das gegebene Phänomen zu beschreiben, sodass der Zeitaufwand und Energieverbrauch der Berechnungen minimiert werden können. Bei zeitvariablen Vorgängen in der Physik und Biologie ermöglicht uns die Theorie dynamischer Systeme, das Verhalten des Systems zu einem späteren Zeitpunkt vorherzusagen, indem die vorherige Entwicklung und zahlreiche mögliche Zustände verfolgt werden. Bei chaotischen Systemen ist dies nicht möglich. Stattdessen nutzen Wissenschaftlerinnen und Wissenschaftler dann die Wahrscheinlichkeitstheorie, um Angaben dazu zu machen, wie wahrscheinlich es ist, dass das System einen bestimmten Zustand annimmt. In „wilden“ chaotischen Systemen gibt es eine unendliche Anzahl statistischer Möglichkeiten, was die Lage noch einmal weiter verkompliziert. Das EU-finanzierte Projekt Emergence führt die erste weltweite Studie dieser Sonderfälle durch, um sie vereinfachen und besser beschreiben zu können.
Ziel
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.
Wissenschaftliches Gebiet
Schlüsselbegriffe
Programm/Programme
Thema/Themen
Finanzierungsplan
ERC-COG - Consolidator GrantGastgebende Einrichtung
75794 Paris
Frankreich