{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T13:31:27Z","timestamp":1787232687532,"version":"3.56.0"},"reference-count":46,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Appl. Dyn. Syst."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>Isochrons are foliations of phase space that extend the notion of phase of a stable periodic orbit to the basin of attraction of this periodic orbit. Each point in the basin of attraction lies on only one isochron, and two points on the same isochron converge to the periodic orbit with the same phase. Global isochrons, that is, isochrons extended into the full basin of attraction rather than just a neighborhood of the periodic orbit, can form remarkable foliations. For example, accumulations of all isochrons can occur in arbitrarily small regions of phase space; the limit of such an accumulation is called the phaseless set, which lies on the boundary of the basin of attraction of the periodic orbit. Since global isochrons must typically be approximated numerically, such complicated geometries are often difficult to realize for actual examples. Indeed, the computation of global isochrons can be challenging, particularly for systems with multiple time scales. We present a novel method for computing isochrons via the continuation of a two-point boundary value problem, which is particularly effective for systems with multiple time scales. We use this method to compute global isochrons for a two-dimensional reduced Hodgkin\u2013Huxley model and illustrate that the one-dimensional isochrons for a planar multiple-time-scale system can accumulate in the interior of the basin of attraction of the periodic orbit in a way similar to two-dimensional isochrons accumulating on the boundary of a three-dimensional basin of attraction.<\/jats:p>","DOI":"10.1137\/090777244","type":"journal-article","created":{"date-parts":[[2010,11,2]],"date-time":"2010-11-02T18:32:28Z","timestamp":1288722748000},"page":"1201-1228","source":"Crossref","is-referenced-by-count":57,"title":["Continuation-based Computation of Global Isochrons"],"prefix":"10.1137","volume":"9","author":[{"given":"Hinke M.","family":"Osinga","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jeff","family":"Moehlis","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,11,2]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1007\/s10441-010-9099-4"},{"key":"R2","first-page":"37","volume":"31","author":"Beno\u00eet \u00c9.","year":"1981","journal-title":"Collect. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-0757","issn-type":"print"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1137\/0710052"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1162\/089976604322860668"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2004.12.003"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1016\/0378-4371(89)90006-X"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1007\/s00422-009-0344-3"},{"key":"R8","unstructured":"M. Desroches, J. M. Guckenheimer, B. Krauskopf, C. Kuehn, H. M. Osinga, and M. Wechselberger,\n                      Mixed-Mode Oscillations with Multiple Time Scales\n                      , preprint, Bristol Centre for Applied Nonlinear Mathematics # BCANM.1594, University of Bristol, Bristol, UK, 2010; SIAM Rev., submitted."},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1016\/j.apnum.2010.07.001"},{"key":"R10","doi-asserted-by":"crossref","unstructured":"E. J. Doedel,\n                      Lecture notes on numerical analysis of nonlinear equations\n                      , in Numerical Continuation Methods for Dynamical Systems: Path Following and Boundary Value Problems, B. Krauskopf, H. M. Osinga, and J. Gal\u00e1n-Vioque, eds., Springer-Verlag, New York, 2007, pp. 1\u201350.","DOI":"10.1007\/978-1-4020-6356-5_1"},{"key":"R11","unstructured":"E. J. Doedel,\n                      AUTO-\n                      07\n                      P: Continuation and Bifurcation Software for Ordinary Differential Equations\n                      , with major contributions by A. R. Champneys, F. Dercole, T. F. Fairgrieve, Yu. A. Kuznetsov, B. E. Oldeman, R. C. Paffenroth, B. Sandstede, X. J. Wang, and C. Zhang; software available via http:\/\/cmvl.cs.concordia.ca\/auto\/ http:\/\/cmvl.cs.concordia.ca\/auto\/."},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127408021439"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127409022804"},{"key":"R14","doi-asserted-by":"crossref","unstructured":"F. Dumortier and R. Roussarie,\n                      Canard Cycles and Center Manifolds\n                      , Mem. Amer. Math. Soc. 121 (577), AMS, Providence, RI, 1996.","DOI":"10.1090\/memo\/0577"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1137\/05062408X"},{"key":"R16","doi-asserted-by":"crossref","unstructured":"B. Ermentrout,\n                      Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students\n                      , Software Environ. Tools 14, SIAM, Philadelphia, 2002.","DOI":"10.1137\/1.9780898718195"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1007\/BF00160535"},{"key":"R18","doi-asserted-by":"crossref","unstructured":"L. Glass and M. C. Mackey,\n                      From Clocks to Chaos: The Rhythms of Life\n                      , Princeton