{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,25]],"date-time":"2026-03-25T14:26:31Z","timestamp":1774448791750,"version":"3.50.1"},"reference-count":27,"publisher":"World Scientific Pub Co Pte Lt","issue":"11","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2005,11]]},"abstract":"<jats:p> Strange hyperbolic attractors are hard to find in real physical systems. This paper provides the first example of a realistic system, a canonical three-dimensional (3D) model of bursting neurons, that is likely to have a strange hyperbolic attractor. Using a geometrical approach to the study of the neuron model, we derive a flow-defined Poincar\u00e9 map giving an accurate account of the system's dynamics. In a parameter region where the neuron system undergoes bifurcations causing transitions between tonic spiking and bursting, this two-dimensional map becomes a map of a disk with several periodic holes. A particular case is the map of a disk with three holes, matching the Plykin example of a planar hyperbolic attractor. The corresponding attractor of the 3D neuron model appears to be hyperbolic (this property is not verified in the present paper) and arises as a result of a two-loop (secondary) homoclinic bifurcation of a saddle. This type of bifurcation, and the complex behavior it can produce, have not been previously examined. <\/jats:p>","DOI":"10.1142\/s0218127405014222","type":"journal-article","created":{"date-parts":[[2006,2,6]],"date-time":"2006-02-06T09:49:18Z","timestamp":1139219358000},"page":"3567-3578","source":"Crossref","is-referenced-by-count":28,"title":["HYPERBOLIC PLYKIN ATTRACTOR CAN EXIST IN NEURON MODELS"],"prefix":"10.1142","volume":"15","author":[{"given":"VLADIMIR","family":"BELYKH","sequence":"first","affiliation":[{"name":"Mathematics Department, Volga State Academy, 5, Nesterov st., Nizhny Novgorod 603 600, Russia"}]},{"given":"IGOR","family":"BELYKH","sequence":"additional","affiliation":[{"name":"Laboratory of Nonlinear Systems, Swiss Federal Institute of Technology Lausanne (EPFL), EPFL-IC-ISC-LANOS, Station 14, 1015 Lausanne, Switzerland"}]},{"given":"ERIK","family":"MOSEKILDE","sequence":"additional","affiliation":[{"name":"Department of Physics, The Technical University of Denmark, 2800 Kongens Lyngby, Denmark"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.16.4860"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1007\/s101890070012"},{"key":"rf4","first-page":"467","volume":"88","author":"Gavrilov N. S.","journal-title":"Math. USSR Sbornik"},{"key":"rf5","unstructured":"M.\u00a0Golubitsky, K.\u00a0Josic and T.\u00a0Kaper, Festschrift Dedicated to Floris Takens, Global Analysis of Dynamical Systems (2001)\u00a0pp. 277\u2013308."},{"key":"rf6","first-page":"1","volume":"6","author":"Gonchenko S. V.","journal-title":"Int. J. Chaos"},{"key":"rf7","series-title":"Applied Mathematical Sciences","volume-title":"Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields","volume":"42","author":"Guckenheimer J.","year":"1990"},{"key":"rf8","doi-asserted-by":"publisher","DOI":"10.1137\/S1111111101394040"},{"key":"rf9","doi-asserted-by":"publisher","DOI":"10.1098\/rspb.1984.0024"},{"key":"rf10","doi-asserted-by":"crossref","first-page":"667","DOI":"10.1017\/S0143385700008117","volume":"14","author":"Homburg A.","journal-title":"Erg. Th. Dyn. Syst."},{"key":"rf11","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/16\/4\/318"},{"key":"rf12","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127400000840"},{"key":"rf13","first-page":"918","volume":"13","author":"Newhouse S. E.","journal-title":"Topology"},{"key":"rf14","doi-asserted-by":"publisher","DOI":"10.1070\/SM1992v073n02ABEH002553"},{"key":"rf15","unstructured":"J.\u00a0Palis and C.\u00a0Pugh, Fifty Problems in Dynamical Systems, Lecture Notes in Mathematics\u00a0486 (1974)\u00a0pp. 34\u2013353."},{"key":"rf16","doi-asserted-by":"publisher","DOI":"10.1070\/SM1974v023n02ABEH001719"},{"key":"rf17","unstructured":"J.\u00a0Rinzel, Mathematical Topics in Population Biology, Morphogenesis, and Neurosciences, Lecture Notes in Biomathematics\u00a071, eds. E.\u00a0Teramoto and M.\u00a0Yamaguti (Springer-Verlag, Berlin, 1987)\u00a0pp. 251\u2013291."},{"key":"rf18","unstructured":"J.\u00a0Rinzel and B.\u00a0Ermentrout, Methods in Neuronal Modeling, eds. C.\u00a0Koch and I.\u00a0Segev (MIT Press, Cambridge, MA, 1989)\u00a0pp. 251\u2013291."},{"key":"rf19","unstructured":"L. P.\u00a0Shilnikov, Bifurcation Theory and Turbulence, Nonlinear and Turbulent Processes in Physics\u00a03 (Harwood Academic Publ., 1984)\u00a0pp. 1627\u20131635."},{"key":"rf20","doi-asserted-by":"publisher","DOI":"10.1142\/9789812798596"},{"key":"rf21","doi-asserted-by":"publisher","DOI":"10.1142\/9789812798558"},{"key":"rf22","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127404010539"},{"key":"rf23","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.71.056214"},{"key":"rf24","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.94.048101"},{"key":"rf25","doi-asserted-by":"publisher","DOI":"10.1137\/0151071"},{"key":"rf26","first-page":"133","volume":"2","author":"Terman D.","journal-title":"J. Nonlin. Sci."},{"key":"rf27","first-page":"596","volume":"342","author":"Turaev D. V.","journal-title":"Soviet Math. Dokl."},{"key":"rf28","doi-asserted-by":"publisher","DOI":"10.1016\/0167-2789(93)90286-A"}],"container-title":["International Journal of Bifurcation and Chaos"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218127405014222","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,7]],"date-time":"2019-08-07T15:03:24Z","timestamp":1565190204000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218127405014222"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,11]]},"references-count":27,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2011,11,20]]},"published-print":{"date-parts":[[2005,11]]}},"alternative-id":["10.1142\/S0218127405014222"],"URL":"https:\/\/doi.org\/10.1142\/s0218127405014222","relation":{},"ISSN":["0218-1274","1793-6551"],"issn-type":[{"value":"0218-1274","type":"print"},{"value":"1793-6551","type":"electronic"}],"subject":[],"published":{"date-parts":[[2005,11]]}}}