{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,20]],"date-time":"2026-05-20T22:11:37Z","timestamp":1779315097772,"version":"3.51.4"},"reference-count":12,"publisher":"World Scientific Pub Co Pte Lt","issue":"03","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2004,3]]},"abstract":"<jats:p> We study several aspects of FitzHugh\u2013Nagumo's (FH\u2013N) equations without diffusion. Some global stability results as well as the boundedness of solutions are derived by using a suitably defined Lyapunov functional. We show the existence of both supercritical and subcritical Hopf bifurcations. We demonstrate that the number of all bifurcation diagrams is 8 but that the possible sequential occurrences of bifurcation events is much richer. We present a numerical study of an example exhibiting a series of various bifurcations, including subcritical Hopf bifurcations, homoclinic bifurcations and saddle-node bifurcations of equilibria and of periodic solutions. Finally, we study periodically forced FH\u2013N equations. We prove that phase-locking occurs independently of the magnitude of the periodic forcing. <\/jats:p>","DOI":"10.1142\/s0218127404009685","type":"journal-article","created":{"date-parts":[[2004,4,14]],"date-time":"2004-04-14T07:46:19Z","timestamp":1081928779000},"page":"913-925","source":"Crossref","is-referenced-by-count":54,"title":["FITZHUGH\u2013NAGUMO REVISITED: TYPES OF BIFURCATIONS, PERIODICAL FORCING AND STABILITY REGIONS BY A LYAPUNOV FUNCTIONAL"],"prefix":"10.1142","volume":"14","author":[{"given":"TANYA","family":"KOSTOVA","sequence":"first","affiliation":[{"name":"Lawrence Livermore National Laboratory,  L-561, Livermore, CA 94550, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"RENUKA","family":"RAVINDRAN","sequence":"additional","affiliation":[{"name":"Department of Mathematics,  Indian Institute of Science, Bangalore 560012, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"MARIA","family":"SCHONBEK","sequence":"additional","affiliation":[{"name":"Department of Mathematics,  University of California at Santa Cruz, Santa Cruz, CA 95064, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1142\/9789812831132_0004"},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1137\/0910062"},{"key":"rf3","volume-title":"Models in Biology: Mathematics, Statistics and Computing","author":"Brown D.","year":"1993"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1007\/BF00280410"},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-4211-4"},{"key":"rf6","volume-title":"Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields","author":"Guckenheimer J.","year":"1997"},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-2421-9"},{"key":"rf8","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-08539-4"},{"key":"rf9","doi-asserted-by":"publisher","DOI":"10.1016\/S0022-5193(88)80335-7"},{"key":"rf10","doi-asserted-by":"publisher","DOI":"10.1016\/0167-2789(88)90091-7"},{"key":"rf11","doi-asserted-by":"publisher","DOI":"10.1016\/0025-5564(92)90026-S"},{"key":"rf12","volume-title":"Nonlinear Dynamics and Chaos","author":"Strogatz S. H.","year":"1994"}],"container-title":["International Journal of Bifurcation and Chaos"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218127404009685","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,7]],"date-time":"2019-08-07T11:01:54Z","timestamp":1565175714000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218127404009685"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,3]]},"references-count":12,"journal-issue":{"issue":"03","published-online":{"date-parts":[[2011,11,20]]},"published-print":{"date-parts":[[2004,3]]}},"alternative-id":["10.1142\/S0218127404009685"],"URL":"https:\/\/doi.org\/10.1142\/s0218127404009685","relation":{},"ISSN":["0218-1274","1793-6551"],"issn-type":[{"value":"0218-1274","type":"print"},{"value":"1793-6551","type":"electronic"}],"subject":[],"published":{"date-parts":[[2004,3]]}}}