Two-to-one resonant hopf bifurcations in a quadratically nonlinear oscillator involving time delay

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dc.contributor.author Li, X.Y. en_US
dc.contributor.author Luo, Zhen en_US
dc.contributor.author Zhang, Nong en_US
dc.contributor.author Ji, Jc en_US
dc.contributor.editor en_US
dc.date.accessioned 2012-10-12T03:34:02Z
dc.date.available 2012-10-12T03:34:02Z
dc.date.issued 2012 en_US
dc.identifier 2011003820 en_US
dc.identifier.citation Ji Jinchen et al. 2012, 'Two-to-one resonant hopf bifurcations in a quadratically nonlinear oscillator involving time delay', World Scientific Publishing Co. Pte. Ltd., vol. 22, no. 3, pp. 1250060-1-1250060-14. en_US
dc.identifier.issn 0218-1274 en_US
dc.identifier.other C1 en_US
dc.identifier.uri http://hdl.handle.net/10453/18415
dc.description.abstract The trivial equilibrium of a weakly nonlinear oscillator having quadratic nonlinearities under a delayed feedback control can change its stability via a single Hopf bifurcation as the time delay increases. Double Hopf bifurcation occurs when the characteristic equation has two pairs of purely imaginary solutions. An interaction of resonant Hopf-Hopf bifurcations may be possible when the two critical time delays corresponding to the two Hopf bifurcations have the same value. With the aid of normal form theory and centre manifold theorem as well as the method of multiple scales, the present paper studies the dynamics of a quadratically nonlinear oscillator involving time delay in the vicinity of the point of two-to-one resonances of Hopf-Hopf bifurcations. The ratio of the frequencies of two Hopf bifurcations is numerically found to be nearly equal to two. The two resonant Hopf bifurcations can generate two respective periodic solutions. Consequently, the centre manifold corresponding to these two solutions is determined by a set of four first-order differential equations under two-to-one internal resonances. It is shown that the amplitudes of the two bifurcating periodic solutions admit the trivial solution and two-mode solutions for the averaged equations on the centre manifolds. Correspondingly, the cumulative behavior of the original nonlinear oscillator exhibits the initial equilibrium and a quasi-periodic motion having two frequencies. Illustrative examples are given to show the unstable zero solution, stable zero solution, and stable two-mode solution of the nonlinear oscillator under the two-to-one resonant Hopf-Hopf interactions. en_US
dc.language English en_US
dc.publisher World Scientific Publishing Co. Pte. Ltd. en_US
dc.relation.isbasedon en_US
dc.relation.isbasedon http://dx.doi.org/10.1142/S0218127412500605 en_US
dc.title Two-to-one resonant hopf bifurcations in a quadratically nonlinear oscillator involving time delay en_US
dc.parent International Journal of Bifurcation and Chaos en_US
dc.journal.volume 22 en_US
dc.journal.number 3 en_US
dc.publocation Singapore en_US
dc.identifier.startpage 1250060-1 en_US
dc.identifier.endpage 1250060-14 en_US
dc.cauo.name FEIT.School of Elec, Mech and Mechatronic Systems en_US
dc.conference Verified OK en_US
dc.for 091300 en_US
dc.personcode 997749 en_US
dc.personcode 0000076432 en_US
dc.personcode 111984 en_US
dc.personcode 950854 en_US
dc.percentage 66 en_US
dc.classification.name Mechanical Engineering en_US
dc.classification.type FOR-08 en_US
dc.edition en_US
dc.custom en_US
dc.date.activity en_US
dc.location.activity en_US
dc.description.keywords Two-to-one internal resonances; time delay; resonant Hopf bifurcations; Hopf-Hopf interactions; nonlinear oscillator; quadratic nonlinearities; feedback control. en_US
dc.staffid en_US
dc.staffid 950854 en_US
utslib.copyright.status Closed Access
utslib.copyright.date 2015-04-15 12:17:09.805752+10


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