{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,15]],"date-time":"2025-12-15T19:51:53Z","timestamp":1765828313935,"version":"3.41.2"},"reference-count":22,"publisher":"World Scientific Pub Co Pte Ltd","issue":"13","funder":[{"name":"Characteristic Innovation Projects of Colleges and Universities in Guangdong Province","award":["2019KTSCX088"],"award-info":[{"award-number":["2019KTSCX088"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11771197"],"award-info":[{"award-number":["11771197"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Key Subject Program of Lingnan Normal University","award":["1171518004"],"award-info":[{"award-number":["1171518004"]}]},{"DOI":"10.13039\/501100003453","name":"Natural Science Foundation of Guangdong Province","doi-asserted-by":"publisher","award":["2022A1515010964"],"award-info":[{"award-number":["2022A1515010964"]}],"id":[{"id":"10.13039\/501100003453","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2023,10]]},"abstract":"<jats:p> In this paper, the bounded invariant surfaces of a generalized Langford system are discussed. Firstly, by the first integrals of systems restricted in the Poincar\u00e9 sections of a periodic orbit, the accurate expressions of a heteroclinic orbit, a family of invariant tori and a heteroclinic invariant ellipsoid are given near a periodic orbit. Then, applying the successor functions to compute the periods of periodic orbits for the systems in the Poincar\u00e9 sections, we present the parameter conditions for the existence of periodic orbits with any periods on these invariant tori. Finally, using the averaging theory and the theory of the Poincar\u00e9 bifurcation and by determining the monotonicity of the ratio of two Abelian integrals, we give the conditions respectively such that the system has a unique invariant torus and a unique heteroclinic invariant ellipsoid near a zero-Hopf equilibrium. <\/jats:p>","DOI":"10.1142\/s0218127423501535","type":"journal-article","created":{"date-parts":[[2023,10,31]],"date-time":"2023-10-31T02:10:07Z","timestamp":1698718207000},"source":"Crossref","is-referenced-by-count":3,"title":["Invariant Tori and Heteroclinic Invariant Ellipsoids of a Generalized Hopf\u2013Langford System"],"prefix":"10.1142","volume":"33","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0116-865X","authenticated-orcid":false,"given":"Jiyu","family":"Zhong","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Lingnan Normal University, Zhanjiang, Guangdong 524048, P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ying","family":"Liang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Lingnan Normal University, Zhanjiang, Guangdong 524048, P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2023,10,30]]},"reference":[{"key":"S0218127423501535BIB001","doi-asserted-by":"crossref","DOI":"10.1887\/0750308036","volume-title":"Global Analysis of Dynamical Systems","author":"Broer H. W.","year":"2001"},{"key":"S0218127423501535BIB002","doi-asserted-by":"crossref","first-page":"4555","DOI":"10.1016\/j.jde.2019.10.031","volume":"268","author":"C\u00e2ndido M. R.","year":"2020","journal-title":"J. Diff. Eqs."},{"key":"S0218127423501535BIB003","first-page":"1","volume":"9","author":"C\u00e2ndido M. 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