Bifurcation control and sound intensities in musical art
type: Item
instance of: scholarly article
title: Bifurcation control and sound intensities in musical art
zbMATH Open document ID: 1476.34140
author: Majid Gazor
author: Ahmad Shoghi
published in: Journal of Differential Equations
publication date: 2021-08-06T00:00:00Z
review text: The authors propose a natural formulation for investigating and generating varieties of pitch simultaneity and sound intensity using Eulerian flows with \(n\)-tuple Hopf singularity for a sufficiently large \(n\). Chord leaves and chord limit sets are defined as two critical flow invariant manifolds. The reduction of flows on chord leaves facilitate the study and bifurcation control through a generic codimension-three scalar bifurcation problem. There is a one to one correspondence between bifurcations of positive equilibria in the scalar equation and bifurcations of invariant chord limit sets. An ordered set of double saddle-node and pitchfork bifurcation control of subcritical and supercritical types can describe and generate the sound intensity bifurcations of a sheet music. These scalar bifurcations are interpreted in terms of appearances and disappearances of various chord leaf-stable and unstable flow-invariant chord limit sets. This topic is interesting and the obtained results are completely novel. This work displays an important application in music.
Mathematics Subject Classification ID: 34H20
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 00A65
zbMATH DE Number: 7379502
zbMATH Keywords: multiple Hopf bifurcation control
zbMATH Keywords: pitch simultaneity
zbMATH Keywords: invariant chord limit set
zbMATH Keywords: chord bundle
zbMATH Keywords: chord leaf
zbMATH Keywords: chord manifold
reviewed by: Changjin Xu
MaRDI profile type: Publication
full work available at URL: j.jde.2021.05.022
OpenAlex ID: W3161378538
cites work: A three-parameter study of a degenerate case of the Hopf-pitchfork bifurcation
cites work: The integrability problem for a class of planar systems
cites work: Analytic integrability around a nilpotent singularity
cites work: Generalized Voice-Leading Spaces
cites work: Bifurcation control. Theory and applications
cites work: BIFURCATION CONTROL: THEORIES, METHODS, AND APPLICATIONS
cites work: Bifurcation phenomena in control flows
cites work: Q3069273
cites work: Controllability and stabilization of one-dimensional systems near bifurcation points
cites work: Bifurcation control and universal unfolding for Hopf-zero singularities with leading solenoidal terms
cites work: Bifurcation Controller Designs for the Generalized Cusp Plants of Bogdanov--Takens Singularity with an Application to Ship Control
cites work: Spectral sequences and parametric normal forms
cites work: Bifurcation Stabilization with Local Output Feedback
cites work: Analysis and Control of Hopf Bifurcations
cites work: The Controlled Center Dynamics
cites work: A characterization of normal forms for control systems
cites work: Bifurcations of 1-parameter Families of Control-affine Systems in the Plane
cites work: Bifurcation Control via State Feedback for Systems with a Single Uncontrollable Mode
cites work: Bifurcation and Normal Form of Nonlinear Control Systems, Part I
cites work: Extended Quadratic Controller Normal Form and Dynamic State Feedback Linearization of Nonlinear Systems
cites work: Controllability and local accessibility - A normal form approach
cites work: Control Bifurcations
cites work: Differential equations in music and dance
cites work: Musical Tonality, Neural Resonance and Hebbian Learning
cites work: Musimathics: the mathematical foundations of music. Vol. II. With a foreword by John Chowning
cites work: A dynamical systems approach to musical tuning
cites work: The Geometry of Musical Chords
DOI: 10.1016/J.JDE.2021.05.022
description: scientific article; zbMATH DE number 7379502