{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T13:31:09Z","timestamp":1787232669687,"version":"build-2736575974"},"reference-count":53,"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 modify the freezing method introduced by Beyn and Th\u00fcmmler in 2004 for analyzing rigidly rotating spiral waves in excitable media. The proposed method is designed to stably determine the rotation frequency and the core radius of rotating spirals, as well as the approximate shape of spiral waves in unbounded domains. In particular, we introduce spiral wave boundary conditions based on geometric approximations of spiral wave solutions by Archimedean spirals and by involutes of circles. We further propose a simple implementation of boundary conditions for the case when the inhibitor is nondiffusive, a case which had previously caused spurious oscillations. We then utilize our method to numerically analyze the large core limit. The proposed method allows us to investigate the case close to criticality where spiral waves acquire infinite core radius $r_c$ and zero rotation frequency $\\omega$, before they begin to develop into retracting fingers. We confirm the linear scaling regime of a drift bifurcation for the rotation frequency and the core radius of spiral wave solutions close to criticality. This regime is unattainable with conventional numerical methods.<\/jats:p>","DOI":"10.1137\/090780055","type":"journal-article","created":{"date-parts":[[2010,6,2]],"date-time":"2010-06-02T18:09:01Z","timestamp":1275502141000},"page":"536-567","source":"Crossref","is-referenced-by-count":11,"title":["The Large Core Limit of Spiral Waves in Excitable Media: A Numerical Approach"],"prefix":"10.1137","volume":"9","author":[{"given":"Sebastian","family":"Hermann","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Georg A.","family":"Gottwald","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,6,2]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/12\/4\/301"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1016\/0167-2789(91)90194-E"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.72.164"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1038\/35036035"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1137\/07070749X"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1137\/030600515"},{"key":"R7","doi-asserted-by":"crossref","unstructured":"W.J. Beyn and V. Th\u00fcmmler,\n                      Phase conditions, symmetries, and PDE continuation\n                      , in Numerical Continuation Methods for Dynamical Systems, B. Krauskopf, H. Osinga, and J. Gal\u00e1n-Vioque, eds., Springer, Dordrecht, The Netherlands, 2007, pp. 301\u2013330.","DOI":"10.1007\/978-1-4020-6356-5_10"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1002\/gamm.200910002"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1038\/355349a0"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127495001149"},{"key":"R11","doi-asserted-by":"crossref","unstructured":"D. R. Durran,\n                      Numerical Methods for Wave Equations in Geophysical Fluid Dynamics\n                      , Springer, New York, 1999.","DOI":"10.1007\/978-1-4757-3081-4"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1016\/S0960-0779(97)00111-2"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1016\/S0960-0779(01)00129-1"},{"key":"R14","doi-asserted-by":"crossref","first-page":"479","DOI":"10.4171\/dm\/11","volume":"1","author":"Fiedler B.","year":"1996","journal-title":"Doc. Math."},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1007\/s002050050126"},{"key":"R16","doi-asserted-by":"crossref","unstructured":"A. J. Foulkes and V. N. Biktashev,\n                      Riding a Spiral Wave: Numerical Simulation of Spiral Waves in a Co-Moving Frame of Reference\n                      , preprint, 2010; arXiv:1001.4454v1.","DOI":"10.1103\/PhysRevE.81.046702"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1007\/s003329900040"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1007\/s003329910004"},{"key":"R19","unstructured":"M. Golubitsky and I. Stewart,\n                      The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space\n                      , Birkh\u00e4user Boston, Cambridge, MA, 2003."},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1063\/1.1772552"},{"key":"R21","doi-asserted-by":"crossref","unstructured":"R. Gray, J. Wikswo, and N. Otani,\n                      Origin choice and petal loss in the flower garden of spiral wave tip trajectories\n                      , Chaos, 19 (2009), paper 033118.","DOI":"10.1063\/1.3204256"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.79.665"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.60.5073"},{"key":"R24","doi-asserted-by":"crossref","unstructured":"S. Hermann,\n                      Numerical Investigation of the Large Core Limit of Spiral Waves in Excitable Media\n                      , Ph.D. thesis, The University of Sydney, Australia, 2010.","DOI":"10.1137\/090780055"},{"key":"R25","doi-asserted-by":"crossref","unstructured":"R. Hoyle,\n                      Pattern Formation: An Introduction to Methods\n                      , Cambridge University Press, Cambridge, UK, 2006.","DOI":"10.1017\/CBO9780511616051"},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1039\/b104485k"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.46.5054"},{"key":"R28","doi-asserted-by":"publisher","DOI":"10.1137\/0521081"},{"key":"R29","doi-asserted-by":"publisher","DOI":"10.1063\/1.166115"},{"key":"R30","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.86.175"},{"key":"R31","doi-asserted-by":"publisher","DOI":"10.1063\/1.1494875"},{"key":"R32","doi-asserted-by":"publisher","DOI":"10.1016\/0167-2789(94)90054-X"},{"key":"R33","doi-asserted-by":"publisher","DOI":"10.1016\/0167-2789(87)90068-6"},{"key":"R34","unstructured":"W. Press, B. Flannery, S. Teukolsky, and W. Vetterling,\n                      Numerical Recipes in C: The Art of Scientific Programming\n                      , Cambridge University Press, Cambridge, UK, 1992."},{"key":"R35","doi-asserted-by":"crossref","unstructured":"B. Sandstede and A. Scheel,\n                      Curvature effects on spiral spectra: Generation of point eigenvalues near branch points\n                      , Phys. Rev. E, 73 (2006), paper 16217.","DOI":"10.1103\/PhysRevE.73.016217"},{"key":"R36","first-page":"153","volume":"324","author":"Sandstede B.","year":"1997","journal-title":"C. R. Acad. Sci. Math."},{"key":"R37","doi-asserted-by":"publisher","DOI":"10.1006\/jdeq.1997.3326"},{"key":"R38","doi-asserted-by":"publisher","DOI":"10.1007\/s003329900076"},{"key":"R39","doi-asserted-by":"publisher","DOI":"10.1016\/0167-2789(91)90210-Z"},{"key":"R40","unstructured":"V. Th\u00fcmmler,\n                      Numerical Analysis of the Method of Freezing Traveling Waves\n                      , Ph.D. thesis, Universit\u00e4t Bielefeld, Bielefeld, Germany, 2005."},{"key":"R41","unstructured":"V. Th\u00fcmmler,\n                      Numerical bifurcation analysis of relative equilibria with Femlab\n                      , in Proceedings of the COMSOL Users Conference (Comsol Anwenderkonferenz), Frankfurt, Femlab GmbH, Goettingen, Germany, 2006; http:\/\/www.comsol.com\/papers\/1544."},{"key":"R42","doi-asserted-by":"publisher","DOI":"10.1137\/070698324"},{"key":"R43","doi-asserted-by":"publisher","DOI":"10.1016\/0167-2789(88)90062-0"},{"key":"R44","first-page":"205","volume":"16","author":"Wiener N.","year":"1946","journal-title":"Arch. Inst. Cardiol. Mexico","ISSN":"https:\/\/id.crossref.org\/issn\/0020-3785","issn-type":"print"},{"key":"R45","doi-asserted-by":"publisher","DOI":"10.1126\/science.175.4022.634"},{"key":"R46","unstructured":"A. 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