Chaos and fractals

Lecturer(s): Christophe BAILLY, Didier DRAGNA
Course ⋅ 22 hAutonomy ⋅ 10 h


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, 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.


Dynamical systems, stability, bifurcations, limit cycle, strange attractor, chaos, fractal dimensions, Lyapunov exponents, control, identification and reconstruction.


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 for maps, Lyapunov exponents for dynamical systems, long-time prediction. 5-Chaos in Hamiltonian systems: illustration in celestial mechanics, two- and restricted three-body problems; some properties of Hamiltonian systems, resonances, KAM theorem, stability of the solar system. 6-Control of chaos: motivation, algorithms and illustrations. 7-Identification and reconstruction from time series.

Learning Outcomes

  • To know the fundamental concepts for studying dynamic systems
  • To appropriate these concepts through simple numerical case studies
  • Be able to explain the behavior of nonlinear dynamic systems


Knowledge 50%, Know-how 50%;
Knowledge = 40% Final written exam + 60% Assigned homework; Know-how = 40% Final written exam + 60% Assigned homework;

No second session for the final exam; Assigned homework: exercises and computer projects (by pairs of students)