{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T13:31:03Z","timestamp":1787232663643,"version":"build-2736575974"},"reference-count":6,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Appl. Dyn. Syst."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>We study the symmetries of periodic solutions obtained from Hopf bifurcation in systems with finite abelian symmetries. The H mod K theorem gives necessary and sufficient conditions for the existence of periodic solutions with spatial symmetries K and spatio-temporal symmetries H in systems with finite symmetry group $\\Gamma$. Our main result, the abelian Hopf H mod K theorem, gives necessary and sufficient conditions for when these H mod K periodic solutions can occur by Hopf bifurcation when $\\Gamma$ is a finite abelian group. We give examples of our results in the case when the symmetry group $\\Gamma=\\mathbf{Z}_l\\times\\mathbf{Z}_k$ acts on $\\mathbf{R}^l\\times\\mathbf{R}^k$ by permutation of coordinates. In this case, we classify the H mod K periodic solutions that are obtainable by a generic Hopf bifurcation and show that there exist families of H mod K periodic solutions that cannot be obtained by Hopf bifurcation.<\/jats:p>","DOI":"10.1137\/090756582","type":"journal-article","created":{"date-parts":[[2010,4,7]],"date-time":"2010-04-07T18:08:11Z","timestamp":1270663691000},"page":"283-291","source":"Crossref","is-referenced-by-count":7,"title":["The Abelian Hopf\n                    <i>H<\/i>\n                    mod\n                    <i>K<\/i>\n                    Theorem"],"prefix":"10.1137","volume":"9","author":[{"given":"Natasha","family":"Filipski","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Martin","family":"Golubitsky","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,4,7]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1007\/s002850000058"},{"key":"R2","unstructured":"N. Filipski,\n                      Periodic Solutions in Systems with Finite Abelian Symmetries\n                      , Ph.D. thesis, Department of Mathematics, University of Houston, Houston, TX, 2008."},{"key":"R3","doi-asserted-by":"crossref","unstructured":"M. Golubitsky and I. Stewart,\n                      The Symmetry Perspective\n                      , Birkh\u00e4user-Verlag, Berlin, 2003.","DOI":"10.1007\/978-3-0348-8167-8"},{"key":"R4","doi-asserted-by":"crossref","unstructured":"M. Golubitsky, I. Stewart, and D. Schaeffer,\n                      Singularities and Groups in Bifurcation Theory\n                      , Vol. II, Springer-Verlag, New York, 1988.","DOI":"10.1007\/978-1-4612-4574-2"},{"key":"R5","unstructured":"P. Lancaster and M. Tismenetsky,\n                      The Theory of Matrices\n                      , 2nd ed., Academic Press, New York, 1985."},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1007\/BF00247751"}],"container-title":["SIAM Journal on Applied Dynamical Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/090756582","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T13:12:25Z","timestamp":1787231545000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/090756582"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,1]]},"references-count":6,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2010,1]]}},"alternative-id":["10.1137\/090756582"],"URL":"https:\/\/doi.org\/10.1137\/090756582","relation":{},"ISSN":["1536-0040"],"issn-type":[{"value":"1536-0040","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,1]]}}}