{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:43:11Z","timestamp":1760146991584},"reference-count":17,"publisher":"World Scientific Pub Co Pte Lt","issue":"03","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2012,3]]},"abstract":"<jats:p> 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\u2013Hopf 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\u2013Hopf 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\u2013Hopf interactions. <\/jats:p>","DOI":"10.1142\/s0218127412500605","type":"journal-article","created":{"date-parts":[[2012,2,15]],"date-time":"2012-02-15T20:41:54Z","timestamp":1329338514000},"page":"1250060","source":"Crossref","is-referenced-by-count":12,"title":["TWO-TO-ONE RESONANT HOPF BIFURCATIONS IN A QUADRATICALLY NONLINEAR OSCILLATOR INVOLVING TIME DELAY"],"prefix":"10.1142","volume":"22","author":[{"given":"J. C.","family":"JI","sequence":"first","affiliation":[{"name":"School of Electrical, Mechanical and Mechatronic Systems, University of Technology, Sydney, PO Box 123, Broadway, NSW 2007, Australia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"X. Y.","family":"LI","sequence":"additional","affiliation":[{"name":"School of Mechanical Engineering, Hebei University of Technology, Tianjin 300130, P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Z.","family":"LUO","sequence":"additional","affiliation":[{"name":"School of Electrical, Mechanical and Mechatronic Systems, University of Technology, Sydney, PO Box 123, Broadway, NSW 2007, Australia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"N.","family":"ZHANG","sequence":"additional","affiliation":[{"name":"School of Electrical, Mechanical and Mechatronic Systems, University of Technology, Sydney, PO Box 123, Broadway, NSW 2007, Australia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2012,5,2]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1023\/A:1022622101608"},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1140-2"},{"key":"rf3","volume-title":"Differential Equations, Stability, Oscillations, Time Lags","author":"Halany A.","year":"1966"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-9892-2"},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-4342-7"},{"key":"rf6","volume-title":"Theory and Applications of Hopf Bifurcations","author":"Hassard B. 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