Self excited oscillations: from Poincaré to Andronov [ Back ]

16:10 to 16:40
UAB - Dept. Matemàtiques (C1/-128)
Jean-Marc Ginoux
Univertsité de Toulon


Till recently, the young Russian mathematician Aleksandr’ Andronov was considered by many scientists as the first to have applied the concept of limit cycle, introduced almost half a century before by Henri Poincaré, in order to state the existence of self-sustained oscillations. Consequently, if the discovery of a series of “forgotten lectures” given by Poincaré at the Ecole Supérieure des Postes et Télégraphes (today Sup’ Telecom ParisTech) in 1908 proves that he had applied his own concept of limit cycle to a problem of Wireless Telegraphy preceding thus Andronov of twenty years, it reopens the discussion of Poincaré’s French legacy in Dynamical System Theory or, more precisely in Nonlinear Oscillations Theory. Poincaré’s “forgotten lectures” will be presented here and their reception in France before the First World War and in the 1920’s in the French engineers community will be analyzed. Then, a special attention will be paid to the role of Jean-Baptiste Pomey who asked Elie Cartan to solve a problem of sustained oscillations in an electrotechnics oscillator and, to the particular case of Alfred Li ́enard who proved, under certain conditions, existence and uniqueness of a periodic solution for such an oscillator without having regarded this periodic solution as a “Poincaré’s limit cycle”. Starting from 1929, Andronov’s note at the Comptes-Rendus seems to have become the reference in terms of connection with Poincaré’s works. So, the reception of Andronov’s results by the French scientific community will be briefly analyzed in considering a selection of works published in France between 1929 to 1943. At last, the fact that Poincaré’s lectures have been forgotten as well as the fact that neither Cartan nor Liénard have made any connection with Poincaré’s works although they have obviously used some of them, will be discussed. Moreover, the role of Jacques Hadamard in the diffusion in France and in Spain of the works of the Russian schools and of Poincaré’s methods will be also pointed out. Thus, the question of Poincaré’s legacy will appear in a new perspective.