Nonresonant Hopf bifurcations of a controlled van der Pol-Duffing oscillator

Ji, JC

Nonresonant Hopf bifurcations of a controlled van der Pol-Duffing oscillator

Ji, JC

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Publication Type:: Journal Article
Citation:: Journal of Sound and Vibration, 2006, 297 (1-2), pp. 183 - 199
Issue Date:: 2006-10-24

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Field	Value	Language
dc.contributor.author	Ji, JC https://orcid.org/0000-0003-3280-5463	en_US
dc.date.issued	2006-10-24	en_US
dc.identifier.citation	Journal of Sound and Vibration, 2006, 297 (1-2), pp. 183 - 199	en_US
dc.identifier.issn	0022-460X	en_US
dc.identifier.uri	http://hdl.handle.net/10453/15268
dc.description.abstract	The trivial equilibrium of a van der Pol-Duffing oscillator with a nonlinear feedback control may lose its stability via Hopf bifurcations, when the time delay involved in the feedback control reaches certain values. Nonresonant Hopf-Hopf interactions may occur in the controlled van der Pol-Duffing oscillator when the corresponding characteristic equation has two pairs of purely imaginary roots. With the aid of normal form theory and centre manifold theorem as well as a perturbation method, the dynamic behaviour of the nonresonant co-dimension two bifurcation is investigated by studying the possible solutions and their stability of the four-dimensional ordinary differential equations on the centre manifold. In the vicinity of the nonresonant Hopf bifurcation, the oscillator may exhibit the initial equilibrium solution, two periodic solutions as well as a quasi-periodic solution on a two-dimensional torus, depending on the dummy unfolding parameters and nonlinear terms. The analytical predictions are found to be in good agreement with the results of numerical integration of the original delay differential equation. © 2006 Elsevier Ltd. All rights reserved.	en_US
dc.relation.ispartof	Journal of Sound and Vibration	en_US
dc.relation.isbasedon	10.1016/j.jsv.2006.03.043	en_US
dc.subject.classification	Acoustics	en_US
dc.title	Nonresonant Hopf bifurcations of a controlled van der Pol-Duffing oscillator	en_US
dc.type	Journal Article
utslib.citation.volume	1-2	en_US
utslib.citation.volume	297	en_US
utslib.for	091304 Dynamics, Vibration and Vibration Control	en_US
utslib.for	02 Physical Sciences	en_US
utslib.for	09 Engineering	en_US
dc.location.activity	Incheon
pubs.embargo.period	Not known	en_US
pubs.organisational-group	/University of Technology Sydney
pubs.organisational-group	/University of Technology Sydney/Faculty of Engineering and Information Technology
pubs.organisational-group	/University of Technology Sydney/Faculty of Engineering and Information Technology/School of Mechanical and Mechatronic Engineering
utslib.copyright.status	closed_access
pubs.issue	1-2	en_US
pubs.publication-status	Published	en_US
pubs.volume	297	en_US

Abstract:

The trivial equilibrium of a van der Pol-Duffing oscillator with a nonlinear feedback control may lose its stability via Hopf bifurcations, when the time delay involved in the feedback control reaches certain values. Nonresonant Hopf-Hopf interactions may occur in the controlled van der Pol-Duffing oscillator when the corresponding characteristic equation has two pairs of purely imaginary roots. With the aid of normal form theory and centre manifold theorem as well as a perturbation method, the dynamic behaviour of the nonresonant co-dimension two bifurcation is investigated by studying the possible solutions and their stability of the four-dimensional ordinary differential equations on the centre manifold. In the vicinity of the nonresonant Hopf bifurcation, the oscillator may exhibit the initial equilibrium solution, two periodic solutions as well as a quasi-periodic solution on a two-dimensional torus, depending on the dummy unfolding parameters and nonlinear terms. The analytical predictions are found to be in good agreement with the results of numerical integration of the original delay differential equation. © 2006 Elsevier Ltd. All rights reserved.

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http://hdl.handle.net/10453/15268