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History of Chaos from a French Perspective

  • Pierre Coullet
  • Yves PomeauEmail author
Chapter
Part of the Understanding Complex Systems book series (UCS)

Abstract

This review tries to explain how Chaos theory developed in France in the late seventies-early eighties, not as the result of a planned attempt to bolster a field but, as often in human matters, as a result of an unlikely convergence of various events, as well as of a long tradition in the study of nonlinear phenomena that can be traced back to Poincaré. Some general reflexions will be presented on the connection between the way Science and research are organized and the way things really work.

Keywords

Lorenz System Period Doubling Chaos Theory Nonlinear Science French System 
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.
    D. Aubin, A. Dahan Dalmonico, Writing the history of dynamical systems and chaos: longue durée and revolution, disciplines and culture. Hist. Math. 29, 1 (2002) and references thereinGoogle Scholar
  2. 2.
    H. Poincaré, Sur les courbes définies par une équation différentielle. Journal de mathématiques pures et appliquées, Série 1 7, 375 (1881); 8, 251 (1882); Série 2 1, 167 (1885) and 2, 151 (1886). Those four papers, based on Poincaré PhD thesis are a monument of the history of MathematicsGoogle Scholar
  3. 3.
    P. Coullet, Bifurcation at the dawn of Modern Science. CR Mecanique 340, 777 (2012). We can only urge interested readers to read this beautiful piece on Science of classical timesGoogle Scholar
  4. 4.
    J.M. Ginoux, History of Nonlinear Oscillations Theory (1880–1940, to appear)Google Scholar
  5. 5.
    Y. Rocard, Dynamique générale des vibrations (Dunod, Paris, 1971)zbMATHGoogle Scholar
  6. 6.
    N. Levinson, Transformation theory of nonlinear differential equations of the second order. Ann. Math. 45, 723 (1944)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    D. Ruelle, F. Takens, On the nature of turbulence. Commun. Math. Phys. 20, 167 (1971)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    M. Hénon, C. Heiles, The applicability of the third integral of motion: some numerical experiments. Astrophys. J. 69, 73 (1964)MathSciNetGoogle Scholar
  9. 9.
    M. Hénon, A two-dimensional mapping with a strange attractor. Commun. Math. Phys. 50, 69 (1976)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    J.L. Ibanez, Y. Pomeau, Simple case of non-periodic (strange) attractor. J. Non-Equilib. Thermodyn. 3, 135 (1978)ADSzbMATHGoogle Scholar
  11. 11.
    J. Laskar, Large-scale Chaos in the solar system. Astron. Astrophys. 287, L9 (1994)ADSGoogle Scholar
  12. 12.
    C. Mira, Nonlinear maps from Toulouse colloquium (1973) to Noma’13, in Nonlinear Maps and Their Applications, edited by R. Lopez-Ruiz et al. Springer Proceedings in Mathematics and Statistics, vol. 112 (Springer, Heidelberg, 2014)Google Scholar
  13. 13.
    R. May, Simple mathematical models with complicated dynamics. Nature 261, 459 (1976)ADSCrossRefGoogle Scholar
  14. 14.
    L.D. Landau, On the problem of turbulence. Dokl. Akad. Nauk SSSR 44, 339 (1944)MathSciNetGoogle Scholar
  15. 15.
    P. Bergé, Y. Pomeau, C. Vidal, Order Within Chaos: Toward a Deterministic Approach to Turbulence (Wiley, New York, 1984)zbMATHGoogle Scholar
  16. 16.
    J.P. Gollub, H.L. Swinney, Onset of turbulence in a rotating fluid. Phys. Rev. Lett. 35, 927 (1975)ADSCrossRefGoogle Scholar
  17. 17.
    J. Maurer, A. Libchaber, Une expérience de Rayleigh-Bénard de geometrie réduite: multiplication, démultiplication et accrochage de fréquences. J. Phys.41, Colloque C3, 51 (1980)Google Scholar
  18. 18.
    P. Bergé, Intermittency in Rayleigh-Bénard convection. J. Phys. Lett. 41, L341 (1980)CrossRefGoogle Scholar
  19. 19.
    B.J. Alder, T.E. Wainwright, Decay of the velocity autocorrelation function. Phys. Rev. A1, 18 (1970)ADSCrossRefGoogle Scholar
  20. 20.
    Y. Pomeau, A new kinetic theory for a dense classical gas. Phys. Lett. A. 27A, 601 (1968); A divergence free kinetic equation for a dense Boltzmann gas. Phys. Lett. A 26A, 336 (1968)Google Scholar
  21. 21.
    Y. Pomeau, P. Manneville, Intermittent transition to turbulence in dissipative dynamical systems. Commun. Math. Phys. 74, 189 (1980)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    M.J. Feigenbaum, Quantitative universality for a class of nonlinear transformations. J. Stat. Phys. 19, 25 (1978)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    P. Coullet, C. Tresser, Iterations d’endomorphismes et groupe de renormalisation. CRAS Série A 287, 577 (1978)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Y. Pomeau et al., Intermittent behaviour in the Belousov-Zhabotinsky reaction. J. Phys. Lett. 42, L271 (1981)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Université de Nice-Sophia AntipolisNiceFrance
  2. 2.Department of MathematicsUniversity of ArizonaTucsonUSA

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