{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,25]],"date-time":"2026-06-25T07:55:26Z","timestamp":1782374126479,"version":"3.54.5"},"reference-count":22,"publisher":"World Scientific Pub Co Pte Lt","issue":"07","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2003,7]]},"abstract":"<jats:p> According to [Yu, 1999], at most two terms remain in the amplitude equation of the normal form of a continuous system, expressed in polar coordinates, with a Hopf or Generalized Hopf singularity, if we (only) apply specific nonlinear transformation to the conventional normal form; but, at least one remains in the phase equation. In this paper we show that, using a particular nonlinear scaling, these terms in the phase equation can be eliminated, which simplifies the (diffeomorphic) normal form given by [Yu, 1999]. Besides, we have also treated the Neimark\u2013Sacker and Generalized Neimark\u2013Sacker bifurcations, the corresponding discrete cases of the Hopf and Generalized Hopf, obtaining results which are similar to the continuous ones. <\/jats:p>","DOI":"10.1142\/s0218127403007667","type":"journal-article","created":{"date-parts":[[2003,8,28]],"date-time":"2003-08-28T05:49:31Z","timestamp":1062049771000},"page":"1831-1839","source":"Crossref","is-referenced-by-count":8,"title":["Simplest Normal Forms of Hopf\u2013Neimark\u2013Sacker Bifurcations"],"prefix":"10.1142","volume":"13","author":[{"given":"J. C.","family":"Valverde","sequence":"first","affiliation":[{"name":"Departamento de Matem\u00e1ticas, Universidad de Castilla-La Mancha, 02071 Albacete, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1216\/rmjm\/1181071677"},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4684-0147-9"},{"key":"rf3","unstructured":"D. K.\u00a0Arrowsmith and C. M.\u00a0Place, An Introduction to Dynamical Systems (Cambridge University Press, Cambridge, 1990)\u00a0pp. 211\u2013215."},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1006\/jmaa.1999.6460"},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1006\/jdeq.2000.3783"},{"key":"rf6","doi-asserted-by":"publisher","DOI":"10.1016\/0377-0427(90)90353-2"},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511665639"},{"key":"rf8","doi-asserted-by":"publisher","DOI":"10.1109\/31.16563"},{"key":"rf9","volume-title":"Nonlinear Models of Fluctuating Growth","author":"Cugno F.","year":"1984"},{"key":"rf10","doi-asserted-by":"publisher","DOI":"10.1016\/0167-2789(87)90049-2"},{"key":"rf11","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1140-2"},{"key":"rf12","unstructured":"G.\u00a0Iooss, Bifurcations of Maps and Applications, Mathematical Studies\u00a036 (North Holland, Amsterdam, 1979)\u00a0pp. 27\u201347."},{"key":"rf13","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-2421-9"},{"key":"rf14","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-6374-6"},{"key":"rf15","volume-title":"Methods of Normal Forms","author":"Nayfeh A. H.","year":"1993"},{"key":"rf17","doi-asserted-by":"publisher","DOI":"10.1016\/0022-0396(73)90062-4"},{"key":"rf18","doi-asserted-by":"publisher","DOI":"10.1007\/BF02684366"},{"key":"rf19","doi-asserted-by":"publisher","DOI":"10.1007\/BF03167860"},{"key":"rf20","doi-asserted-by":"publisher","DOI":"10.1137\/0134013"},{"key":"rf21","doi-asserted-by":"publisher","DOI":"10.1112\/blms\/15.3.177"},{"key":"rf22","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-4067-7"},{"key":"rf23","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127499001401"}],"container-title":["International Journal of Bifurcation and Chaos"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218127403007667","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T20:16:52Z","timestamp":1565122612000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218127403007667"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,7]]},"references-count":22,"journal-issue":{"issue":"07","published-online":{"date-parts":[[2011,11,20]]},"published-print":{"date-parts":[[2003,7]]}},"alternative-id":["10.1142\/S0218127403007667"],"URL":"https:\/\/doi.org\/10.1142\/s0218127403007667","relation":{},"ISSN":["0218-1274","1793-6551"],"issn-type":[{"value":"0218-1274","type":"print"},{"value":"1793-6551","type":"electronic"}],"subject":[],"published":{"date-parts":[[2003,7]]}}}