{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,8,26]],"date-time":"2025-08-26T06:33:42Z","timestamp":1756190022686},"reference-count":23,"publisher":"World Scientific Pub Co Pte Lt","issue":"01","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2015,1]]},"abstract":"<jats:p> We give the normal forms of all polynomial differential systems in \u211d<jats:sup>3<\/jats:sup> which have a nondegenerate or degenerate quadric as an invariant algebraic surface. We also characterize among these systems those which have a Darboux invariant constructed uniquely using the invariant quadric, giving explicitly their expressions. As an example, we apply the obtained results in the determination of the Darboux invariants for the Chen system with an invariant quadric. <\/jats:p>","DOI":"10.1142\/s0218127415500157","type":"journal-article","created":{"date-parts":[[2015,2,2]],"date-time":"2015-02-02T05:03:54Z","timestamp":1422853434000},"page":"1550015","source":"Crossref","is-referenced-by-count":9,"title":["Normal Forms for Polynomial Differential Systems in \u211d<sup>3<\/sup> Having an Invariant Quadric and a Darboux Invariant"],"prefix":"10.1142","volume":"25","author":[{"given":"Jaume","family":"Llibre","sequence":"first","affiliation":[{"name":"Departament de Matem\u00e0tiques, Universitat Aut\u00f2noma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain"}]},{"given":"Marcelo","family":"Messias","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica e Computa\u00e7\u00e3o, Faculdade de Ci\u00eancias e Tecnologia, UNESP \u2013 Univ Estadual Paulista, 19060-900 Presidente Prudente, S\u00e3o Paulo, Brazil"}]},{"given":"Alisson","family":"de Carvalho Reinol","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica e Computa\u00e7\u00e3o, Faculdade de Ci\u00eancias e Tecnologia, UNESP \u2013 Univ Estadual Paulista, 19060-900 Presidente Prudente, S\u00e3o Paulo, Brazil"}]}],"member":"219","published-online":{"date-parts":[[2015,2,2]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127499001024"},{"key":"rf2","volume-title":"Limit Cycles of Differential Equations","author":"Christopher C.","year":"2007"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.2007.229.63"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1990-0998352-5"},{"key":"rf5","volume-title":"Qualitative Theory of Planar Differential Systems","author":"Dumortier F.","year":"2006"},{"key":"rf6","volume-title":"Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields","author":"Guckenheimer J.","year":"2002"},{"key":"rf7","volume-title":"Lectures Notes in Mathematics","volume":"708","author":"Jouanolou J. P.","year":"1979"},{"key":"rf8","doi-asserted-by":"publisher","DOI":"10.1088\/1751-8113\/41\/27\/275210"},{"key":"rf9","doi-asserted-by":"publisher","DOI":"10.1016\/j.physd.2008.10.011"},{"key":"rf10","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2008.07.020"},{"key":"rf11","doi-asserted-by":"publisher","DOI":"10.1016\/j.bulsci.2007.12.001"},{"key":"rf12","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127410027593"},{"key":"rf13","first-page":"12","volume":"52","author":"Llibre J.","year":"2011","journal-title":"J. Math. Phys."},{"key":"rf14","first-page":"125","volume":"22","author":"Llibre J.","year":"2012","journal-title":"Int. J. Bifurcation and Chaos"},{"key":"rf15","doi-asserted-by":"publisher","DOI":"10.1007\/s12346-011-0053-x"},{"key":"rf16","first-page":"1250154-1","volume":"22","author":"Llibre J.","year":"2012","journal-title":"Int. J. Bifurcation and Chaos"},{"key":"rf17","first-page":"1127","volume":"41","author":"Llibre J.","year":"2013","journal-title":"Z. Angew. Math. Phys. ZAMP"},{"key":"rf18","author":"Llibre J.","year":"2014","journal-title":"Commun. Contemp. Math."},{"key":"rf20","doi-asserted-by":"publisher","DOI":"10.1175\/1520-0469(1963)020<0130:DNF>2.0.CO;2"},{"key":"rf21","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127407018725"},{"key":"rf22","volume-title":"Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering","author":"Strogatz S. H.","year":"2001"},{"key":"rf23","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1042-9"},{"key":"rf24","volume-title":"Introduction to Applied Nonlinear Dynamical Systems and Chaos","author":"Wiggins S.","year":"2003"}],"container-title":["International Journal of Bifurcation and Chaos"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218127415500157","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,7]],"date-time":"2019-08-07T13:55:22Z","timestamp":1565186122000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218127415500157"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,1]]},"references-count":23,"journal-issue":{"issue":"01","published-online":{"date-parts":[[2015,2,2]]},"published-print":{"date-parts":[[2015,1]]}},"alternative-id":["10.1142\/S0218127415500157"],"URL":"https:\/\/doi.org\/10.1142\/s0218127415500157","relation":{},"ISSN":["0218-1274","1793-6551"],"issn-type":[{"value":"0218-1274","type":"print"},{"value":"1793-6551","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,1]]}}}