8. The Paradigm of Relaxation Oscillations in France

Actually, the non-algebricity of the limit cycle of Van der Pol’s equation (3.​8) considered by Y. A. Mitropolskii (1965) was demonstrated by Odani (1995).

The curve (\(\Gamma\)) corresponds to the oscillation characteristic of the triode modeled by Van der Pol (1920, 703) with a cubic function (see supra Part I).

Nevertheless, it was in the Mécanique section that all of Haag’s notes will be published in the C.R.A.S.

The function λ is a graphical representation of the curve (\(\Gamma\)), i.e. the cube root Van der Pol (1920, 703) used to schematize the triode’s property (see supra Part I).

See Haag (1934b).

Haag referred to his previous note to the C.R.A.S. See Haag (1936).

External ballistics is the branch that studies the free flight of the projectiles without internal propulsion, as it was the case for the artillery during WWI.

According to Robert D’Adhémar (1934, 24) it was Félix Hélie (1795–1885) and Pierre Henri Hugoniot (1851–1887) who recommended successive arc calculations, derived from the Euler method, in ballistics.

Haag has chosen a parameter ɛ which is exactly the inverse of that used by Van der Pol (1926d) in order to allow asymptotic expansions. So, he obtained relaxtion oscillations for ɛ ≪ 1 and not for ɛ ≫ 1 as Van der Pol (1926d, 981).

In the case of van der Pol’s equation (3.​8) (1926d, 979), Haag chose \(\lambda = \frac{x^{3}} {3} - x\) (see infra).

This type of representation was previously used by Rocard (1937a). Later, it was used by Stoker (1950, 141) who used Van der Pol’s “piecewise linear model”. See infra.

Actually, if we compare the development in Haag’s essays (1943, 1944) with the results of his notes published in the C.R.A.S. (Haag 1934b, 1936, 1937, 1938) we see that as early as 1938 he had already calculated the amplitude, if not the period of relaxation oscillations.

The second volume was published after his death in 1955 and written by Jean Chazy (1882–1955).

See Haag (1952–1955, 204).

Even though he did not mention them, he found Blondel’s results (1919b) for the amplitude and period of the oscillations (see supra Part I).

See supra, Rocard (1935a, 402).

The complex case is the only one likely to produce undamped oscillations.

It was established, however, that Rocard attended the first Conférence Internationale de Non-linéaire in Paris in 1933 under the supervision of Van der Pol. See infra.

Self-oscillations.

It seems it was the only occurrence of this expression in Rocard’s whole book (1937a).

“Theory of oscillations”.

Rocard’s Figure 2 (1941, 32) represents a device analogous to the one presented by Van der Pol (1930, 307) (see supra) and implemented by Le Corbeiller (1931a, 42) (see supra).

With dimensions superior or equal to three.

Similarly to Jean Abelé, see Rocard’s example (1943, 218).

A printing error occurred in the first original version entitled “Stabilisation des oscillations de relation” (“stabilization of relation oscillations”).

Telecom ParisTech, Alumni, Directory 2009 (internal document).

Quartz is a mineral component presenting the particularity of oscillating with a stable frequency when it is traversed by an electric current. Piezoelectric properties of the quartz allow the possibility to obtain very accurate oscillation frequencies.

Panc-Tcheng Kao referred to Van der Pol (1930).

This corresponds to the case of relaxation oscillations. The case of ɛ ≪ 1 is the one for sinusoidal oscillations.

The action potential consists in a sudden modification of the resting potential. This is an electrical phenomenon produced by an electrical stimulation of a nerve cell.

Fessard refers to Adrian (1930). Edgar Douglas Adrian (1889–1977) was an English physiologist, who obtained the Nobel Prize in Medicine in 1932.

Fessard refers to Rayleigh (1883).

Fessard refers to Van der Pol (1926d, 1930).

It is an ammeter with a mobile frame, for which the pointer was replaced with a mirror reflecting a beam, indicating the value of the electric current’s voltage.

Hochard referred to Van der Pol (1930).

However, it was established that he presented in 1929 an essay entitled “Étude expérimentale de l’effet Peltier et de l’effet Thomson” at the science faculty of the University of Paris to obtain his Diplôme d’Études Supérieures Sciences Physiques.

It is a discharge chamber filled with gas, containing a coil at the cathode, a plate at the anode, and one or more grids. An inert gas or metallic vapor fills the discharge chamber. In the case of the RCA 885, the gas used is Xenon. The grid only controls the current’s start, and thus provides a triggering effect.

Hose adapted an opening for the flow of a fluid under pressure in order to modify the output or form of a jet.

Bourrières refers to Le Corbeiller (1931a).

From a theory of dynamical systems point of view, this would mean that for a same given initial condition, a trajectory may have two different futures. See Bergé (1988, 120).

