{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T13:31:55Z","timestamp":1787232715305,"version":"build-2736575974"},"reference-count":45,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Appl. Dyn. Syst."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>We investigate the bifurcation structure of stationary localized patterns of the two-dimensional Swift\u2013Hohenberg equation on an infinitely long cylinder and on the plane. On cylinders, we find localized roll, square, and stripe patches that exhibit snaking and nonsnaking behavior on the same bifurcation branch. Some of these patterns snake between four saddle-node limits; in this case, recent analytical results predict the existence of a rich bifurcation structure to asymmetric solutions, and we trace out these branches and the PDE spectra along these branches. On the plane, we study the bifurcation structure of fully localized roll structures, which are often referred to as worms. In all the above cases, we use geometric ideas and spatial-dynamics techniques to explain the phenomena that we encounter.<\/jats:p>","DOI":"10.1137\/100782747","type":"journal-article","created":{"date-parts":[[2010,7,1]],"date-time":"2010-07-01T18:03:28Z","timestamp":1278007408000},"page":"704-733","source":"Crossref","is-referenced-by-count":98,"title":["To Snake or Not to Snake in the Planar Swift\u2013Hohenberg Equation"],"prefix":"10.1137","volume":"9","author":[{"given":"Daniele","family":"Avitabile","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"David J. B.","family":"Lloyd","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"John","family":"Burke","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Edgar","family":"Knobloch","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bj\u00f6rn","family":"Sandstede","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,7,1]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.85.756"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.79.2983"},{"key":"R3","unstructured":"D. Avitabile,\n                      Computation of Planar Patterns and Their Stability\n                      , Ph.D. thesis, Department of Mathematics, University of Surrey, Surrey, UK, 2008."},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1017\/S0022112006000759"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1137\/080713306"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.60.3910"},{"key":"R7","doi-asserted-by":"crossref","unstructured":"U. Bortolozzo, M. G. Clerc, and S. Residori,\n                      Solitary localized structures in a liquid crystal light-valve experiment\n                      , New J. Phys., 11 (2009), paper 093037.","DOI":"10.1088\/1367-2630\/11\/9\/093037"},{"key":"R8","doi-asserted-by":"crossref","unstructured":"J. Burke and E. Knobloch,\n                      Localized states in the generalized Swift-Hohenberg equation\n                      , Phys. Rev. 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Crawford and E. Knobloch,\n                      Symmetry and symmetry-breaking bifurcations in fluid mechanics\n                      , in Ann. Rev. Fluid Mech. 23, Annual Reviews, Palo Alto, 1991, pp. 341\u2013387.","DOI":"10.1146\/annurev.fl.23.010191.002013"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.65.851"},{"key":"R16","unstructured":"H. Dankowicz and F. Schilder,\n                      A new development platform for parameter continuation and bifurcation analysis in nonlinear dynamical systems\n                      , in Proceedings of the 8th World Congress on Computational Mechanics (WCCM8), Venice, Italy, 2008."},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.69.2511"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.77.2475"},{"key":"R19","unstructured":"E. J. Doedel and B. Oldeman, AUTO07P\n                      : Continuation and Bifurcation Software for Ordinary Differential Equations\n                      , Technical report, Faculty of Engineering and Computer Science, Concordia University, Montreal, 2009."},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1142\/S021812740902444X"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1145\/1089014.1089021"},{"key":"R22","unstructured":"R. B. 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Lett., 97 (2006), paper 044502.","DOI":"10.1103\/PhysRevLett.97.044502"},{"key":"R29","doi-asserted-by":"publisher","DOI":"10.1038\/369215a0"},{"key":"R30","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.83.3190"},{"key":"R31","doi-asserted-by":"publisher","DOI":"10.1137\/070707622"},{"key":"R32","doi-asserted-by":"publisher","DOI":"10.1006\/jdeq.1997.3303"},{"key":"R33","doi-asserted-by":"publisher","DOI":"10.1080\/02678299308026314"},{"key":"R34","doi-asserted-by":"crossref","unstructured":"L. M. Pismen,\n                      Patterns and Interfaces in Dissipative Dynamics\n                      , Springer-Verlag, Berlin, 2006.","DOI":"10.1007\/3-540-30431-2"},{"key":"R35","doi-asserted-by":"publisher","DOI":"10.1016\/0167-2789(86)90104-1"},{"key":"R36","doi-asserted-by":"crossref","unstructured":"R. Richter and I. V. Barashenkov,\n                      Two-dimensional solitons on the surface of magnetic fluids\n                      , Phys. Rev. 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Trefethen,\n                      Spectral Methods in\n                      MATLAB, Software Environ. Tools 10, SIAM, Philadelphia, 2000.","DOI":"10.1137\/1.9780898719598"},{"key":"R44","doi-asserted-by":"publisher","DOI":"10.1038\/382793a0"},{"key":"R45","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-2789(98)00309-1"}],"container-title":["SIAM Journal on Applied Dynamical Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/100782747","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T13:19:17Z","timestamp":1787231957000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/100782747"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,1]]},"references-count":45,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2010,1]]}},"alternative-id":["10.1137\/100782747"],"URL":"https:\/\/doi.org\/10.1137\/100782747","relation":{},"ISSN":["1536-0040"],"issn-type":[{"value":"1536-0040","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,1]]}}}