{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T15:32:28Z","timestamp":1759937548296,"version":"3.37.3"},"reference-count":72,"publisher":"World Scientific Pub Co Pte Ltd","issue":"09","funder":[{"name":"NSF of Zhejiang Province","award":["LQ18A010001"],"award-info":[{"award-number":["LQ18A010001"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61473340"],"award-info":[{"award-number":["61473340"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"NSF of Zhejiang University of Science and Technology","award":["F701108G14"],"award-info":[{"award-number":["F701108G14"]}]},{"name":"NSF of Zhejiang University of Science and Technology","award":["F701108G13"],"award-info":[{"award-number":["F701108G13"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2018,8]]},"abstract":"<jats:p>This note revisits an extended Lorenz system, which was presented in the paper entitled \u201cHopf bifurcations in an extended Lorenz system\u201d by Zhou et al. [2017]. On the one hand, one points out and corrects some wrong results in that paper on the Hopf bifurcation at the symmetric equilibria [Formula: see text] and [Formula: see text]. On the other hand, combining Lyapunov function and the concepts of [Formula: see text]- and [Formula: see text]-limit sets, it is rigorously proved that there exists two and only two heteroclinic trajectories but no homoclinic trajectories under some certain conditions of its parameters and initial values. In addition, numerical simulations illustrate the consistence with the theoretical conclusions. The results together not only improve and complement the known ones, but also provide support in some future applications.<\/jats:p>","DOI":"10.1142\/s0218127418501110","type":"journal-article","created":{"date-parts":[[2018,9,6]],"date-time":"2018-09-06T03:36:43Z","timestamp":1536205003000},"page":"1850111","source":"Crossref","is-referenced-by-count":2,"title":["Heteroclinic Trajectory and Hopf Bifurcation in an Extended Lorenz System"],"prefix":"10.1142","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8784-115X","authenticated-orcid":false,"given":"Xianyi","family":"Li","sequence":"first","affiliation":[{"name":"Institute of Nonlinear Analysis and Department of Big Data Science, School of Science, Zhejiang University of Science and Technology, Hangzhou, Zhejiang 310023, P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Haijun","family":"Wang","sequence":"additional","affiliation":[{"name":"Institute of Nonlinear Analysis and Department of Big Data Science, School of Science, Zhejiang University of Science and Technology, Hangzhou, Zhejiang 310023, P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2018,9,5]]},"reference":[{"key":"S0218127418501110BIB001","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2004.09.038"},{"volume-title":"Qualitative Theory of Second-Order Dynamical System","year":"1966","author":"Andronov A. A.","key":"S0218127418501110BIB002"},{"volume-title":"Theory of Bifurcations of Dynamic Systems on a Plane","year":"1973","author":"Andronov A. A.","key":"S0218127418501110BIB003"},{"volume-title":"Geometrical Methods in the Theory of Ordinary Differential Equations","year":"1982","author":"Arnold V. I.","key":"S0218127418501110BIB004"},{"key":"S0218127418501110BIB005","doi-asserted-by":"publisher","DOI":"10.1007\/BF00276104"},{"key":"S0218127418501110BIB006","doi-asserted-by":"publisher","DOI":"10.1146\/annurev.fl.27.010195.002003"},{"key":"S0218127418501110BIB007","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127413500120"},{"volume-title":"Controlling Chaos and Bifurcations in Engineer System","year":"1999","author":"Chen