Thin reed or metal strip placed on various wind instruments which are made to vibrate from the passage of the air produces the sound.

Auger referred to chapter VI of Foch’s work (1935, 55) dedicated to relaxation oscillations.

Defended on the 17th of June 1943 before Misters Bruhat (President), Foch and Rocard (Examiners).

In 1944, he was called to sit at the Commission de Mécanique du C.N.R.S. After WWII, he became a member of the Academy of Science in 1949. He was honored with many distinctions: Ancel prize (1929), prix Montyon de Mécanique de l’Institut de France (1930) then Chevallier (1946) of the Société des Ingénieurs Civils which he later presided over.

The numbers of the equations correspond to their validity domain. See Fig. 8.20.

Summarized by Ginoux (2009, Ch. 5, 85).

In the case of a first order equation, it is a general integral. See Ginoux (2009, 94).

Hamburger refers to the original dutch version of Van der Pol’s article ( 1928a,b) on heart beats: De hartslag als relaxatietrillingen en een electrisch model van het hart, Werk. Genoot. Nat.-, Genees- en Heelk, 12 (1928), 614–618.

This principle states that populations of two species exploiting a same resource cannot coexist indefinitely in a stable and homogenous environment, and the most competitive species will eliminate the other sooner or later.

Periodic cycle law, law of conservation of means, law of perturbation of means (Volterra 1931, 19–27). From Israel (1996, 64), Gause, who seemed to be able to verify the first law, wrote to Volterra, who showed support. Unfortunately, what followed was disappointing.

Alpatov (1898–1979) was the student of the statistician Raymond Pearl (1879–1940), and greatly influenced Gause.

See Gause (1934, 1935) and Gause et al. (1936).

He also proposed models containing at least three species.

Which is expressed by a saturation of the predation.

We were informed of the existence of this film by Prof. Giorgio Israel (1996, 63), and a copy was kindly sent to us by Prof. Pascal Acot and Drouin (1997, 47).

Indeed, it was shown that Lefschetz’s translation differed strongly from Andronov and Khaikin’s original work (1937).

It was also translated by Lefschetz and was more faithful to the original.

These essays have been “found” based on bibliographical elements. The French dissertations found between these two wars have been digitalized and are yet accessible at: http://​www.​numdam.​org.

The examining committee was not specified in the document sent by the I.N.I.S.T.

Personal communication with Mr. L. Zadeh.

See Camichel (1927) and Camichel et al. (1927).

See Fremont (1915).

Nowadays, this question is still open.

Bénard-Kàrmàn vortex street is a periodic pattern of vortices cause by the unstable separation of a flow around a body (generally a cylinder), and forming in its wake. See Fig. 8.25.

Castagnetto refers to Bénard (1908).

Castagnetto refers to Karman and Rubach (1912).

Castagnetto refers to Foch (1935).

Castagnetto refers to ( 1926c, 1927c).

D represents the diameter of the obstacle (cylinder) and V the fluid velocity.

Jean Abelé (1886–1961) was a physicist, philosopher and writer. He first studied at the Collège Saint-Joseph in Reims before joining the Compagnie de Jésus. He then obtained a Licence ès Lettres and studied philosophy and science at Saint-Hélier (Jersey isle) then at the École d’Antoing (Belgium). He completed his specialization in Physics in Paris, under the supervision of Édouard Branly, and became licencié ès Sciences. He was mobilized during WWI along with General Gustave Ferrié in the section de radiogogniométrie (wireless telegraphy). He then studied theology at Enghien and was appointed Physics professor in Vals (1923–1961). He worked in a modest laboratory but nonetheless had several of his inventions patented, such as a device to receive and amplify the high frequency of electric oscillations (1924), and a receiver for telegraphy and telephony (1926). He obtained a doctorat ès Sciences de la Faculté des Sciences at the University of Paris in 1943. His thesis dissertation won in 1948 the Prix du Général Ferrié. In 1948 he was appointed director of studies in Vals. In 1945, he became member of the Centre national de la Recherche Scientifique. In 1949, he presented his work at the Congrès International de Philosophie des Sciences held in Paris. Aside from the many studies he published in the periodic Études, he colaborated with the Archives de Philosophie, he was a member of its executive committee.

Abelé refers to Van der Pol (1934).

Abelé refers to Andronov (1929a).

A simple permutation of the variables x and y leads to the equation corresponding to the shunt circuit.

In 1955, he was Chef de travaux of the Laboratoire de Radioélectricité. In 1965 he was appointed Director of studies of the École d’ingénieurs de Grenoble. He then became the very first director of the Institut Universitaire Technologique (I.U.T.) of Nice in 1970 and participated in the creation of the Laboratoire d’Électronique de Nice in 1973.

It is strange that his work was not mentioned by Moussiegt.