Generic behaviours of conservative dynamical systems

Abstract

Dynamical systems is the study of the long-term behaviour of evolving systems. The foundations were set by the innovative work of Henri Poincaré (1854 - 1912), Les Méthodes Nouvelles de la Mécanique Céleste (1892-1899), with fundamental questions concerning the stability of the solar system. Attempts to answer those questions revealed the incapacity of solving exactly the mathematical questions arising from physical systems. It appeared that understanding typical systems, or systems in general, was mathematically more fruitful. The aim of the modern theory of dynamical systems is thus to describe the behaviour of typical trajectories, for typical evolution laws. Furthermore, when dealing with real-world systems, neither the initial data nor the evolution law are known exactly. For these reasons, one is most interested in properties that are stable, i.e., that are persistent under small perturbations of the evolution law.