{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,19]],"date-time":"2026-03-19T18:09:48Z","timestamp":1773943788745,"version":"3.50.1"},"reference-count":37,"publisher":"World Scientific Pub Co Pte Ltd","issue":"06","funder":[{"DOI":"10.13039\/501100001809","name":"The National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["11431008"],"award-info":[{"award-number":["11431008"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100001809","name":"The National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["11571318"],"award-info":[{"award-number":["11571318"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100001809","name":"The National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["11162020"],"award-info":[{"award-number":["11162020"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100000038","name":"the Natural Sciences and Engineering Research Council of Canada","doi-asserted-by":"crossref","award":["R2686A02"],"award-info":[{"award-number":["R2686A02"]}],"id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2018,6,15]]},"abstract":"<jats:p> In the two articles in Appl. Math. Comput., J. Gin\u00e9 [2012a, 2012b] studied the number of small limit cycles bifurcating from the origin of the system: [Formula: see text], [Formula: see text], where [Formula: see text] and [Formula: see text] are homogeneous polynomials of degree [Formula: see text]. It is shown that the maximal number of the small limit cycles, denoted by [Formula: see text], satisfies [Formula: see text] for [Formula: see text]; and [Formula: see text], [Formula: see text]. It seems that the correct answer for their case [Formula: see text] should be [Formula: see text]. In this paper, we apply Hopf bifurcation theory and normal form computation, and perturb the isolated, nondegenerate center (the origin) to prove that [Formula: see text] for [Formula: see text]; and [Formula: see text] for [Formula: see text], which improve Gin\u00e9\u2019s results with one more limit cycle for each case. <\/jats:p>","DOI":"10.1142\/s0218127418500785","type":"journal-article","created":{"date-parts":[[2018,6,21]],"date-time":"2018-06-21T03:03:03Z","timestamp":1529550183000},"page":"1850078","source":"Crossref","is-referenced-by-count":13,"title":["An Improvement on the Number of Limit Cycles Bifurcating from a Nondegenerate Center of Homogeneous Polynomial Systems"],"prefix":"10.1142","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8848-0584","authenticated-orcid":false,"given":"Pei","family":"Yu","sequence":"first","affiliation":[{"name":"Department of Mathematics, Shanghai Normal University, Shanghai 200234, P. R. China"},{"name":"Department of Applied Mathematics, Western University, London, Ontario, N6A 5B7, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Maoan","family":"Han","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Shanghai Normal University, Shanghai 200234, P. R. China"},{"name":"School of Mathematics Sciences, Qufu Normal University, Qufu 273165, P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jibin","family":"Li","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Huaqiao University, Quanzhou, Fujian 362021, P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2018,6,21]]},"reference":[{"key":"S0218127418500785BIB001","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4684-0147-9"},{"key":"S0218127418500785BIB002","first-page":"181","volume":"30","author":"Bautin N.","year":"1952","journal-title":"Matem. Sb."},{"key":"S0218127418500785BIB003","doi-asserted-by":"publisher","DOI":"10.1017\/S030821050001341X"},{"key":"S0218127418500785BIB004","doi-asserted-by":"publisher","DOI":"10.1134\/S0012266108020158"},{"key":"S0218127418500785BIB005","doi-asserted-by":"publisher","DOI":"10.1016\/S0007-4497(02)01111-9"},{"key":"S0218127418500785BIB006","doi-asserted-by":"publisher","DOI":"10.1007\/s10114-005-0614-5"},{"key":"S0218127418500785BIB007","first-page":"1350154-1","volume":"23","author":"Chen C.","year":"2013","journal-title":"Int. J. Bifurcation and Chaos"},{"key":"S0218127418500785BIB008","first-page":"23","volume-title":"Trends in Mathematics: Differential Equations with Symbolic Computation","author":"Christopher C.","year":"2006"},{"key":"S0218127418500785BIB009","doi-asserted-by":"publisher","DOI":"10.1007\/s10114-005-0623-4"},{"key":"S0218127418500785BIB010","first-page":"1870","volume":"188","author":"Gin\u00e9 J.","year":"2007","journal-title":"Appl. Math. Comput."