Iterated maps for clarinet-like systems

TL;DR

This paper analyzes the non-linear iterated maps modeling clarinet-like systems, revealing complex behaviors such as period tripling, intermittency, and chaotic regimes, with insights gained from high-order iterates.

Contribution

It provides a detailed study of the geometrical properties of the iterated maps and introduces methods to classify oscillation regimes in clarinet-like systems.

Findings

Identification of period tripling regimes

Conditions for intermittency in the system

High-order iterates reveal oscillation characteristics

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.).