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From Nonlinear Oscillations to Chaos Theory

  • Jean-Marc Ginoux
Chapter
Part of the Understanding Complex Systems book series (UCS)

Abstract

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”.

Keywords

Periodic Solution Nonlinear Oscillation Closed Curve Relaxation Oscillation Stable Limit Cycle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    A.A. Andronov, Open image in new window in IVs’ezd ruskikh fizikov (5-16.08, p. 23–24). (Poincaré’s limit cycles and the theory oscillations), this report has been read guring the IVth congress of Russian physicists in Moscow between 5 to 16 August 1928, p. 23–24Google Scholar
  2. 2.
    A.A. Andronov, Les cycles limites de Poincaré et la théorie des oscillations auto-entretenues. C. R. Acad. Sci. 189, 559–561 (1929)Google Scholar
  3. 3.
    A. Blondel, Amplitude du courant oscillant produit par les audions générateurs. C. R. Acad. Sci. 169, 943–948 (1919)Google Scholar
  4. 4.
    E. Cartan, H. Cartan, Note sur la génération des oscillations entretenues. Ann. P. T. T. 14, 1196–1207 (1925)Google Scholar
  5. 5.
    M.L. Cartwright, J. Littlewood, On non-linear differential equations of the second order, I: the equation \(\ddot{y} - k\left (1 - y^{2}\right )\dot{y} + y = b\lambda kcos\left (\lambda t + a\right )\), k large. J. Lond. Math. Soc. 20, 180–189 (1945)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    M.L. Cartwright, J. Littlewood, On nonlinear differential equations of the second order, II: the equation \(\ddot{y} - kf\left (y\right )\dot{y} + g\left (y,k\right ) = p\left (t\right ) = p_{1}\left (t\right ) + kp_{2}\left (t\right )\); k > 0, \(f\left (y\right ) \geq 1\). Ann. Math. 48, 472–494 (1947)Google Scholar
  7. 7.
    M.L. Cartwright, J. Littlewood, Errata. Ann. Math. 49, 1010 (1948)MathSciNetCrossRefGoogle Scholar
  8. 8.
    M.L. Cartwright, J. Littlewood, Addendum. Ann. Math. 50, 504–505 (1949)MathSciNetCrossRefGoogle Scholar
  9. 9.
    M.L. Cartwright, Non-linear vibrations: a chapter in mathematical history. Presidential address to the mathematical association, January 3, 1952. Math. Gazette 36(316), 81–88 (1952)Google Scholar
  10. 10.
    W. du Bois Duddell, On rapid variations in the current through the direct-current arc. J. Inst. Electr. Eng. 30(148), 232–283 (1900)Google Scholar
  11. 11.
    W. du Bois Duddell, On rapid variations in the current through the direct-current arc. J. Inst. Electr. Eng. 46, 269–273, 310–313 (1900)Google Scholar
  12. 12.
    Th. du Moncel, Réactions réciproques des machines dynamo-électriques et magnéto-électriques. La Lumière Électrique 2(17), 352 (1880)Google Scholar
  13. 13.
    J.M. Ginoux, L. Petitgirard, Poincaré’s forgotten conferences on wireless telegraphy. Int. J. Bifurcation Chaos 20(11), 3617–3626 (2010)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    J.M. Ginoux, Analyse mathématiques des phénomènes oscillatoires non linéaires. Thèse, Université Pierre & Marie Curie, Paris VI (2011)Google Scholar
  15. 15.
    J.M. Ginoux, The first “lost” international conference on nonlinear oscillations (I.C.N.O.). Int. J. Bifurcation Chaos 4(22), 3617–3626 (2012)Google Scholar
  16. 16.
    J.M. Ginoux, C. Letellier, Van der Pol and the history of relaxation oscillations: toward the emergence of a concepts. Chaos 22, 023120 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    J.M. Ginoux, Self-excited oscillations: from Poincaré to Andronov. Nieuw Archief voor Wiskunde (New Archive for Mathematics). Journal published by the Royal Dutch Mathematical Society (Koninklijk Wiskundig Genootschap) 13(3), 170–177 (2012)Google Scholar
  18. 18.
    J.M. Ginoux, R. Lozi, Blondel et les oscillations auto-entretenues. Arch. Hist. Exact Sci. 66(5), 485–530 (2012)CrossRefzbMATHGoogle Scholar
  19. 19.
    J.M. Ginoux, Histoire de la théorie des oscillations non linéaires (Hermann, Paris, 2015)Google Scholar
  20. 20.
    J.M. Ginoux, History of Nonlinear Oscillations Theory. Archimede, New Studies in the History and Philosophy of Science and Technology (Springer, New York, 2016)Google Scholar
  21. 21.
    J.M.A. Gérard-Lescuyer, Sur un paradoxe électrodynamique. C. R. Acad. Sci. 168, 226–227 (1880)Google Scholar
  22. 22.
    J.M.A. Gérard-Lescuyer, On an electrodynamical paradox. Philos. Mag V 10, 215–216 (1880)CrossRefGoogle Scholar
  23. 23.
    J. Guckenheimer, K. Hoffman, W. Weckesser, The forced Van der Pol equation i: the slow flow and its bifurcations. SIAM J. Appl. Dyn. Syst. 2(1), 1–35 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    P. Janet, Sur les oscillations électriques de période moyenne. J. Phys. Theor. Appl. 2(1), 337–352 (1893)CrossRefGoogle Scholar
