{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,30]],"date-time":"2025-06-30T05:28:53Z","timestamp":1751261333067},"reference-count":27,"publisher":"World Scientific Pub Co Pte Lt","issue":"10","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2014,10]]},"abstract":"<jats:p> In this paper, periodic motions in a periodically excited, Duffing oscillator with a time-delayed displacement are investigated through the Fourier series, and the stability and bifurcation of such periodic motions are discussed through eigenvalue analysis. The time-delayed displacement is from the feedback control of displacement. The analytical bifurcation trees of period-1 motions to chaos in the time-delayed Duffing oscillator are presented through asymmetric period-1 to period-4 motions. Stable and unstable periodic motions are illustrated through numerical and analytical solutions. From numerical illustrations, the analytical solutions of stable and unstable period-m motions are relatively accurate with A<jats:sub>N\/m<\/jats:sub> &lt; 10<jats:sup>-6<\/jats:sup> compared to numerical solutions. From such analytical solutions, any complicated solutions of period-m motions can be obtained for any prescribed accuracy. Because time-delay may cause discontinuity, the appropriate time-delay inputs (or initial conditions) in the initial time-delay interval should satisfy the analytical solution of periodic motions in the time-delayed dynamical systems. Otherwise, periodic motions in such a time-delayed system cannot be obtained directly. <\/jats:p>","DOI":"10.1142\/s0218127414501260","type":"journal-article","created":{"date-parts":[[2014,10,31]],"date-time":"2014-10-31T05:52:45Z","timestamp":1414734765000},"page":"1450126","source":"Crossref","is-referenced-by-count":23,"title":["Period-m Motions to Chaos in a Periodically Forced, Duffing Oscillator with a Time-Delayed Displacement"],"prefix":"10.1142","volume":"24","author":[{"given":"Albert C. J.","family":"Luo","sequence":"first","affiliation":[{"name":"Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL 62026-1805, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hanxiang","family":"Jin","sequence":"additional","affiliation":[{"name":"Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL 62026-1805, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2014,10,30]]},"reference":[{"key":"rf1","volume-title":"Asymptotic Methods in the Theory of Nonlinear Oscillations","author":"Bogoliubov N. N.","year":"1961"},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01176982"},{"key":"rf3","first-page":"98","volume":"56","author":"Fatou P.","year":"1928","journal-title":"Bull. Soc. 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