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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

Abstract

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.).

Keywords

Bifurcations Iterated maps Reed musical instruments Clarinet Acoustics 

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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

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