{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,8,27]],"date-time":"2023-08-27T07:20:34Z","timestamp":1693120834434},"reference-count":22,"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":[[2014,3]]},"abstract":"<jats:p> In this paper, period-3 motions to chaos in the periodically forced, softening Duffing oscillator are investigated analytically. The analytical solutions for period-3 and period-6 motions are approximated through the generalized harmonic balance method. The bifurcation trees of period-3 motions to chaos are presented analytically. The symmetric and asymmetric period-3 motions are found. The symmetric to asymmetric period-3 motions experience the saddle-node bifurcation. From the Hopf bifurcation of the asymmetric period-3 motion, period-6 motions are determined analytically from the bifurcation tree of period-3 motions. Such an investigation provides a complete picture of period-3 motions to period-6 motions. With such bifurcation tree, the chaotic motions relative to period-3 motions in such a softening Duffing oscillator can be determined analytically. In a similar fashion, other bifurcation trees of period-m motions to chaos can be determined analytically. <\/jats:p>","DOI":"10.1142\/s0218127414300109","type":"journal-article","created":{"date-parts":[[2014,3,25]],"date-time":"2014-03-25T05:36:36Z","timestamp":1395725796000},"page":"1430010","source":"Crossref","is-referenced-by-count":8,"title":["Period-3 Motions to Chaos in a Softening Duffing Oscillator"],"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, IL62026-1805, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jianzhe","family":"Huang","sequence":"additional","affiliation":[{"name":"Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL62026-1805, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2014,3,25]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01176982"},{"key":"rf2","volume-title":"Erzwunge Schweingungen bei Veranderlicher Eigenfrequenz","author":"Duffing G.","year":"1918"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1006\/jsvi.1996.0478"},{"key":"rf4","volume-title":"Nonlinear Oscillations in Physical Systems","author":"Hayashi G.","year":"1964"},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1090\/qam\/492551"},{"key":"rf6","doi-asserted-by":"publisher","DOI":"10.1098\/rsta.1979.0068"},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.1006\/jsvi.1995.0078"},{"key":"rf8","doi-asserted-by":"publisher","DOI":"10.1016\/0022-460X(92)90448-7"},{"key":"rf9","doi-asserted-by":"publisher","DOI":"10.1016\/S0016-0032(96)00089-0"},{"key":"rf10","doi-asserted-by":"publisher","DOI":"10.1006\/jsvi.1999.2386"},{"key":"rf11","volume-title":"Continuous Dynamical Systems","author":"Luo A. 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