Asymptotic behavior in dynamic systems

In general, it is impossible to express the flow of a dynamic system using functions that would enable us to study its asymptotic behavior. Liouville" is credited with giving examples of very simple differential equations whose solutions cannot be represented by "elementary functions" alone, but by finite expressions using basic operators (sum, product, composition, integration). The first consequence of this is that a quantitative study will have to rely on means of approximate resolution, which have long been lacking. On the other hand, experience shows that, even when equipped with powerful computational tools, fundamental phenomena can still be missed. There are several reasons for this. Firstly, certain boundary phenomena only occur for a set of initial values of zero measurement: all the chances of avoiding these values are therefore present in a numerical simulation. This is the case for a frictionless pendulum, for which it is difficult to predict that the (unstable) vertical equilibrium is the limit of exactly two trajectories. On the other hand, boundary phenomena can be extremely complicated, and sensitive to initial conditions: this is particularly true of certain so-called chaotic systems, which even today, while being the subject of assiduous study, are by no means circumscribed. In fact, it was this type of problem (the problem of the stability of the solar system) that led "Poincaré", with obvious effectiveness, to be the first to propose powerful study techniques. Finally, dynamical systems are merely models of a physical situation, models that are sometimes consciously simplified. We need to measure how this approximation of the system by a simpler system can influence the results obtained. Here again, no matter how useful it may be to use an efficient calculator, a preliminary study is essential. In fact, dynamical systems, in their almost pure state, are a feature of many areas of mathematics today: by enabling increased numerical experimentation, computers are calling on theorists to account for ever more complex phenomena.

This article presents the elementary techniques for the asymptotic study of the flow of a dynamic system. It begins with linearization, the limits and successes of which are described in the first section. It also covers the use of "Lyapounov" functions, which enable us to study equilibrium stability efficiently in frequent cases.

The second technique is based on a more geometric study. It is effective within the restricted framework of plane dynamical systems. Within this framework, it highlights both behavioral constraints that are not found in higher dimensions (the "Poincaré" and "Bendixson" theorems), and difficulties that are characteristic of the study of complex dynamical systems, in particular the possibility that...