Order, chaos and fractals

The concept of deterministic chaos has profoundly changed the way we approach the modeling of many problems. Poincaré's three body problem in celestial mechanics and Lorenz' work in meteorology are two now famous emblematic examples. The course introduces the main ideas and theoretical notions used to describe the behavior of these chaotic, and therefore nonlinear dynamical systems. A small number of effective degrees of freedom is very often sufficient to observe chaos, which makes the mathematical analysis affordable. The field of application was historically rather that of mechanics, but all fields of physics and even beyond (biology, medicine, economics, social sciences) are concerned, as will be illustrated in the course as well as in the case studies.

1. Introduction to dynamical systems.
2. Stability of equilibria. Lyapunov stability, fixed points, limit cycles, Poincaré-Bendixon theorem, canonical bifurcations, attractor.
3. Fractals: introduction, generation, percolation, dimensions.
4. Sensitivity to initial conditions: introduction, Lyapunov exponents (maps, dynamical systems), spectrum computation, long-time prediction.
5. Chaos in Hamiltonian systems: some properties of Hamiltonian systems, canonical form, resonances and small divisors, KAM theorem, illustrations taken from celestial mechanics: two- and restricted three-body problems, stability of the solar system).
6. Control of chaos: motivation, algorithms and illustrations.
7. Identification and reconstruction from time series.
8. Philosophical final remarks on modeling.