Unveiling bifurcation scenarios using automatic continuation: application to a two-mode saxophone model
Abstract
A nonlinear dynamical system can exhibit a wide variety of different behaviors when its control parameters are changed. This is particularly true for musical instruments for which these different behaviors condition the produced sound. An automatic procedure is designed for a numerical continuation method (Harmonic Balance Method associated with Asymptotic Numerical Method) to explore all of the bifurcation diagrams for every tested parameter combination, which allows to test a very wide range of system parameters. This method is applied to a saxophone model with a single degree-of-freedom reed model and two acoustic modes. As a result, a finite list of the possible behaviors of a two-mode saxophone model is established. These behaviors are grouped in "bifurcation scenarios", i.e. the order in which each specific bifurcation appears on the branches of the bifurcation diagram, and how these branches are interconnected. These bifurcation scenarios are well explained by combining known scenarios tied to the normal forms of nearby high codimension bifurcations: the 2:1 resonant double Hopf bifurcation of codimension 3, and the Cusp and Bautin bifurcations of codimension 2. We highlight interesting cases where periodic solution branches interconnect through branching points.
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BibTeX
@misc{colinot2026unveiling,
title = {{Unveiling bifurcation scenarios using automatic continuation: application to a two-mode saxophone model}},
author = {Tom Colinot and Etienne Gourc},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05772012v1},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Unveiling bifurcation scenarios using automatic continuation: application to a two-mode saxophone model
AU - Tom Colinot
AU - Etienne Gourc
PY - 2026
DA - 2026/09/30
PB - HAL
UR - https://hal.science/hal-05772012v1
AB - A nonlinear dynamical system can exhibit a wide variety of different behaviors when its control parameters are changed. This is particularly true for musical instruments for which these different behaviors condition the produced sound. An automatic procedure is designed for a numerical continuation method (Harmonic Balance Method associated with Asymptotic Numerical Method) to explore all of the bifurcation diagrams for every tested parameter combination, which allows to test a very wide range of system parameters. This method is applied to a saxophone model with a single degree-of-freedom reed model and two acoustic modes. As a result, a finite list of the possible behaviors of a two-mode saxophone model is established. These behaviors are grouped in "bifurcation scenarios", i.e. the order in which each specific bifurcation appears on the branches of the bifurcation diagram, and how these branches are interconnected. These bifurcation scenarios are well explained by combining known scenarios tied to the normal forms of nearby high codimension bifurcations: the 2:1 resonant double Hopf bifurcation of codimension 3, and the Cusp and Bautin bifurcations of codimension 2. We highlight interesting cases where periodic solution branches interconnect through branching points.
ER -