Poincare's forgotten conferences on wireless telegraphy

TL;DR

This paper uncovers that Henri Poincaré's 1908 lectures contained early insights into the concept of limit cycles in nonlinear differential equations, predating and aligning with later developments in dynamical systems theory.

Contribution

It demonstrates that Poincaré's 1908 lectures contained the first known connection between limit cycles and stable oscillations in radio engineering devices.

Findings

Poincaré's lectures predate and align with Andronov's work on limit cycles.

He identified the role of stable limit cycles in maintained oscillations.

The work links early wireless telegraphy theory with modern dynamical systems concepts.

Abstract

At the beginning of the twentieth century while Henri Poincar\'e (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 Sup\'erieure des Postes et T\'el\'egraphes (today Sup'T\'elecom). 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. 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 Poincar\'e's concept of limit cycle has been carried out by Poincar\'e himself, twenty years before in these forgotten conferences of 1908.