Henri Poincar\'e et l'\'emergence du concept de cycle limite

TL;DR

This paper explores Henri Poincaré's early development of the limit cycle concept, its significance in nonlinear oscillations, and its application to wireless technology, challenging previous attribution to A. Andronov.

Contribution

It uncovers Poincaré's prior use of the limit cycle concept in 1908, emphasizing its historical importance in nonlinear dynamics and oscillation theory.

Findings

Poincaré introduced the limit cycle concept in 1882.

He applied it to wireless oscillations in 1908.

The concept predates and informs later work by Andronov.

Abstract

The concept of "limit cycle" was introduced by Henri Poincar\'e in his second memoir "On curves defined by a differential equation" in 1882. From the point of view of physics, a stable limit cycle (or attractive) is the representation of the periodic solution of a (mechanical or electrical) dissipative system whose oscillations are maintained by the system itself. Conversely, the existence of a stable limit cycle ensures the maintenance of the oscillations. So far, the historiography considered that the Russian mathematician Aleksandr Andronov was the first to establish such a correspondence between the periodic solution of a self-oscillating system and Poincar\'e's concept of limit cycle. The recent discovery of a series of lectures given by Henri Poincar\'e in 1908 at the School of Post and Telegraph (now Telecom Paris Tech) proves that he had already used his limit cycle concept to establish the existence of a regime of maintained waves in a wireless device of radiotechnics. This article aims on the one hand to trace the emergence of this concept since its creation by Poincar\'e and the other hand to highlight its importance and role in the history of non-linear oscillations.