{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,19]],"date-time":"2025-10-19T15:31:15Z","timestamp":1760887875396},"reference-count":11,"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":[[2006,3]]},"abstract":"<jats:p> In this paper, we consider the planar differential system associated with the potential Hamiltonian H(x,y) = (1\/2)y<jats:sup>2<\/jats:sup>+V(x) where V(x) = (1\/2)x<jats:sup>2<\/jats:sup>+(a\/4)x<jats:sup>4<\/jats:sup>+(b\/6)x<jats:sup>6<\/jats:sup> with b \u2260 0. This family of differential systems always has a center at the origin and, eventually, three other period annuli. We prove that the corresponding period functions can have at most one critical period altogether. More precisely, we show that this critical period is simple and that it corresponds to the period function associated to the center at the origin. To prove the result we use that the period function verifies a Picard\u2013Fuchs equation. Finally we describe the bifurcation diagram of the period function of the center at the origin. <\/jats:p>","DOI":"10.1142\/s0218127406015155","type":"journal-article","created":{"date-parts":[[2006,5,23]],"date-time":"2006-05-23T07:19:24Z","timestamp":1148368764000},"page":"765-774","source":"Crossref","is-referenced-by-count":17,"title":["A NOTE ON THE CRITICAL PERIODS OF POTENTIAL SYSTEMS"],"prefix":"10.1142","volume":"16","author":[{"given":"F.","family":"MA\u00d1OSAS","sequence":"first","affiliation":[{"name":"Departament de Matem\u00e0tiques, Facultat de Ci\u00e8ncies, Universitat Aut\u00f2noma de Barcelona, 08193 Bellaterra, Barcelona, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"J.","family":"VILLADELPRAT","sequence":"additional","affiliation":[{"name":"Departament d'Enginyeria Inform\u00e0tica i Matem\u00e0tiques, ETSE, Universitat Rovira i Virgili, 43007 Tarragona, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1016\/0022-0396(87)90122-7"},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1989-0930075-2"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1016\/0022-0396(86)90071-9"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1006\/jdeq.1999.3635"},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1006\/jdeq.1997.3296"},{"key":"rf6","doi-asserted-by":"publisher","DOI":"10.1006\/jdeq.1993.1004"},{"key":"rf7","first-page":"1157","volume":"1","author":"Givental A. B","journal-title":"Leningrad Math. J."},{"key":"rf8","doi-asserted-by":"publisher","DOI":"10.1006\/jdeq.1996.3212"},{"key":"rf10","doi-asserted-by":"publisher","DOI":"10.4310\/MRL.1996.v3.n3.a6"},{"key":"rf11","doi-asserted-by":"publisher","DOI":"10.1016\/0362-546X(87)90082-4"},{"key":"rf12","doi-asserted-by":"publisher","DOI":"10.1006\/jdeq.2002.4175"}],"container-title":["International Journal of Bifurcation and Chaos"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218127406015155","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T20:10:24Z","timestamp":1565122224000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218127406015155"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,3]]},"references-count":11,"journal-issue":{"issue":"03","published-online":{"date-parts":[[2011,11,20]]},"published-print":{"date-parts":[[2006,3]]}},"alternative-id":["10.1142\/S0218127406015155"],"URL":"https:\/\/doi.org\/10.1142\/s0218127406015155","relation":{},"ISSN":["0218-1274","1793-6551"],"issn-type":[{"value":"0218-1274","type":"print"},{"value":"1793-6551","type":"electronic"}],"subject":[],"published":{"date-parts":[[2006,3]]}}}