{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,20]],"date-time":"2025-09-20T21:10:19Z","timestamp":1758402619306},"reference-count":32,"publisher":"World Scientific Pub Co Pte Lt","issue":"04","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2015,4]]},"abstract":"<jats:p> In this paper, we investigate the codimension-two double Hopf bifurcation in delay-coupled van der Pol\u2013Duffing oscillators. By using normal form theory of delay differential equations, the normal form associated with the codimension-two double Hopf bifurcation is calculated. Choosing appropriate values of the coupling strength and the delay can result in nonresonance and weak resonance double Hopf bifurcations. The dynamical classification near these bifurcation points can be explicitly determined by the corresponding normal form. Periodic, quasi-periodic solutions and torus are found near the bifurcation point. The numerical simulations are employed to support the theoretical results. <\/jats:p>","DOI":"10.1142\/s0218127415500583","type":"journal-article","created":{"date-parts":[[2015,5,4]],"date-time":"2015-05-04T02:52:15Z","timestamp":1430707935000},"page":"1550058","source":"Crossref","is-referenced-by-count":21,"title":["Delay-Induced Double Hopf Bifurcations in a System of Two Delay-Coupled van der Pol\u2013Duffing Oscillators"],"prefix":"10.1142","volume":"25","author":[{"given":"Heping","family":"Jiang","sequence":"first","affiliation":[{"name":"Department of Mathematics, Tongji University, Shanghai 200092, P. R. China"},{"name":"School of Mathematics and Statistics, Huangshan University, Huangshan 245041, P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tonghua","family":"Zhang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Swinburne University of Technology, VIC 3122, Australia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yongli","family":"Song","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Tongji University, Shanghai 200092, P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2015,5,3]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1137\/120887898"},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1023\/A:1022622101608"},{"key":"rf3","first-page":"217","volume":"17","author":"Campbell S. A.","year":"1999","journal-title":"Canad. Appl. Math. Quart."},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1137\/S0036139903434833"},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127413500144"},{"key":"rf6","doi-asserted-by":"publisher","DOI":"10.1006\/jdeq.1995.1144"},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.1006\/jdeq.1995.1145"},{"key":"rf8","doi-asserted-by":"publisher","DOI":"10.1016\/S0006-3495(61)86902-6"},{"key":"rf9","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2004.12.029"},{"key":"rf10","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1140-2"},{"key":"rf11","doi-asserted-by":"publisher","DOI":"10.1360\/012010-1047"},{"key":"rf12","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-9892-2"},{"key":"rf13","volume-title":"Theory and Applications of Hopf Bifurcation","author":"Hassard B.","year":"1981"},{"key":"rf14","doi-asserted-by":"publisher","DOI":"10.1016\/j.cnsns.2012.05.001"},{"key":"rf15","doi-asserted-by":"publisher","DOI":"10.1016\/j.physd.2007.01.003"},{"key":"rf16","doi-asserted-by":"publisher","DOI":"10.1016\/j.na.2010.07.043"},{"key":"rf17","volume-title":"Elements of Applied Nonlinear Dynamical Systems and Chaos","author":"Kuznetsov Y. A.","year":"2004"},{"key":"rf18","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2011.10.023"},{"key":"rf19","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2007.05.072"},{"key":"rf20","doi-asserted-by":"publisher","DOI":"10.1007\/s00332-006-0802-1"},{"key":"rf21","doi-asserted-by":"publisher","DOI":"10.1007\/s11071-010-9799-y"},{"key":"rf22","doi-asserted-by":"publisher","DOI":"10.1109\/TNNLS.2012.2209459"},{"key":"rf23","volume-title":"Nonlinear Dynamics and Chaos: Geometrical Methods for Engineers and Scientists","author":"Thompson J. M. T.","year":"1986"},{"key":"rf24","doi-asserted-by":"crossref","unstructured":"K.\u00a0Tomita, Chaos, ed. A. V.\u00a0Holden (Princeton University Press, Princeton, NJ, 1986)\u00a0pp. 211\u2013236.","DOI":"10.1515\/9781400858156.211"},{"key":"rf25","first-page":"701","volume":"1","author":"van der Pol B.","year":"1920","journal-title":"Radio Rev."},{"key":"rf26","first-page":"1","volume":"7","author":"Wang W.","year":"2007","journal-title":"Int. J. Bifurcation and Chaos"},{"key":"rf27","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2010.03.012"},{"key":"rf28","doi-asserted-by":"publisher","DOI":"10.1007\/s11071-010-9909-x"},{"key":"rf29","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2010.11.011"},{"key":"rf30","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2007.07.032"},{"key":"rf32","doi-asserted-by":"publisher","DOI":"10.1137\/040614207"},{"key":"rf33","doi-asserted-by":"publisher","DOI":"10.1007\/s11071-013-1047-9"}],"container-title":["International Journal of Bifurcation and Chaos"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218127415500583","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T14:59:05Z","timestamp":1565103545000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218127415500583"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,4]]},"references-count":32,"journal-issue":{"issue":"04","published-online":{"date-parts":[[2015,5,3]]},"published-print":{"date-parts":[[2015,4]]}},"alternative-id":["10.1142\/S0218127415500583"],"URL":"https:\/\/doi.org\/10.1142\/s0218127415500583","relation":{},"ISSN":["0218-1274","1793-6551"],"issn-type":[{"value":"0218-1274","type":"print"},{"value":"1793-6551","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,4]]}}}