Project description
Taming the (mathematically) wild
Finding simplicity in complexity or chaos is the key to making sense of the world around us. Mathematics can help us do that. Models strive to retain the level of complexity necessary and sufficient to describe a given phenomenon, to reduce both computation time and computing energy consumption. When it comes to time-varying processes in physics and biology, dynamical systems theory allows us to predict the system’s behaviour at a later point in time by following its evolution through various possible states. For chaotic systems for which that is not possible, scientists use probabilities and the likelihood that the system will be in a given state. In 'wild' chaotic systems, an infinite number of statistical possibilities exists, complicating matters once again. The EU-funded Emergence project is undertaking the first global study of these special cases, in an effort to simplify and describe them.
Objective
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.
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Programme(s)
Funding Scheme
ERC-COG - Consolidator GrantHost institution
75794 Paris
France