Additive resonances of a controlled van der Pol-Duffing oscillator

Ji, JC; Zhang, N

Additive resonances of a controlled van der Pol-Duffing oscillator

Ji, JC

Zhang, N

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Publication Type:: Journal Article
Citation:: Journal of Sound and Vibration, 2008, 315 (1-2), pp. 22 - 33
Issue Date:: 2008-08-05

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Field	Value	Language
dc.contributor.author	Ji, JC https://orcid.org/0000-0003-3280-5463	en_US
dc.contributor.author	Zhang, N https://orcid.org/0000-0001-6408-0044	en_US
dc.date.issued	2008-08-05	en_US
dc.identifier.citation	Journal of Sound and Vibration, 2008, 315 (1-2), pp. 22 - 33	en_US
dc.identifier.issn	0022-460X	en_US
dc.identifier.uri	http://hdl.handle.net/10453/9328
dc.description.abstract	The trivial equilibrium of a controlled van der Pol-Duffing oscillator with nonlinear feedback control may lose its stability via a non-resonant interaction of two Hopf bifurcations when two critical time delays corresponding to two Hopf bifurcations have the same value. Such an interaction results in a non-resonant bifurcation of co-dimension two. In the vicinity of the non-resonant Hopf bifurcations, the presence of a periodic excitation in the controlled oscillator can induce three types of additive resonances in the forced response, when the frequency of the external excitation and the frequencies of the two Hopf bifurcations satisfy a certain relationship. With the aid of centre manifold theorem and the method of multiple scales, three types of additive resonance responses of the controlled system are investigated by studying the possible solutions and their stability of the four-dimensional ordinary differential equations on the centre manifold. The amplitudes of the free-oscillation terms are found to admit three solutions; two non-trivial solutions and the trivial solution. Of two non-trivial solutions, one is stable and the trivial solution is unstable. A stable non-trivial solution corresponds to a quasi-periodic motion of the original system. It is also found that the frequency-response curves for three cases of additive resonances are an isolated closed curve. It is shown that the forced response of the oscillator may exhibit quasi-periodic motions on a three-dimensional torus consisting of three frequencies; the frequencies of two bifurcating solutions and the frequency of the excitation. Illustrative examples are given to show the quasi-periodic motions. © 2008 Elsevier Ltd. All rights reserved.	en_US
dc.relation.ispartof	Journal of Sound and Vibration	en_US
dc.relation.isbasedon	10.1016/j.jsv.2008.01.052	en_US
dc.subject.classification	Acoustics	en_US
dc.title	Additive resonances of a controlled van der Pol-Duffing oscillator	en_US
dc.type	Journal Article
utslib.citation.volume	1-2	en_US
utslib.citation.volume	315	en_US
utslib.for	0913 Mechanical Engineering	en_US
utslib.for	02 Physical Sciences	en_US
utslib.for	09 Engineering	en_US
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	open_access
pubs.issue	1-2	en_US
pubs.publication-status	Published	en_US
pubs.volume	315	en_US

Abstract:

The trivial equilibrium of a controlled van der Pol-Duffing oscillator with nonlinear feedback control may lose its stability via a non-resonant interaction of two Hopf bifurcations when two critical time delays corresponding to two Hopf bifurcations have the same value. Such an interaction results in a non-resonant bifurcation of co-dimension two. In the vicinity of the non-resonant Hopf bifurcations, the presence of a periodic excitation in the controlled oscillator can induce three types of additive resonances in the forced response, when the frequency of the external excitation and the frequencies of the two Hopf bifurcations satisfy a certain relationship. With the aid of centre manifold theorem and the method of multiple scales, three types of additive resonance responses of the controlled system are investigated by studying the possible solutions and their stability of the four-dimensional ordinary differential equations on the centre manifold. The amplitudes of the free-oscillation terms are found to admit three solutions; two non-trivial solutions and the trivial solution. Of two non-trivial solutions, one is stable and the trivial solution is unstable. A stable non-trivial solution corresponds to a quasi-periodic motion of the original system. It is also found that the frequency-response curves for three cases of additive resonances are an isolated closed curve. It is shown that the forced response of the oscillator may exhibit quasi-periodic motions on a three-dimensional torus consisting of three frequencies; the frequencies of two bifurcating solutions and the frequency of the excitation. Illustrative examples are given to show the quasi-periodic motions. © 2008 Elsevier Ltd. All rights reserved.

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http://hdl.handle.net/10453/9328