At the beginning of the twentieth century while Henri Poincaré (1854-1912) was already deeply involved in the developments of wireless telegraphy, he was invited, in 1908, to give a series of lectures at the Ecole Supérieure des Postes et Télégraphes (today Sup’Télecom). In the last part of his presentation he established that the necessary condition for the existence of a stable regime of maintained oscillations in a device of radio engineering completely analogous to the triode: the singing arc, is the presence in the phase plane of stable limit cycle.This work was intended for engineers, and was not included in his complete works.
The aim of this work is to prove that the correspondence highlighted by Andronov between the periodic solution of a non-linear second order differential equation and Poincaré’s concept of limit cycle has been carried out by Poincaré himself, twenty years before in these forgotten conferences of 1908.
Physics books often discount Poincare as a mathematician who did not really understand physics like electromagnetism. However, the record shows that he understood it better than Einstein or anyone else in Europe:
During the last two decades of his life, Poincaré had been involved in many research on the propagation of electromagnetic waves. In 1890, he wrote to Hertz to report a miscalculation in his famous experiments. Three years later, he solved the telegraphists equation [Poincaré, 1893]. The following year he published a book entitled: “Oscillations électriques” [Poincaré, 1894] and in 1899 another one: “La Théorie de Maxwell et les oscillations hertziennes” [Poincaré, 1899]. This book, also published in English and German in 1904 and reprinted in French in 1907, has been considered as a reference. In Chapter XIII, page 79 Poincaré stated that the singing arc and the Hertz spark gap transmitter were also analogous except that oscillations are maintained in the first and damped in the second. Thus, from the early twentieth century until his death Poincaré continued his research on wireless telegraphy and on maintained waves and oscillations [Poincaré, 1901, 1902, 1903, 1904, 1907, 1908, 1909abcde, 1910abc, 1911,1912].