{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,1]],"date-time":"2022-04-01T20:33:49Z","timestamp":1648845229246},"reference-count":9,"publisher":"World Scientific Pub Co Pte Lt","issue":"09","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2007,9]]},"abstract":"<jats:p> In this paper we will find continuous periodic orbits passing near infinity for a class of polynomial vector fields in \u211d<jats:sup>3<\/jats:sup>. We consider polynomial vector fields that are invariant under a symmetry with respect to a plane \u03a3 and that possess a \"generalized heteroclinic loop\" formed by two singular points e<jats:sup>+<\/jats:sup> and e<jats:sup>-<\/jats:sup> at infinity and their invariant manifolds \u0393 and \u039b. \u0393 is an invariant manifold of dimension 1 formed by an orbit going from e<jats:sup>-<\/jats:sup> to e<jats:sup>+<\/jats:sup>, \u0393 is contained in \u211d<jats:sup>3<\/jats:sup> and is transversal to \u03a3. \u039b is an invariant manifold of dimension 2 at infinity. In fact, \u039b is the two-dimensional sphere at infinity in the Poincar\u00e9 compactification minus the singular points e<jats:sup>+<\/jats:sup> and e<jats:sup>-<\/jats:sup>. The main tool for proving the existence of such periodic orbits is the construction of a Poincar\u00e9 map along the generalized heteroclinic loop together with the symmetry with respect to \u03a3. <\/jats:p>","DOI":"10.1142\/s0218127407019056","type":"journal-article","created":{"date-parts":[[2007,11,30]],"date-time":"2007-11-30T09:59:08Z","timestamp":1196416748000},"page":"3295-3302","source":"Crossref","is-referenced-by-count":1,"title":["GENERATION OF SYMMETRIC PERIODIC ORBITS BY A HETEROCLINIC LOOP FORMED BY TWO SINGULAR POINTS AND THEIR INVARIANT MANIFOLDS OF DIMENSIONS 1 AND 2 IN \u211d<sup>3<\/sup>"],"prefix":"10.1142","volume":"17","author":[{"given":"MONTSERRAT","family":"CORBERA","sequence":"first","affiliation":[{"name":"Departament de Tecnologies Digitals i de la Informaci\u00f3, Universitat de Vic, 08500 Vic, Barcelona, Spain"}]},{"given":"JAUME","family":"LLIBRE","sequence":"additional","affiliation":[{"name":"Departament de Matem\u00e0tiques, Universitat Aut\u00f2noma de Barcelona, 08193 Bellaterra, Barcelona, Spain"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1990-0998352-5"},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127406016884"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1088\/0305-4470\/39\/50\/001"},{"key":"rf4","volume-title":"Introdution aux Fonctions Analysables et Preuve Constructive de la Conjecture de Dulac","author":"\u00c9calle J.","year":"1992"},{"key":"rf5","series-title":"Transl. Math. Monographs","doi-asserted-by":"crossref","DOI":"10.1090\/mmono\/094","volume-title":"Finiteness Theorems for Limit Cycles","volume":"94","author":"Ilyasenko Yu.","year":"1991"},{"key":"rf6","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-5703-5"},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.1007\/BF02584827"},{"key":"rf8","first-page":"163","volume":"6","author":"Shil'nikov L. P.","journal-title":"Sov. Math. Dokl."},{"key":"rf9","doi-asserted-by":"publisher","DOI":"10.1070\/SM1970v010n01ABEH001588"}],"container-title":["International Journal of Bifurcation and Chaos"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218127407019056","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,7]],"date-time":"2019-08-07T00:00:44Z","timestamp":1565136044000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218127407019056"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,9]]},"references-count":9,"journal-issue":{"issue":"09","published-online":{"date-parts":[[2011,11,20]]},"published-print":{"date-parts":[[2007,9]]}},"alternative-id":["10.1142\/S0218127407019056"],"URL":"https:\/\/doi.org\/10.1142\/s0218127407019056","relation":{},"ISSN":["0218-1274","1793-6551"],"issn-type":[{"value":"0218-1274","type":"print"},{"value":"1793-6551","type":"electronic"}],"subject":[],"published":{"date-parts":[[2007,9]]}}}