Secondary resonances of a quadratic nonlinear oscillator following two-to-one resonant Hopf bifurcations

Ji, JC

Secondary resonances of a quadratic nonlinear oscillator following two-to-one resonant Hopf bifurcations

Ji, JC

Permalink

Publication Type:: Journal Article
Citation:: Nonlinear Dynamics, 2014, 78 (3), pp. 2161 - 2184
Issue Date:: 2014-01-01

Closed Access

	Filename	Description	Size
	Ji-ND-2014-v78i3-2161-2184.pdf	Published Version	2.41 MB	Adobe PDF	View/Open

Copyright Clearance Process

Recently Added
In Progress
Closed Access

This item is closed access and not available.

Full metadata record

Field	Value	Language
dc.contributor.author	Ji, JC https://orcid.org/0000-0003-3280-5463	en_US
dc.date.issued	2014-01-01	en_US
dc.identifier.citation	Nonlinear Dynamics, 2014, 78 (3), pp. 2161 - 2184	en_US
dc.identifier.issn	0924-090X	en_US
dc.identifier.uri	http://hdl.handle.net/10453/33142
dc.description.abstract	© 2014, Springer Science+Business Media Dordrecht. Stable bifurcating solutions may appear in an autonomous time-delayed nonlinear oscillator having quadratic nonlinearity after the trivial equilibrium loses its stability via two-to-one resonant Hopf bifurcations. For the corresponding non-autonomous time-delayed nonlinear oscillator, the dynamic interactions between the periodic excitation and the stable bifurcating solutions can induce resonant behaviour in the forced response when the forcing frequency and the frequencies of Hopf bifurcations satisfy certain relationships. Under hard excitations, the forced response of the time-delayed nonlinear oscillator can exhibit three types of secondary resonances, which are super-harmonic resonance at half the lower Hopf bifurcation frequency, sub-harmonic resonance at two times the higher Hopf bifurcation frequency and additive resonance at the sum of two Hopf bifurcation frequencies. With the help of centre manifold theorem and the method of multiple scales, the secondary resonance response of the time-delayed nonlinear oscillator following two-to-one resonant Hopf bifurcations is studied based on a set of four averaged equations for the amplitudes and phases of the free-oscillation terms, which are obtained from the reduced four-dimensional ordinary differential equations for the flow on the centre manifold. The first-order approximate solutions and the nonlinear algebraic equations for the amplitudes and phases of the free-oscillation terms in the steady state solutions are derived for three secondary resonances. Frequency-response curves, time trajectories, phase portraits and Poincare sections are numerically obtained to show the secondary resonance response. Analytical results are found to be in good agreement with those of direct numerical integrations.	en_US
dc.relation.ispartof	Nonlinear Dynamics	en_US
dc.relation.isbasedon	10.1007/s11071-014-1588-6	en_US
dc.subject.classification	Acoustics	en_US
dc.title	Secondary resonances of a quadratic nonlinear oscillator following two-to-one resonant Hopf bifurcations	en_US
dc.type	Journal Article
utslib.citation.volume	3	en_US
utslib.citation.volume	78	en_US
utslib.for	090205 Hybrid Vehicles and Powertrains	en_US
utslib.for	0913 Mechanical Engineering	en_US
utslib.for	01 Mathematical 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	closed_access
pubs.issue	3	en_US
pubs.publication-status	Published	en_US
pubs.volume	78	en_US

Abstract:

© 2014, Springer Science+Business Media Dordrecht. Stable bifurcating solutions may appear in an autonomous time-delayed nonlinear oscillator having quadratic nonlinearity after the trivial equilibrium loses its stability via two-to-one resonant Hopf bifurcations. For the corresponding non-autonomous time-delayed nonlinear oscillator, the dynamic interactions between the periodic excitation and the stable bifurcating solutions can induce resonant behaviour in the forced response when the forcing frequency and the frequencies of Hopf bifurcations satisfy certain relationships. Under hard excitations, the forced response of the time-delayed nonlinear oscillator can exhibit three types of secondary resonances, which are super-harmonic resonance at half the lower Hopf bifurcation frequency, sub-harmonic resonance at two times the higher Hopf bifurcation frequency and additive resonance at the sum of two Hopf bifurcation frequencies. With the help of centre manifold theorem and the method of multiple scales, the secondary resonance response of the time-delayed nonlinear oscillator following two-to-one resonant Hopf bifurcations is studied based on a set of four averaged equations for the amplitudes and phases of the free-oscillation terms, which are obtained from the reduced four-dimensional ordinary differential equations for the flow on the centre manifold. The first-order approximate solutions and the nonlinear algebraic equations for the amplitudes and phases of the free-oscillation terms in the steady state solutions are derived for three secondary resonances. Frequency-response curves, time trajectories, phase portraits and Poincare sections are numerically obtained to show the secondary resonance response. Analytical results are found to be in good agreement with those of direct numerical integrations.

Please use this identifier to cite or link to this item:

http://hdl.handle.net/10453/33142