Relaxation oscillations are commonly associated with the name of Balthazar van der Pol via his paper (Philosophical Magazine, 1926) in which he apparently introduced this terminology to describe the nonlinear oscillations produced by self-sustained oscillating systems such as a triode circuit. Our aim is to investigate how relaxation oscillations were actually discovered. Browsing the literature from the late 19th century, we identified four self-oscillating systems in which relaxation oscillations have been observed: (i) the series dynamo machine conducted by Gérard-Lescuyer (1880), (ii) the musical arc discovered by Duddell (1901) and investigated by Blondel (1905), (iii) the triode invented by de Forest (1907), and (iv) the multivibrator elaborated by Abraham and Bloch (1917). The differential equation describing such a self-oscillating system was proposed by Poincaré for the musical arc (1908), by Janet for the series dynamo machine (1919), and by Blondel for the triode (1919). Once Janet (1919) established that these three self-oscillating systems can be described by the same equation, van der Pol proposed (1926) a generic dimensionless equation which captures the relevant dynamical properties shared by these systems. Van der Pol’s contributions during the period of 1926-1930 were investigated to show how, with Le Corbeiller’s help, he popularized the “relaxation oscillations” using the previous experiments as examples and, turned them into a concept.
REFERENCES
1.
2.
M.-L.
Cartwright
, J. Lond. Math. Soc.
35
, 367
(1960
).3.
B.
van der Pol
, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science
, 7
(2
), 978
(1926
).4.
M.-L.
Cartwright
and J. E.
Littlewood
, 20, 180 (1945
).5.
G.
Israel
, “Technological innovations and new mathematics: Van der Pol and the birth of non-linear dynamics
,” in Technological concepts and mathematical models in the evolution of modern engineering systems
, edited by M.
Lucertini
, A. M.
Gasca
, and F.
Nicolo
(Birkhäuser
, Basel
, 2004
), pp. 52
–78
.6.
D.
Aubin
and A.
Dahan
, Hist. Math.
29
, 273
(2002
).7.
A.
Dahan Dalmedico
, Science in Context
, 17
, 235
(2004
).8.
S.
Diner
, “Les voies du chaos déterministe dans l’école russe
,” in Chaos et déterminisme
, edited by A.
Dahan-Dalmedico
, J.-L.
Chabert
, and K.
Chemla
(Seuil
, Paris
, 1992
), pp. 331
–368
.9.
R.
Merton
, Science
159
, 56
(1968
).10.
J. M.
Ginoux
, “Analyse mathématiques des phénomènes oscillatoires non linéaires
,” Ph.D. thesis, Université of Paris vi
, 2011
.11.
12.
P.
Janet
, C. R. Acad. Sci.
168
, 764
(1919
).13.
P.
Janet
, Ann. des P.T.T.
14
, 1193
(1925
).14.
W.
Duddell
, J. Inst. Electr. Eng.
30
(148
), 232
(1900
).15.
W.
Duddell
, Science
26
, 815
(1907
).16.
17.
H.
Luggin
, Centralblatt für Elektrotech.
10
, 567
(1888
).18.
W.
Duddell
, The Electrician
52
, 902
(1901
).19.
A.
Blondel
, La Lumière électrique
43
(2
), 51
(1892
).20.
A.
Blondel
, Éclairage Électrique
44
(28
), 41
–81
(1905
).21.
H.
Poincaré
, Journal de Mathématiques Pures et Appliquées
, 2
, 151
(1886
).22.
H.
Poincaré
, Lumière électrique
, 4
, 259
–291
323
–355
(1908
).23.
A. A.
Andronov
, C. R. Acad. Sci.
189
, 559
(1929
).24.
J. M.
Ginoux
and L.
Petitgirard
, Int. J. Bifurcation Chaos
11
(20
), 3617
(2010
).25.
L.
de Forest
, US patent 836070 (1906
);L.
de Forest
, US patent 841386 (1907
).26.
H.
Abraham
and E.
Bloch
, Ann. Phys. (Paris)
9
, 237
(1919
).27.
E.
Cartan
and H.
Cartan
, Annales des P.T.T.
14
, 1196
(1925
).28.
A.
Witz
, C. R. Acad. Sci.
108
, 1243
(1889
).29.
A.
Blondel
, C. R. Acad. Sci.
169
, 943
(1919
).30.
B.
Van der Pol
, Radio Rev. (London)
, 1
, 701
–754
(1920
).31.
32.
E. B.
Appleton
, Proc. Cambridge Philos. Soc.
21
, 231
(1922
).33.
34.
B.
Van der Pol
, Tijdschr. van het Ned. Radiogenootschap
3
, 25
(1926
).35.
B.
Van der Pol
, Jahrbuch der drahtlosen Telegraphie und Telephonie
28
, 178
(1926
).36.
R.
Mesny
, Radio-électricité générale. i Étude des circuits, de la propagation
(E. Chiron
, Paris
, 1935
).37.
A. A.
Andronov
S. E.
Khaikin
, Theory of oscillations
(O.N.T.I
, Moscow/Leningrad
, 1937
), in Russian; It should be understood that this edition differs considerably from the one translated by Lefschetz as well as the later editions highly revised and highly augmented by N. A. Zheleztsov.38.
A. A.
Andronov
and A.
Witt
, Bull. Acad. Sci. SSSR
8
, 189
(1930
).39.
40.
B.
van der Pol
, Polytech. Weekl.
19
, 791
(1925
).41.
B.
Van der Pol
and J.
Van der Mark
, L’Onde Électrique
7
, 365
(1928
).42.
J.C.
Maxwell
, Philos. Trans. Roy. Soc. London
157
, 49
(1866
).43.
B.
Van der Pol
, Physica
6
, 154
(1926
).44.
A.
