{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,29]],"date-time":"2026-05-29T19:28:20Z","timestamp":1780082900108,"version":"3.54.0"},"reference-count":57,"publisher":"World Scientific Pub Co Pte Ltd","issue":"15","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2023,12,15]]},"abstract":"<jats:p> In this paper, we deal with a classical object, namely, a nonhyperbolic limit cycle in a system of smooth autonomous ordinary differential equations. While the existence of a center manifold near such a cycle was assumed in several studies on cycle bifurcations based on periodic normal forms, no proofs were available in the literature until recently. The main goal of this paper is to give an elementary proof of the existence of a periodic smooth locally invariant center manifold near a nonhyperbolic cycle in finite-dimensional ordinary differential equations by using the Lyapunov\u2013Perron method. In addition, we provide several explicit examples of analytic vector fields admitting (non)-unique, (non)-[Formula: see text]-smooth and (non)-analytic periodic center manifolds. <\/jats:p>","DOI":"10.1142\/s0218127423501845","type":"journal-article","created":{"date-parts":[[2023,12,12]],"date-time":"2023-12-12T06:08:11Z","timestamp":1702361291000},"source":"Crossref","is-referenced-by-count":4,"title":["Periodic Center Manifolds for Nonhyperbolic Limit Cycles in ODEs"],"prefix":"10.1142","volume":"33","author":[{"given":"Bram","family":"Lentjes","sequence":"first","affiliation":[{"name":"Department of Mathematics, Hasselt University, 3590 Diepenbeek, Belgium"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mattias","family":"Windmolders","sequence":"additional","affiliation":[{"name":"Department of Mathematics, KU Leuven, 3000 Leuven, Belgium"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2289-0801","authenticated-orcid":false,"given":"Yuri A.","family":"Kuznetsov","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Utrecht University, 3508 TA Utrecht, The Netherlands"},{"name":"Department of Applied Mathematics, University of Twente, 7500 AE Enschede, The Netherlands"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"219","published-online":{"date-parts":[[2023,12,11]]},"reference":[{"key":"S0218127423501845BIB001","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-11080-6"},{"key":"S0218127423501845BIB002","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-322-96657-5_1"},{"key":"S0218127423501845BIB003","doi-asserted-by":"publisher","DOI":"10.1137\/19M1243993"},{"key":"S0218127423501845BIB004","doi-asserted-by":"publisher","DOI":"10.1007\/BF01845701"},{"key":"S0218127423501845BIB005","doi-asserted-by":"publisher","DOI":"10.1512\/iumj.2003.52.2245"},{"key":"S0218127423501845BIB006","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-5929-9"},{"key":"S0218127423501845BIB007","doi-asserted-by":"publisher","DOI":"10.1016\/0022-247X(71)90221-6"},{"key":"S0218127423501845BIB008","doi-asserted-by":"publisher","DOI":"10.1080\/10236198.2015.1045889"},{"key":"S0218127423501845BIB009","doi-asserted-by":"publisher","DOI":"10.1080\/14689367.2018.1531972"},{"key":"S0218127423501845BIB010","doi-asserted-by":"publisher","DOI":"10.1016\/0022-0396(88)90007-1"},{"key":"S0218127423501845BIB011","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2018.04.021"},{"key":"S0218127423501845BIB012","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-64533-5"},{"key":"S0218127423501845BIB013","doi-asserted-by":"publisher","DOI":"10.1007\/BF01457866"},{"key":"S0218127423501845BIB014","doi-asserted-by":"publisher","DOI":"10.1017\/S0308210500026731"},{"key":"S0218127423501845BIB015","first-page":"67","volume-title":"Volterra Integrodifferential Equations in Banach Spaces and Applications (Trento, 1987)","author":"Cl\u00e9ment P.","year":"1989"},{"key":"S0218127423501845BIB016","first-page":"95","volume-title":"Semigroup Theory and Applications (Trieste, 1987)","author":"Cl\u00e9ment P.","year":"1989"},{"key":"S0218127423501845BIB017","doi-asserted-by":"publisher","DOI":"10.1137\/120874904"},{"key":"S0218127423501845BIB018","doi-asserted-by":"publisher","DOI":"10.1016\/j.physd.2013.12.002"},{"key":"S0218127423501845BIB019","doi-asserted-by":"publisher","DOI":"10.1145\/779359.779362"},{"key":"S0218127423501845BIB020","doi-asserted-by":"publisher","DOI":"10.1080\/13873950701742754"},{"key":"S0218127423501845BIB021","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0085429"},{"key":"S0218127423501845BIB022","first-page":"107","volume-title":"Semigroup Theory and Evolution Equations (Delft, 1989)","author":"Diekmann O.","year":"1991"},{"key":"S0218127423501845BIB023","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-4206-2"},{"key":"S0218127423501845BIB024","doi-asserted-by":"publisher","DOI":"10.1137\/060659211"},{"key":"S0218127423501845BIB026","doi-asserted-by":"publisher","DOI":"10.2991\/978-94-6239-003-4"},{"key":"S0218127423501845BIB027","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1140-2"},{"key":"S0218127423501845BIB028","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-4342-7"},{"key":"S0218127423501845BIB029","volume-title":"Ordinary Differential Equations","author":"Hale J. 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