Catastrophes and chaos in dynamic systems

The origins of chaos studies go back to the beginning of the last century, with Henri Poincaré's work on the N-body problem. Paragraph 5.3 deals with the restricted 3-body problem in gravitational interaction, a simple example of chaos in celestial mechanics. These systems are Hamiltonian systems, and we devote a section to Hamiltonian chaos (section 5), which is observed and studied, often in order to control it, in many fields such as particle gas pedals (beam collimation) or plasma physics (magnetic confinement of a fusion plasma).

The other major class of dynamic systems are dissipative systems. They have been extensively studied since the 1960s, following the work of E. Lorenz, M. Hénon, D. Ruelle, R. Thom and M. Feigenbaum. Lorenz, M. Hénon, D. Ruelle, R. Thom and M. Feigenbaum. This led to the introduction of the notions of "strange reactors" and "catastrophes". These concepts have many applications. These include fluid mechanics (instabilities and turbulence), electronics, astrophysics, chemical reactions, ecology, biology, etc. We devote two sections to these systems, depending on whether they are continuous in time (section 2) or iterated applications (section 3). Readers interested in the scientific progress in this field, from Kepler to the present day, may wish to refer to C. Letellier's book Chaos in Nature.