Blondel
, C. R. Acad. Sci.
128
, 877
(1899
).45.
K. T.
Compton
, Phys. Rev.
30
, 161
(1910
).46.
P.
Curie
, La Lumière Électrique
31
, 201
(1891
).47.
P.
Le Corbeiller
, Les systèmes auto-entretenus et les oscillations de relaxation
(Hermann
, Paris
, 1931
).48.
N.
Krylov
and B.
Bogoliubov
, Izd. Akad. Nauk SSSR (1937
) [in Russian, translated by S. Lefschetz, Princeton University Press, 1943].49.
B.
Van der Pol
, Onde lectrique
9
, 245
–293
(1930
).50.
A.
Liénard
, Rev. générale de l’Electricité
23
, 901
–946
(1928
).51.
B.
van der Pol
, Jahrbuch der drahtlosen Telegraphic und Telephonie
29
, 114
(1927
).52.
J.
Haag
, Annales de l’École Normale Supérieure
, III
60
–35
65
–289
(errata) (1943
).53.
J.
Haag
, Annales de l’École Normale Supérieure
, III
61
–73
(errata) (1944
).54.
B.
van der Pol
and J.
van der Mark
, Nature (London)
120
, 363
(1927
).55.
56.
G.
Duffing
, Erzwungene Schwingungen bei veranderlicher Eigenfrequenz und ihre technische Bedeutung
(Friedr. Vieweg & Sohn
, Braunschweig
, 1918
).57.
H.
Barkausen
and K.
Kurz
, Phys. Z.
21
, 1
(1920
).58.
E.
Friedländer
, Archiv fur Elektrotechnik
16
, 273
(1926
).59.
E.
Friedländer
, Archiv fur Elektrotechnik
17
(2
), 103
(1926
).60.
E.
Hudec
, Archiv fur Elektrotechnik
22
, 459
(1929
).61.
F.
Kirschstein
, Archiv fur Elektrotechnik
24
, 731
(1930
).62.
63.
Y.
Rocard
, “Sur certains types nouveaux d’oscillations mécaniques
,” in Livre jubilaire de
, edited by M.
Brillouin
(Gauthier-Villars
, Paris
, 1935
), pp. 400
–408
.64.
L. I.
Mandel’shtam
, N. D.
Papaleksi
, A. A.
Andronov
, S. E.
Chaikin
, and A. A.
Witt
, J. Phys. Tech. l’URSS
2
, 81
(1935
).65.
R. E.
O’Malley
, “Singular perturbation theory for ordinary differential equations
,” in Applied Mathematical Sciences
(Springer-Verlag
, New York
, 1991
), Vol. 89
.66.
H.
Poincaré
, Les Méthodes Nouvelles de la Mécanique Céleste
(Gauthier-Villars
, Paris
, 1892, 1895, and 1899
), 3 volumes.67.
L.
Prandtl
, “Über Fluüssigkeits bewegung bei kleiner Reibung
,” in Verhndlungen des dritten internationalen Mathematik-Kongresses
(Tübner
, Leipzig
, 1905
), pp. 484
–491
.68.
Ph.
Le Corbeiller
, Les systèmes auto-entretenus et les oscillations de relaxation
(Hermann
, Paris
, 1931
).69.
A.
Fessard
, L’Année psychologique
32
49
(1931
).70.
F.-J.
Bourrières
, “Sur un phénomène d’oscillation auto-entretenue en mécanique des fluides
,” in Publications Scientifiques et Techniques du Ministère de l’air
, edited by E.
Blondel La Rougery
(Gauthier-Villars
, Paris
, 1939
), Vol. 147
.71.
B.
van der Pol
, Acta Med. Scand.
103
(S108
), 76
(1940
).73.
L.
Hamburger
, De Economist
79
(1
), 1
(1930
); Institut de Statistique de l’Université de Paris, Supplément aux Indices du Mouvement des Affaires 9, 1 (1931).74.
A.
Kleinhoonte
, Arch. Neer. Sci. Exca. Natur. B
5
, 1
(1929
).75.
J. C.
Bose
, Transactions of the Bose Institute
, 2
, 255
(1919
).76.
A. J.
Lotka
, Proc. Natl. Acad. Sci. U.S.A.
6
(7
), 410
(1920
).77.
V.
Volterra
, Mem. R. Accad. Lincei S.
6
(2
), 31
(1926
).78.
H. A.
Boot
and J. T.
Randall
, IEEE Trans. Electron Devices
23
(7
), 724
(1976
).79.
B.
van der Pol
, Proc. Inst. Radio Eng.
22
(9
), 1051
(1934
).80.
81.
D. G.
Tucker
, J. Inst. Electr. Eng.
93
(1
), 57
(1945
).82.
R.
Adler
, Proc. Inst. Radio Engineers
34
(6
), 351
(1946
).83.
84.
S.
Smale
, Bull. Am. Math. Soc.
73
(6
), 747
(1967
).85.
86.
M.
Levi
, Mem. Am. Math. Soc.
244
, 1
(1981
).87.
Y.
Ueda
and N.
Akamatsu
, IEEE Trans. Circuit Syst.
28
(3
), 217
(1981
).88.
T. S.
Parker
and L. O.
Chua
, IEEE Trans. Circuit Syst.
30
(8
), 518
(1983
).89.
M. P.
Kennedy
and L. O.
Chua
, IEEE Trans. Circuit Syst.
33
(10
), 974
(1986
).90.
U.
Parlitz
and W.
Lauterborn
, Phys. Rev. A
36
, 1428
(1987
).91.
C.
Letellier
, V.
Messager
, and R.
Gilmore
, Phys. Rev. E
77
(4
), 046203
(2008
).© 2012 American Institute of Physics.
2012
American Institute of Physics
You do not currently have access to this content.