University Press, Princeton, NJ, 1988.","DOI":"10.1515\/9780691221793"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1162\/neco.2006.18.4.817"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1109\/67.738317"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1007\/BF01273747"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1137\/080737666"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1152\/jn.00359.2004"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1137\/040602894"},{"key":"R25","doi-asserted-by":"crossref","unstructured":"M. W. Hirsch, C. C. Pugh, and M. Shub,\n                      Invariant Manifolds\n                      , Lecture Notes in Math. 583, Springer-Verlag, New York, 1977.","DOI":"10.1007\/BFb0092042"},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1113\/jphysiol.1952.sp004764"},{"key":"R27","doi-asserted-by":"crossref","unstructured":"F. C. Hoppensteadt and E. M. Izhikevich,\n                      Weakly Connected Neural Networks\n                      , Springer-Verlag, New York, 1997.","DOI":"10.1007\/978-1-4612-1828-9"},{"key":"R28","doi-asserted-by":"crossref","unstructured":"E. M. Izhikevich,\n                      Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting\n                      , MIT Press, Cambridge, MA, 2007.","DOI":"10.7551\/mitpress\/2526.001.0001"},{"key":"R29","doi-asserted-by":"crossref","unstructured":"J. Keener and J. Sneyd,\n                      Mathematical Physiology\n                      , Springer-Verlag, New York, 1998.","DOI":"10.1007\/b98841"},{"key":"R30","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1998.6059"},{"key":"R31","doi-asserted-by":"crossref","unstructured":"B. Krauskopf and H. M. Osinga,\n                      Computing invariant manifolds via the continuation of orbit segments\n                      , in Numerical Continuation Methods for Dynamical Systems: Path Following and Boundary Value Problems, B. Krauskopf, H. M. Osinga, and J. Gal\u00e1n-Vioque, eds., Springer-Verlag, New York, 2007, pp. 117\u2013154.","DOI":"10.1007\/978-1-4020-6356-5_4"},{"key":"R32","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127405012533"},{"key":"R33","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/21\/8\/001"},{"key":"R34","doi-asserted-by":"publisher","DOI":"10.1007\/s00285-005-0347-1"},{"key":"R35","doi-asserted-by":"crossref","unstructured":"H. M. Osinga and G. R. Rokni Lamooki,\n                      Numerical study of manifold computations\n                      , in Proceedings of the International Conference on Differential Equations (Equadiff 2003), F. Dumortier, H. W. Broer, J. Mawhin, A. Vanderbauwhede, and S. Verduyn Lunel, eds., Hasselt, World Scientific, Singapore, 2005, pp. 190\u2013195.","DOI":"10.1142\/9789812702067_0020"},{"key":"R36","doi-asserted-by":"publisher","DOI":"10.1529\/biophysj.104.046193"},{"key":"R37","doi-asserted-by":"crossref","unstructured":"J. Palis and W. De Melo,\n                      Geometric Theory of Dynamical Systems\n                      , Springer-Verlag, Berlin, 1982.","DOI":"10.1007\/978-1-4612-5703-5"},{"key":"R38","unstructured":"T. Pavlidis,\n                      Biological Oscillators: Their Mathematical Analysis\n                      , Academic Press, New York, 1973."},{"key":"R39","doi-asserted-by":"publisher","DOI":"10.1137\/040611240"},{"key":"R40","doi-asserted-by":"publisher","DOI":"10.1137\/090773519"},{"key":"R41","unstructured":"C. Sim\u00f3,\n                      On the analytical and numerical approximation of invariant manifolds\n                      , in Les M\u00e9thodes Modernes de la M\u00e9canique C\u00e9leste (Goutelas, 1989), D. Benest and C. Froeschl\u00e9, eds., Editions Frontieres, Gif-sur-Yvettes, 1990, pp. 285\u2013330."},{"key":"R42","doi-asserted-by":"publisher","DOI":"10.1109\/61.484014"},{"key":"R43","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/23\/6\/004"},{"key":"R44","doi-asserted-by":"crossref","unstructured":"A. Timbus, R. Teodorescu, F. Blaabjerg, and M. Liserre,\n                      Synchronization methods for three phase distributed power generation systems. An overview and evaluation\n                      , in Proceedings of the 36th IEEE Power Electronics Specialists Conference (PESC '05), Recife, 2005, Piscataway, NJ, pp. 2474\u20132481.","DOI":"10.1109\/PESC.2005.1581980"},{"key":"R45","doi-asserted-by":"publisher","DOI":"10.1007\/BF02339491"},{"key":"R46","doi-asserted-by":"crossref","unstructured":"A. T. Winfree,\n                      The Geometry of Biological Time\n                      , 2nd ed., Springer-Verlag, New York, 2001.","DOI":"10.1007\/978-1-4757-3484-3"}],"container-title":["SIAM Journal on Applied Dynamical Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/090777244","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T13:15:59Z","timestamp":1787231759000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/090777244"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,1]]},"references-count":46,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2010,1]]}},"alternative-id":["10.1137\/090777244"],"URL":"https:\/\/doi.org\/10.1137\/090777244","relation":{},"ISSN":["1536-0040"],"issn-type":[{"value":"1536-0040","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,1]]}}}