From Nonlinear Oscillations to Chaos Theory

Part of the Understanding Complex Systems book series (UCS)


In this work we propose to reconstruct the historical road leading from nonlinear oscillations to chaos theory by analyzing the research performed on the following three devices: the series-dynamo machine, the singing arc and the triode, over a period ranging from the end of the nineteenth century till the end of the Second World War.

Thus, it will be shown that the series-dynamo machine, i.e. an electromechanical device designed in 1880 for experiments, enabled to highlight the existence of sustained oscillations caused by the presence in the circuit of a component analogous to a “negative resistance”.

The singing arc, i.e. a spark-gap transmitter used in Wireless Telegraphy to produce oscillations and so to send messages, allowed to prove that, contrary to what has been stated by the historiography till recently, Poincaré made application of his mathematical concept of limit cycle in order to state the existence of sustained oscillations representing a stable regime of sustained waves necessary for radio communication.

During the First World War, the singing arc was progressively replaced by the triode and in 1919, an analogy between series-dynamo machine, singing arc and triode was highlighted. Then, in the following decade, many scientists such as André Blondel, Jean-Baptiste Pomey, Élie and Henri Cartan, Balthasar Van der Pol and Alfred Liénard provided fundamental results concerning these three devices. However, the study of these research has shown that if they made use of Poincaré’s methods, they did not make any connection with his works.

In the beginning of the 1920s, Van der Pol started to study the oscillations of two coupled triodes and then, the forced oscillations of a triode. This led him to highlight some oscillatory phenomena which have never been observed previously. It will be then recalled that this new kind of behavior considered as “bizarre” at the end of the Second World War by Mary Cartwright and John Littlewood was later identified as “chaotic”.


Periodic Solution Nonlinear Oscillation Closed Curve Relaxation Oscillation Stable Limit Cycle 
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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Archives Henri Poincaré, CNRSNancyFrance
  2. 2.Laboratoire LSIS, CNRSToulonFrance

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