G. R.","key":"S0218127418501110BIB008"},{"key":"S0218127418501110BIB009","doi-asserted-by":"publisher","DOI":"10.1007\/s11071-014-1318-0"},{"key":"S0218127418501110BIB010","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4613-8159-4"},{"key":"S0218127418501110BIB011","doi-asserted-by":"publisher","DOI":"10.1016\/j.nonrwa.2009.12.010"},{"key":"S0218127418501110BIB012","doi-asserted-by":"publisher","DOI":"10.24033\/bsmf.1031"},{"key":"S0218127418501110BIB013","doi-asserted-by":"publisher","DOI":"10.1017\/S0143385700004119"},{"key":"S0218127418501110BIB014","volume-title":"Chaos in Dynamical Systems","author":"Edward O.","year":"2002","edition":"2"},{"key":"S0218127418501110BIB015","doi-asserted-by":"publisher","DOI":"10.1007\/BF01020332"},{"key":"S0218127418501110BIB016","doi-asserted-by":"publisher","DOI":"10.1007\/BF02683825"},{"key":"S0218127418501110BIB017","first-page":"16","volume":"3","author":"Feng B.","year":"2003","journal-title":"Appl. Math. E-Notes"},{"key":"S0218127418501110BIB018","doi-asserted-by":"publisher","DOI":"10.1016\/0167-2789(93)90284-8"},{"key":"S0218127418501110BIB019","doi-asserted-by":"publisher","DOI":"10.1115\/1.4000318"},{"key":"S0218127418501110BIB020","doi-asserted-by":"publisher","DOI":"10.1007\/BF01010828"},{"key":"S0218127418501110BIB021","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1140-2"},{"volume-title":"Ordinary Diferential Equations","year":"1969","author":"Hale J. K.","key":"S0218127418501110BIB022"},{"key":"S0218127418501110BIB023","doi-asserted-by":"publisher","DOI":"10.1086\/296466"},{"key":"S0218127418501110BIB024","doi-asserted-by":"publisher","DOI":"10.1007\/BF00288433"},{"key":"S0218127418501110BIB025","doi-asserted-by":"publisher","DOI":"10.1023\/A:1008398006403"},{"key":"S0218127418501110BIB026","unstructured":"Il\u2019yashenko, Y. S. [1982] Singular Points and Limit Cycles of Differential Equations in the Real and Complex Plane, Preprint (NIVTS AN SSSR, Pushchino), p. 38."},{"key":"S0218127418501110BIB027","doi-asserted-by":"publisher","DOI":"10.1007\/s10884-004-4290-4"},{"key":"S0218127418501110BIB028","doi-asserted-by":"publisher","DOI":"10.1063\/1.166509"},{"key":"S0218127418501110BIB029","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-3978-7"},{"key":"S0218127418501110BIB030","doi-asserted-by":"publisher","DOI":"10.1016\/j.cnsns.2015.04.007"},{"key":"S0218127418501110BIB031","doi-asserted-by":"publisher","DOI":"10.1142\/S021812740601663X"},{"key":"S0218127418501110BIB032","doi-asserted-by":"publisher","DOI":"10.1007\/s11071-010-9887-z"},{"key":"S0218127418501110BIB033","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127411030039"},{"key":"S0218127418501110BIB034","doi-asserted-by":"crossref","first-page":"865","DOI":"10.1002\/cta.673","volume":"39","author":"Li Y.","year":"2011","journal-title":"Int. J. Circ. Theor. Appl."},{"key":"S0218127418501110BIB035","doi-asserted-by":"publisher","DOI":"10.1007\/s11071-013-0815-x"},{"key":"S0218127418501110BIB036","doi-asserted-by":"publisher","DOI":"10.1115\/1.2754307"},{"key":"S0218127418501110BIB037","doi-asserted-by":"publisher","DOI":"10.1016\/j.nonrwa.2009.09.001"},{"key":"S0218127418501110BIB038","doi-asserted-by":"publisher","DOI":"10.1007\/s11071-011-0090-7"},{"key":"S0218127418501110BIB039","doi-asserted-by":"publisher","DOI":"10.1063\/1.1435078"},{"key":"S0218127418501110BIB040","doi-asserted-by":"publisher","DOI":"10.1175\/1520-0469(1963)020<0130:DNF>2.0.CO;2"},{"key":"S0218127418501110BIB041","doi-asserted-by":"publisher","DOI":"10.1137\/0129022"},{"key":"S0218127418501110BIB042","doi-asserted-by":"publisher","DOI":"10.1109\/TCS.1979.1084636"},{"key":"S0218127418501110BIB043","doi-asserted-by":"publisher","DOI":"10.1088\/1751-8113\/42\/11\/115101"},{"key":"S0218127418501110BIB044","doi-asserted-by":"publisher","DOI":"10.1038\/333419a0"},{"key":"S0218127418501110BIB045","doi-asserted-by":"publisher","DOI":"10.1088\/1361-6544\/30\/1\/115"},{"key":"S0218127418501110BIB046","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0089983"},{"key":"S0218127418501110BIB047","doi-asserted-by":"publisher","DOI":"10.1007\/BF01646553"},{"key":"S0218127418501110BIB048","doi-asserted-by":"publisher","DOI":"10.1142\/4221"},{"key":"S0218127418501110BIB049","doi-asserted-by":"publisher","DOI":"10.1007\/BF03025291"},{"volume-title":"Curvas