Dynamical Systems, Chaotic Behaviour – Uncertainty,
Linear Cocycles and Lyapunov Exponents
He will coordinate his Balzan Research Project together with Jean-Christophe Yoccoz at the National Institute for Pure and Applied Mathematics, IMPA, Rio de Janeiro, RJ, Brazil.
The creation of the modern theory of dynamical systems, towards the end of the nineteenth century, is attributed to Henri Poincaré. It is the principal mathematical approach used to model the evolution of many phenomena in nature. Classical examples are population growth of species, weather and climate prediction. Perhaps the same theory can be applied to understand certain aspects of turbulence in physics. Since Poincaré we have been wondering if it is possible to understand the typical behaviour of a typical dynamical system, where typical should be understood in a probabilistic sense to cover almost all possibilities.
Starting from a selected initial position of the system, one tries to describe the behaviour of its future trajectory, defined by its successive positions as time evolves. For example, the motion of the atmosphere is governed by a very complicated evolution equation, which cannot be solved explicitly. In 1963, Edward Norton Lorenz, a theoretical meteorologist, proposed a “toy” weather model, involving only three dimensions and intended to be much easier to understand. The question of knowing whether this oversimplified model still captures the main properties of the actual atmospheric motion is controversial among physicists and meteorologists. However, Lorenz was able to observe “chaotic behaviour” in his “toy” model. Minute changes in the initial data used, were shown to produce extremely radical changes in the outcome. This was very surprising at the time. Jacob Palis’ research project proposes to tackle several conjectures which would imply that the phenomenon witnessed by Lorenz is not an exception but, on the contrary, may capture some fundamental features of general dynamics. The research project will study (and hopefully prove) a set of conjectures for dynamical systems that leads to a global perspective in this important branch of Mathematics.
The Research Project will take place in the period 2011-2015. Part of the funds of the project will support the activities of young researchers at IMPA in research on Dynamical Systems, Chaotic Behaviour and Uncertainty. Also, as part of the project, three Balzan Symposia will take place, two of them at IMPA and one at the Institut Henri Poincaré in Paris, in subsequent years, starting in 2012. They are designed to review advances and to stimulate further progress along the lines of the research project. There will be a number of fellowships (3-12 months).
Some basic references:
[BDV] C. Bonatti, L. Diaz and M. Viana, Dynamics beyond uniform hyperbolicity. A global geometric and probabilistic perspective, Encyclopaedia of Mathematical Sciences, vol. 102, Springer Verlag, 2004;
[P00] J. Palis, A global view of dynamics and a conjecture on the denseness of finitude of attractors, Astérisque No. 261, 335–347, 2000;
[P05] J. Palis, A global perspective for non-conservative dynamics, Ann. IHP, vol. 22, 485–507, 2005;
[PY09] J. Palis and J.-C. Yoccoz, Non-uniformly hyperbolic horseshoes arising from bifurcations of Poincaré heteroclinic cycles, Publications IHES No. 110, 1–217, 2009.