},{"key":"S0218127418500785BIB011","first-page":"4703","volume":"218","author":"Gin\u00e9 J.","year":"2012","journal-title":"Appl. Math. Comput."},{"key":"S0218127418500785BIB012","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2012.02.044"},{"key":"S0218127418500785BIB013","volume-title":"Bifurcation of Limit Cycles of Planar Systems, Handbook of Differential Equations, Ordinary Differential Equations","volume":"3","author":"Han M.","year":"2006"},{"key":"S0218127418500785BIB014","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127409025250"},{"key":"S0218127418500785BIB015","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4471-2918-9"},{"key":"S0218127418500785BIB016","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2017.11.025"},{"key":"S0218127418500785BIB017","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1902-00923-3"},{"key":"S0218127418500785BIB018","doi-asserted-by":"publisher","DOI":"10.1007\/BF02810590"},{"key":"S0218127418500785BIB019","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127403006352"},{"key":"S0218127418500785BIB020","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2009.01.038"},{"key":"S0218127418500785BIB021","doi-asserted-by":"publisher","DOI":"10.1007\/s12346-010-0024-7"},{"key":"S0218127418500785BIB022","first-page":"293","volume":"2","author":"Lloyd N.","year":"2012","journal-title":"J. Appl. Anal. Comput."},{"key":"S0218127418500785BIB023","first-page":"1051","volume":"11","author":"Shi S.","year":"1979","journal-title":"Sci. Sin."},{"key":"S0218127418500785BIB024","first-page":"153","volume":"23","author":"Shi S.","year":"1980","journal-title":"Sci. Sin."},{"key":"S0218127418500785BIB025","first-page":"36","volume":"1","author":"Sibirskii K. S.","year":"1965","journal-title":"Diff. Eqs."},{"key":"S0218127418500785BIB026","doi-asserted-by":"publisher","DOI":"10.1007\/BF03025291"},{"key":"S0218127418500785BIB027","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127413501046"},{"key":"S0218127418500785BIB028","doi-asserted-by":"publisher","DOI":"10.1016\/j.cnsns.2013.11.019"},{"key":"S0218127418500785BIB029","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2015.09.016"},{"key":"S0218127418500785BIB030","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2018.01.022"},{"key":"S0218127418500785BIB031","doi-asserted-by":"publisher","DOI":"10.1006\/jsvi.1997.1347"},{"key":"S0218127418500785BIB032","doi-asserted-by":"publisher","DOI":"10.1016\/S0960-0779(04)00599-5"},{"key":"S0218127418500785BIB033","doi-asserted-by":"publisher","DOI":"10.1016\/j.cnsns.2008.10.010"},{"key":"S0218127418500785BIB034","first-page":"143","volume":"1","author":"Yu P.","year":"2011","journal-title":"J. Appl. Anal. Comput."},{"key":"S0218127418500785BIB035","doi-asserted-by":"publisher","DOI":"10.1016\/j.cnsns.2013.12.014"},{"key":"S0218127418500785BIB036","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/7\/1\/013"},{"key":"S0218127418500785BIB037","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/8\/5\/011"}],"container-title":["International Journal of Bifurcation and Chaos"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218127418500785","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T16:38:31Z","timestamp":1565109511000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218127418500785"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,6,15]]},"references-count":37,"journal-issue":{"issue":"06","published-online":{"date-parts":[[2018,6,21]]},"published-print":{"date-parts":[[2018,6,15]]}},"alternative-id":["10.1142\/S0218127418500785"],"URL":"https:\/\/doi.org\/10.1142\/s0218127418500785","relation":{},"ISSN":["0218-1274","1793-6551"],"issn-type":[{"value":"0218-1274","type":"print"},{"value":"1793-6551","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,6,15]]}}}