Definidas for Equagoes Diferencials No Plano","year":"1981","author":"Sotomayor J.","key":"S0218127418501110BIB050"},{"key":"S0218127418501110BIB051","doi-asserted-by":"publisher","DOI":"10.1590\/S0101-82052007000100002"},{"key":"S0218127418501110BIB052","doi-asserted-by":"publisher","DOI":"10.2991\/jnmp.2008.15.s3.28"},{"key":"S0218127418501110BIB053","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-5767-7"},{"key":"S0218127418501110BIB054","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2008.10.024"},{"key":"S0218127418501110BIB055","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127416500723"},{"key":"S0218127418501110BIB056","doi-asserted-by":"publisher","DOI":"10.1080\/00423118308965288"},{"key":"S0218127418501110BIB057","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127414501338"},{"key":"S0218127418501110BIB058","first-page":"061021-1","volume":"10","author":"Wang H.","year":"2015","journal-title":"J. Comput. Nonlin. Dyn."},{"key":"S0218127418501110BIB059","doi-asserted-by":"publisher","DOI":"10.1007\/s11071-015-1921-8"},{"key":"S0218127418501110BIB060","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127416501947"},{"key":"S0218127418501110BIB061","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127417501103"},{"key":"S0218127418501110BIB062","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2017.10.029"},{"key":"S0218127418501110BIB063","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127414501272"},{"key":"S0218127418501110BIB064","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1042-9"},{"volume-title":"Introduction to Applied Nonlinear Dynamical System and Chaos","year":"2003","author":"Wiggins S.","key":"S0218127418501110BIB065"},{"key":"S0218127418501110BIB066","doi-asserted-by":"publisher","DOI":"10.1007\/s00220-002-0709-0"},{"key":"S0218127418501110BIB067","doi-asserted-by":"publisher","DOI":"10.1007\/s00220-005-1374-x"},{"key":"S0218127418501110BIB068","first-page":"031012-1","volume":"10","author":"Wu X.","year":"2015","journal-title":"J. Comput. Nonlin. Dyn."},{"key":"S0218127418501110BIB069","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127414500552"},{"key":"S0218127418501110BIB070","volume-title":"Theory of Limit Cycle","author":"Ye Y. Q.","year":"1984","edition":"2"},{"key":"S0218127418501110BIB071","first-page":"024501-1","volume":"9","author":"Zhang Z.","year":"2014","journal-title":"J. Comput. Nonlin. Dyn."},{"key":"S0218127418501110BIB072","volume":"2017","author":"Zhou Z.","year":"2017","journal-title":"Adv. Diff. Eqs."}],"container-title":["International Journal of Bifurcation and Chaos"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218127418501110","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,7,10]],"date-time":"2024-07-10T00:24:31Z","timestamp":1720571071000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218127418501110"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,8]]},"references-count":72,"journal-issue":{"issue":"09","published-online":{"date-parts":[[2018,9,5]]},"published-print":{"date-parts":[[2018,8]]}},"alternative-id":["10.1142\/S0218127418501110"],"URL":"https:\/\/doi.org\/10.1142\/s0218127418501110","relation":{},"ISSN":["0218-1274","1793-6551"],"issn-type":[{"type":"print","value":"0218-1274"},{"type":"electronic","value":"1793-6551"}],"subject":[],"published":{"date-parts":[[2018,8]]}}}