  25. 25.
    P. Janet, Sur une analogie électrotechnique des oscillations entretenues. C. R. Acad. Sci. 168, 764–766 (1919)Google Scholar
  26. 26.
    Ph. Le Corbeiller, Non-linear theory of maintenance of oscillations. J. Inst. Electr. Eng. 79, 361–378 (1936)Google Scholar
  27. 27.
    A. Liénard, Étude des oscillations entretenues. Revue générale de l’Electricité 23, 901–912, 946–954 (1928)Google Scholar
  28. 28.
    A. Lyapounov, Problème général de la stabilité du mouvement. Annales de la faculté des sciences de Toulouse, Sér. 2 9, 203–474 (1907) [Originally published in Russian in 1892. Translated by M. Édouard Davaux, Engineer in the French Navy à Toulon]Google Scholar
  29. 29.
    N. Papaleksi, Open image in new window in Internatinal Conference On Nonlinear Process, Paris, 28–30 January 1933. Z. Tech. Phys. 4, 209–213 (1934)Google Scholar
  30. 30.
    H. Poincaré, Sur les courbes définies par une équation différentielle. Journal de mathématiques pures et appliquées 3(7), 375–422 (1881)zbMATHGoogle Scholar
  31. 31.
    H. Poincaré, Sur les courbes définies par une équation différentielle. Journal de mathématiques pures et appliquées 3(8), 251–296 (1882)zbMATHGoogle Scholar
  32. 32.
    H. Poincaré, Notice sur les Travaux Scientifiques de Henri Poincaré (Gauthier-Villars, Paris, 1884)Google Scholar
  33. 33.
    H. Poincaré, Sur les courbes définies par une équation différentielle. Journal de mathématiques pures et appliquées 4(1), 167–244 (1885)zbMATHGoogle Scholar
  34. 34.
    H. Poincaré, Sur les courbes définies par une équation différentielle. Journal de mathématiques pures et appliquées 4(2), 151–217 (1886)zbMATHGoogle Scholar
  35. 35.
    H. Poincaré, Les Méthodes Nouvelles de la Mécanique Céleste, vols. I–III (Gauthier-Villars, Paris, 1892, 1893, 1899)Google Scholar
  36. 36.
    H. Poincaré, La théorie de Maxwell et les oscillations hertziennes: la télégraphie sans fil, 3rd edn. (Gauthier-Villars, Paris, 1907)zbMATHGoogle Scholar
  37. 37.
    H. Poincaré, Sur la télégraphie sans fil. La Lumière Électrique 2(4), 259–266, 291–297, 323–327, 355–359, 387–393 (1908)Google Scholar
  38. 38.
    H. Poincaré, in Conférences sur la télégraphie sans fil (La Lumière Électrique éd., Paris, 1909)Google Scholar
  39. 39.
    J.B. Pomey, Introduction à la théorie des courants téléphoniques et de la radiotélégraphie (Gauthier-Villars, Paris, 1920)Google Scholar
  40. 40.
    B. Van der Pol, A theory of the amplitude of free and forced triode vibrations. Radio Rev. 1, 701–710, 754–762 (1920)Google Scholar
  41. 41.
    B. Van der Pol, Gedwongen trillingen in een systeem met nietlineairen weerstand (Ontvangst met teruggekoppelde triode). Tijdschrift van het Nederlandsch Radiogenootschap 2, 57–73 (1924)Google Scholar
  42. 42.
    B. Van der Pol, Over Relaxatietrillingen. Physica 6, 154–157 (1926)Google Scholar
  43. 43.
    B. Van der Pol, Over “Relaxatie-trillingen”. Tijdschrift van het Nederlandsch Radiogenootschap 3, 25–40 (1926)Google Scholar
  44. 44.
    B. Van der Pol, Über “Relaxationsschwingungen”. Jahrbuch der drahtlosen Telegraphie und Telephonie 28, 178–184 (1926)Google Scholar
  45. 45.
    B. Van der Pol, On “relaxation-oscillations”. Lond. Edinb. Dublin Philos. Mag. J. Sci. VII 2, 978–992 (1926)CrossRefGoogle Scholar
  46. 46.
    B. Van der Pol, Über Relaxationsschwingungen II. Jahrbuch der drahtlosen Telegraphic und Telephonie 29, 114–118 (1927)Google Scholar
  47. 47.
    B. Van der Pol, Forced oscillations in a circuit with non-linear resistance (reception with reactive triode). Lond. Edinb. Dublin Philos. Mag. J. Sci. VII 3, 65–80 (1927)CrossRefGoogle Scholar
  48. 48.
    B. Van der Pol, J. van der Mark, Frequency demultiplication. Nature 120, 363–364 (1927)ADSCrossRefGoogle Scholar
  49. 49.
    B. Van der Pol, Oscillations sinusoïdales et de relaxation. Onde Électrique 9, 245–256, 293–312 (1930)Google Scholar
  50. 50.
    A. Witz, Des inversions de polarités dans les machines série-dynamos. C. R. Acad. Sci. 108, 1243–1246 (1889)Google Scholar
  51. 51.
    A. Witz, Recherches sur les inversions de polarité des série-dynamos. J. Phys. Theor. Appl. 8(1), 581–586 (1889)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Jean-Marc Ginoux
    • 1
    • 2
  1. 1.Archives Henri Poincaré, CNRSNancyFrance
  2. 2.Laboratoire LSIS, CNRSToulonFrance

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