Équipe Commande des Systèmes - EA 3649
6, avenue du Ponceau
95014 Cergy-Pontoise Cedex
According to a quite global definition, dynamical systems gather causal, evolving systems/processes and phenomena whose evolution is governed by deterministic or probabilistic laws (the models), with respect to, at least, one varying independent parameter (usually the time). With regard to such a rather general definition, many artificial and natural processes, ranging from the spread of epidemics to stock market fluctuations, heart fibrillations, chaotic circuits or even the agents of social networks, can be considered as dynamical systems. Then, apart from economical considerations or purely scientific motivations, understanding the behavior of such systems often represents an objective of crucial importance, so as to be able to predict their short or, if possible, long term evolution. Thus, for many years, such an objective has motivated the effort of mathematicians, physicists, and many other research communities to investigate some fundamental concepts and methods for caracterising the intrinsic (mathematical) structures and behavioral properties of systems. As a result, a rather unified theoretical framework (including analysis tools and methods) has emerged and is today commonly referred to as dynamical system theory.
In order to discover this exciting scientific domain, this chapter aims at giving a brief but comprehensive overview on some central notions - mainly coming from differential geometry and the control theory - as an introduction to some properties of continuous-time (resp. discrete-time) dynamical systems defined by ordinary differential equations (resp. difference equations). To this aim, particular attention will be paid to the characterization of systems trajectories from a qualitative viewpoint, by considering various notions such as limit points and sets, closed orbits and limit cycles, trapping sets and attractors, ...