Descripción del proyecto
Domesticar lo (matemáticamente) salvaje
Encontrar simplicidad en la complejidad o en el caos resulta esencial para comprender el mundo que nos rodea. Las matemáticas nos pueden ayudar a lograrlo. Los modelos tratan de conservar el nivel de complejidad suficiente y necesaria para describir un determinado fenómeno, a fin de reducir los tiempos de cálculo y el consumo energético de computación. En cuanto a los procesos que varían a lo largo del tiempo en la física y la biología, la teoría de sistemas dinámicos nos permite predecir la conducta del sistema posteriormente al seguir su evolución a través de diversos estados posibles. Para los sistemas caóticos en que eso no es posible, los científicos usan probabilidades y la posibilidad de que el sistema se encuentre en un estado determinado. En los sistemas caóticos «salvajes», existe un número infinito de posibilidades estadísticas, lo cual complica aún más la cuestión. El proyecto Emergence, financiado con fondos europeos, lleva a cabo el primer estudio mundial de estos casos especiales, a fin de simplificarlos y describirlos.
Objetivo
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.
Ámbito científico
Palabras clave
Programa(s)
Régimen de financiación
ERC-COG - Consolidator GrantInstitución de acogida
75794 Paris
Francia