Nonlinear Dynamics

, Volume 62, Issue 1–2, pp 253–271 | Cite as

Iterated maps for clarinet-like systems

  • P.-A. Taillard
  • J. Kergomard
  • F. Laloë
Original Paper


The dynamical equations of clarinet-like systems are known to be reducible to a non-linear iterated map within reasonable approximations. This leads to time oscillations that are represented by square signals, analogous to the Raman regime for string instruments. In this article, we study in more detail the properties of the corresponding non-linear iterations, with emphasis on the geometrical constructions that can be used to classify the various solutions (for instance with or without reed beating) as well as on the periodicity windows that occur within the chaotic region. In particular, we find a regime where period tripling occurs and examine the conditions for intermittency. We also show that, while the direct observation of the iteration function does not reveal much on the oscillation regime of the instrument, the graph of the high order iterates directly gives visible information on the oscillation regime (characterization of the number of period doubligs, chaotic behaviour, etc.).


Bifurcations Iterated maps Reed musical instruments Clarinet Acoustics 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    May, R.M.: Simple mathematical models with very complicated dynamics. Nature 261, 459–467 (1976) CrossRefGoogle Scholar
  2. 2.
    Bergé, P., Pomeau, Y., Vidal, C.: Order within Chaos. Wiley/Hermann, New York (1986) zbMATHGoogle Scholar
  3. 3.
    Bergé, P., Pomeau, Y., Vidal, C.: L’ordre dans le chaos. Hermann, Paris (1984) zbMATHGoogle Scholar
  4. 4.
    Collet, P., Eckmann, J.P.: Properties of continuous maps of the interval to itself. In: Osterwalder, K. (ed.) Mathematical Problems in Theoretical Physics. Springer, Heidelberg (1979) Google Scholar
  5. 5.
    Collet, P., Eckmann, J.P.: Iterated Maps on the Interval as Dynamical Systems. Birkhäuser, Basel (1980) zbMATHGoogle Scholar
  6. 6.
    Feigenbaum, J.: The universal metric properties of nonlinear transformations. J. Stat. Phys. 21, 669–706 (1979) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Feigenbaum, J.: The metric universal properties of period doubling bifurcations and the spectrum for a route to turbulence. Ann. N.Y. Acad. Sci. 357, 330–336 (1980) CrossRefGoogle Scholar
  8. 8.
    McIntyre, M.E., Schumacher, R.T., Woodhouse, J.: On the oscillations of musical instruments. J. Acoust. Soc. Am. 74, 1325–1345 (1983) CrossRefGoogle Scholar
  9. 9.
    Maganza, C., Caussé, R., Laloë, F.: Bifurcations, period doubling and chaos in clarinetlike systems. Europhys. Lett. 1, 295–302 (1986) CrossRefGoogle Scholar
  10. 10.
    Brod, K.: Die Klarinette als Verzweigungssytem: eine Anwendung der Methode des iterierten Abbildungen. Acustica 72, 72–78 (1990) Google Scholar
  11. 11.
    Kergomard, J.: Elementary considerations on reed-instrument oscillations. In: Hirschberg, A., Kergomard, J., Weinreich, G. (eds.) Mechanics of Musical Instruments. CISM Courses and Lectures, vol. 335, pp. 229–290. Springer, Wien (1995) Google Scholar
  12. 12.
    Lizée, A.: Doublement de période dans les instruments à anche simple de type clarinette, Master degree thesis, Paris (2004).
  13. 13.
    Idogawa, T., Kobata, T., Komuro, K., Masakazu, I.: Nonlinear vibrations in the air column of a clarinet artificially blown. J. Acoust. Soc. Am. 93, 540–551 (1993) CrossRefGoogle Scholar
  14. 14.
    Gibiat, V., Castellengo, M.: Period doubling occurences in wind instrument musical performances. Acustica United Acta Acustica 86, 746–754 (2000) Google Scholar
  15. 15.
    Kergomard, J., Dalmont, J.P., Gilbert, J., Guillemain, P.: Period doubling on cylindrical reed instruments. In: Proceedings ot the Joint Congress CFA/DAGA’04, pp. 113–114, Strasbourg, 22–25th March 2004 Google Scholar
  16. 16.
    Ollivier, S., Kergomard, J., Dalmont, J.-P.: Idealized models of reed woodwinds. Part II: On the Stability of “Two-Step” Oscillations. Acta Acustica United Acustica 91, 166–179 (2005) Google Scholar
  17. 17.
    Dalmont, J.-P., Gilbert, J., Kergomard, J., Ollivier, S.: An analytical prediction of the oscillation and extinction thresholds of a clarinet. J. Acoust. Soc. Am. 118, 3294–3305 (2005) CrossRefGoogle Scholar
  18. 18.
    Vallée, R., Delisle, C.: Periodicity windows in a dynamical system with delayed feedback. Phys. Rev. A 34, 309–3018 (1986) CrossRefGoogle Scholar
  19. 19.
    Stefanski, K.: Universality of succession of periodic windows in families of 1D-maps. Open Syst. Inf. Dyn. 6, 309–324 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Wilson, T.A., Beavers, G.S.: Operating modes of the clarinet. J. Acoust. Soc. Am. 56, 653–658 (1974) CrossRefGoogle Scholar
  21. 21.
    Hirschberg, A., Van de Laar, R.W.A., Marrou-Maurires, J.P., Wijnands, A.P.J., Dane, H.J., Kruijswijk, S.G., Houtsma, A.J.M.: A quasi-stationary model of air flow in the reed channel of single-reed woodwind instruments. Acustica 70, 146–154 (1990) Google Scholar
  22. 22.
    Dalmont, J.-P., Gilbert, J., Ollivier, S.: Nonlinear characteristics of single-reed instruments: quasistatic volume flow and reed opening measurements. J. Acoust. Soc. Am. 114, 2253–2262 (2003) CrossRefGoogle Scholar
  23. 23.
    Dalmont, J.-P., Frappé, C.: Oscillation and extinction thresholds of the clarinet: Comparison of analytical results and experiments. J. Acoust. Soc. Am. 122, 1173–1179 (2007) CrossRefGoogle Scholar
  24. 24.
    Caussé, R., Kergomard, J., Lurton, X.: Input impedance of brass musical instruments—comparison between experiment and numerical model. J. Acoust. Soc. Am. 75, 241–254 (1984) CrossRefGoogle Scholar
  25. 25.
    Raman, C.V.: On the mechanical theory of vibrations of bowed strings [etc.]. Indian Assoc. Cultiv. Sci. Bull. 15, 1–158 (1918) Google Scholar
  26. 26.
    Mayer-Kress, G., Haken, H.: Attractors of convex maps with positive Schwarzian derivative in the presence of noise. Physica D 10, 329–339 (1984) zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Parlitz, U., Englisch, V., Scheffczyk, C., Lauterborn, W.: Bifurcation structure of bubble oscillators. J. Acoust. Soc. Am. 88, 1061–1077 (1990) CrossRefMathSciNetGoogle Scholar
  28. 28.
    Scheffczyk, C., Parlitz, U., Kurz, T., Knop, W., Lauterborn, W.: Comparison of bifurcation structures of driven dissipative nonlinear oscillators. Phys. Rev. A 43, 6495–6502 (1991) CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Conservatoire de Musique NeuchâteloisLa Chaux-de-FondsSwitzerland
  2. 2.Laboratoire de Mécanique et d’AcoustiqueCNRS UPR 7051Marseille Cedex 20France
  3. 3.Laboratoire Kastler-BrosselENS, UMR 8552 CNRS, ENS, et UMPCParisFrance

Personalised recommendations