{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T14:37:14Z","timestamp":1772289434886,"version":"3.50.1"},"reference-count":12,"publisher":"World Scientific Pub Co Pte Lt","issue":"02","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2006,2]]},"abstract":"<jats:p> We construct examples of vector fields on a three-sphere, amenable to analytic proof of properties that guarantee the existence of complex behavior. The examples are restrictions of symmetric polynomial vector fields in R<jats:sup>4<\/jats:sup> and possess heteroclinic networks producing switching and nearby suspended horseshoes. The heteroclinic networks in our examples are persistent under symmetry preserving perturbations. We prove that some of the connections in the networks are the transverse intersection of invariant manifolds. The remaining connections are symmetry-induced. The networks lie in an invariant three-sphere and may involve connections exclusively between equilibria or between equilibria and periodic trajectories. The same construction technique may be applied to obtain other examples with similar features. <\/jats:p>","DOI":"10.1142\/s021812740601485x","type":"journal-article","created":{"date-parts":[[2006,4,10]],"date-time":"2006-04-10T07:53:16Z","timestamp":1144655596000},"page":"369-381","source":"Crossref","is-referenced-by-count":20,"title":["SIMPLE VECTOR FIELDS WITH COMPLEX BEHAVIOR"],"prefix":"10.1142","volume":"16","author":[{"given":"MANUELA A. D.","family":"AGUIAR","sequence":"first","affiliation":[{"name":"Centro de Matem\u00e1tica da Universidade do Porto (CMUP is supported by Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia through Programa Operacional Ci\u00eancia, Tecnologia e Inova\u00e7\u00e3o (POCTI) and Programa Operacional Sociedade da Informa\u00e7\u00e3o (POSI) of Quadro Comunit\u00e1rio de Apoio III (2000-2006) with European union fundings (FEDER) and national fundings.), Rua do Campo Alegre, 687, 4169-007 Porto, Portugal"},{"name":"Faculdade de Economia, Universidade do Porto, Rua Dr. Roberto Frias, 4200-464 Porto, Portugal"}]},{"given":"SOFIA B. S. D.","family":"CASTRO","sequence":"additional","affiliation":[{"name":"Centro de Matem\u00e1tica da Universidade do Porto (CMUP is supported by Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia through Programa Operacional Ci\u00eancia, Tecnologia e Inova\u00e7\u00e3o (POCTI) and Programa Operacional Sociedade da Informa\u00e7\u00e3o (POSI) of Quadro Comunit\u00e1rio de Apoio III (2000-2006) with European union fundings (FEDER) and national fundings.), Rua do Campo Alegre, 687, 4169-007 Porto, Portugal"},{"name":"Faculdade de Economia, Universidade do Porto, Rua Dr. Roberto Frias, 4200-464 Porto, Portugal"}]},{"given":"ISABEL S.","family":"LABOURIAU","sequence":"additional","affiliation":[{"name":"Centro de Matem\u00e1tica da Universidade do Porto (CMUP is supported by Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia through Programa Operacional Ci\u00eancia, Tecnologia e Inova\u00e7\u00e3o (POCTI) and Programa Operacional Sociedade da Informa\u00e7\u00e3o (POSI) of Quadro Comunit\u00e1rio de Apoio III (2000-2006) with European union fundings (FEDER) and national fundings.), Rua do Campo Alegre, 687, 4169-007 Porto, Portugal"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/18\/1\/019"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1137\/0519093"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1023\/A:1013970014204"},{"key":"rf5","series-title":"Pitman Research Notes in Mathematics Series","volume-title":"Lectures on Bifurcations, Dynamics and Symmetry","volume":"356","author":"Field M. 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Syst."},{"key":"rf11","volume-title":"Stability by Liapunov's Direct Method","author":"La Salle J.","year":"1961"},{"key":"rf12","doi-asserted-by":"publisher","DOI":"10.1017\/S0308210500024173"},{"key":"rf13","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/1\/2\/003"}],"container-title":["International Journal of Bifurcation and Chaos"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S021812740601485X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T20:07:31Z","timestamp":1565122051000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S021812740601485X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,2]]},"references-count":12,"journal-issue":{"issue":"02","published-online":{"date-parts":[[2011,11,20]]},"published-print":{"date-parts":[[2006,2]]}},"alternative-id":["10.1142\/S021812740601485X"],"URL":"https:\/\/doi.org\/10.1142\/s021812740601485x","relation":{},"ISSN":["0218-1274","1793-6551"],"issn-type":[{"value":"0218-1274","type":"print"},{"value":"1793-6551","type":"electronic"}],"subject":[],"published":{"date-parts":[[2006,2]]}}}