{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,6]],"date-time":"2026-10-06T17:41:33Z","timestamp":1791308493196,"version":"4.3.1"},"reference-count":19,"publisher":"World Scientific Pub Co Pte Ltd","issue":"03","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2003,3]]},"abstract":"<jats:p>It is well known that a nonorientable manifold in a three-dimensional vector field is topologically equivalent to a M\u00f6bius strip. The most frequently used example is the unstable manifold of a periodic orbit that just lost its stability in a period-doubling bifurcation. However, there are not many explicit studies in the literature in the context of dynamical systems, and so far only qualitative sketches could be given as illustrations. We give an overview of the possible bifurcations in three-dimensional vector fields that create nonorientable manifolds. We mainly focus on nonorientable manifolds of periodic orbits, because they are the key building blocks. This is illustrated with invariant manifolds of three-dimensional vector fields that arise from applications. These manifolds were computed with a new algorithm for computing two-dimensional manifolds.<\/jats:p>","DOI":"10.1142\/s0218127403006777","type":"journal-article","created":{"date-parts":[[2003,4,8]],"date-time":"2003-04-08T07:12:24Z","timestamp":1049785944000},"page":"553-570","source":"Crossref","is-referenced-by-count":22,"title":["NONORIENTABLE MANIFOLDS IN THREE-DIMENSIONAL VECTOR FIELDS"],"prefix":"10.1142","volume":"13","author":[{"given":"HINKE M.","family":"OSINGA","sequence":"first","affiliation":[{"name":"Department of Engineering Mathematics, University of Bristol, Bristol BS8 1TR, UK"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1016\/0167-2789(85)90093-4"},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01213607"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1007\/s000330050044"},{"key":"rf4","volume-title":"The Graphic Work","author":"Escher M. C.","year":"2000"},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1512\/iumj.1972.21.21017"},{"key":"rf6","doi-asserted-by":"publisher","DOI":"10.1007\/BF01010828"},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.1007\/BF01608556"},{"key":"rf8","series-title":"Mathematical Surveys and Monographs","volume-title":"Nonlocal Bifurcations","volume":"66","author":"Ilyashenko Yu. S.","year":"1999"},{"key":"rf9","doi-asserted-by":"publisher","DOI":"10.1023\/A:1019104828180"},{"key":"rf10","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1998.6059"},{"key":"rf11","doi-asserted-by":"publisher","DOI":"10.1063\/1.166450"},{"key":"rf12","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1208-9_9"},{"key":"rf13","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-2421-9"},{"key":"rf14","doi-asserted-by":"publisher","DOI":"10.1175\/1520-0469(1963)020<0130:DNF>2.0.CO;2"},{"key":"rf15","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-2789(00)00133-0"},{"key":"rf18","doi-asserted-by":"publisher","DOI":"10.1080\/14786440108564176"},{"key":"rf19","doi-asserted-by":"publisher","DOI":"10.1007\/BF02219223"},{"key":"rf21","volume-title":"Calculus on Manifolds","author":"Spivak M.","year":"1965"},{"key":"rf22","doi-asserted-by":"publisher","DOI":"10.1016\/S0030-4018(99)00603-3"}],"container-title":["International Journal of Bifurcation and Chaos"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218127403006777","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T20:13:47Z","timestamp":1565122427000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218127403006777"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,3]]},"references-count":19,"journal-issue":{"issue":"03","published-online":{"date-parts":[[2011,11,20]]},"published-print":{"date-parts":[[2003,3]]}},"alternative-id":["10.1142\/S0218127403006777"],"URL":"https:\/\/doi.org\/10.1142\/s0218127403006777","relation":{},"ISSN":["0218-1274","1793-6551"],"issn-type":[{"value":"0218-1274","type":"print"},{"value":"1793-6551","type":"electronic"}],"subject":[],"published":{"date-parts":[[2003,3]]}}}