{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,20]],"date-time":"2026-01-20T09:57:02Z","timestamp":1768903022295,"version":"3.49.0"},"reference-count":8,"publisher":"World Scientific Pub Co Pte Ltd","issue":"07","funder":[{"name":"Grant-in-Aid for Young Scientists (KAKENHI) (B)","award":["50738383"],"award-info":[{"award-number":["50738383"]}]},{"name":"ImPACT Program of Council for Science, Technology and Innovation"},{"DOI":"10.13039\/501100003382","name":"Core Research for Evolutional Science and Technology, Japan Science and Technology Agency","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100003382","id-type":"DOI","asserted-by":"publisher"}]},{"name":"the Aihara Project, the FIRST program"},{"name":"the Aihara Project, the FIRST program"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2016,6,30]]},"abstract":"<jats:p> We classify the local bifurcations of quasi-periodic [Formula: see text]-dimensional tori in maps (abbr. MT[Formula: see text]) and in flows (abbr. FT[Formula: see text]) for [Formula: see text]. It is convenient to classify these bifurcations into normal bifurcations and resonance bifurcations. Normal bifurcations of MT[Formula: see text] can be classified into four classes: namely, saddle-node, period doubling, double covering, and Neimark\u2013Sacker bifurcations. Furthermore, normal bifurcations of FT[Formula: see text] can be classified into three classes: saddle-node, double covering, and Neimark\u2013Sacker bifurcations. These bifurcations are determined by the type of the dominant Lyapunov bundle. Resonance bifurcations are well known as phase locking of quasi-periodic solutions. These bifurcations are classified into two classes for both MT[Formula: see text] and FT[Formula: see text]: namely, saddle-node cycle and heteroclinic cycle bifurcations of the [Formula: see text]-dimensional tori. The former is reversible, while the latter is irreversible. In addition, we propose a method for analyzing higher-dimensional tori, which uses one-dimensional tori in sections (abbr. ST[Formula: see text]) and zero-dimensional tori in sections (abbr. ST[Formula: see text]). The bifurcations of ST[Formula: see text] can be classified into five classes: saddle-node, period doubling, component doubling, double covering, and Neimark\u2013Sacker bifurcations. The bifurcations of ST[Formula: see text] can be classified into four classes: saddle-node, period doubling, component doubling, and Neimark\u2013Sacker bifurcations. Furthermore, we clarify the relationship between the bifurcations of ST[Formula: see text]\/ST[Formula: see text] and the bifurcations of MT[Formula: see text]\/FT[Formula: see text]. We present examples of all of these bifurcations. <\/jats:p>","DOI":"10.1142\/s0218127416300160","type":"journal-article","created":{"date-parts":[[2016,7,15]],"date-time":"2016-07-15T06:00:50Z","timestamp":1468562450000},"page":"1630016","source":"Crossref","is-referenced-by-count":21,"title":["Quasi-Periodic Bifurcations of Higher-Dimensional Tori"],"prefix":"10.1142","volume":"26","author":[{"given":"Motomasa","family":"Komuro","sequence":"first","affiliation":[{"name":"Center for Fundamental Education, Teikyo University of Science, 2525 Yatsusawa, Uenohara-shi, Yamanashi 409-0193, Japan"}]},{"given":"Kyohei","family":"Kamiyama","sequence":"additional","affiliation":[{"name":"Department of Electronics and Bioinformatics, Meiji University, 1-1-1 Higashi-Mita, Tama-ku, Kawasaki-shi, Kanagawa 214-8571, Japan"}]},{"given":"Tetsuro","family":"Endo","sequence":"additional","affiliation":[{"name":"Department of Electronics and Bioinformatics, Meiji University, 1-1-1 Higashi-Mita, Tama-ku, Kawasaki-shi, Kanagawa 214-8571, Japan"}]},{"given":"Kazuyuki","family":"Aihara","sequence":"additional","affiliation":[{"name":"Institute of Industrial Science, The University of Tokyo, Komaba, Meguro-ku, Tokyo 153-8505, Japan"}]}],"member":"219","published-online":{"date-parts":[[2016,7,15]]},"reference":[{"key":"S0218127416300160BIB001","doi-asserted-by":"publisher","DOI":"10.1016\/j.physd.2008.01.026"},{"key":"S0218127416300160BIB002","series-title":"Lecture Notes in Mathematics","volume-title":"Quasi-Periodic Motions in Families of Dynamical Systems","volume":"1645","author":"Broer H. W.","year":"1996"},{"key":"S0218127416300160BIB003","first-page":"828","volume":"73","author":"Endo T.","year":"1990","journal-title":"IEICE Trans. E"},{"key":"S0218127416300160BIB004","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127414300341"},{"key":"S0218127416300160BIB005","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-3978-7"},{"key":"S0218127416300160BIB006","first-page":"179","volume":"19","author":"Oseledets V. I.","year":"1968","journal-title":"Trudy MMO"},{"key":"S0218127416300160BIB007","doi-asserted-by":"publisher","DOI":"10.1007\/BF02684768"},{"key":"S0218127416300160BIB008","doi-asserted-by":"publisher","DOI":"10.1007\/BF01646553"}],"container-title":["International Journal of Bifurcation and Chaos"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218127416300160","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,9,24]],"date-time":"2019-09-24T15:04:32Z","timestamp":1569337472000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218127416300160"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,6,30]]},"references-count":8,"journal-issue":{"issue":"07","published-online":{"date-parts":[[2016,7,15]]},"published-print":{"date-parts":[[2016,6,30]]}},"alternative-id":["10.1142\/S0218127416300160"],"URL":"https:\/\/doi.org\/10.1142\/s0218127416300160","relation":{},"ISSN":["0218-1274","1793-6551"],"issn-type":[{"value":"0218-1274","type":"print"},{"value":"1793-6551","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,6,30]]}}}