{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,15]],"date-time":"2025-11-15T16:52:25Z","timestamp":1763225545479},"reference-count":24,"publisher":"World Scientific Pub Co Pte Lt","issue":"02","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2005,2]]},"abstract":"<jats:p> Glass networks have been proposed as a model framework for gene regulation, chemical kinetics and neural networks. Their main distinguishing feature is that although the network variables evolve continuously in time, interactions between them depend discontinuously on their sign (i.e. above or below a threshold). While this is a simplification, it has tremendous analytic advantages if the approximation is reasonable in an application. This study explores and classifies bifurcations in Glass networks, and relates them to bifurcations of smooth systems. These bifurcations can often not be studied with traditional bifurcation theory, as the vector fields are discontinuous. However, the theory that has been developed for periodic orbits of Glass networks allows a natural classification for bifurcations of periodic orbits. Some of these are shown to correspond to smooth-system bifurcations, others are shown to fit into the framework of \"C-bifurcations\" or \"border-collision bifurcations\" and others are shown to allow truly ambiguous behavior, for which Filippov's theory for discontinuous vector fields is an appropriate tool. Routes to chaos are also explored, and it is demonstrated that period-doubling cascades do not occur. However, sudden transitions to chaos, which are common in Glass networks, can result in a limiting sense from compression to a point of a period-doubling cascade in corresponding networks with sigmoidal interactions as the sigmoid's gain is increased. Other phenomena such as intermittency and multistability are also discussed. <\/jats:p>","DOI":"10.1142\/s0218127405012302","type":"journal-article","created":{"date-parts":[[2005,5,10]],"date-time":"2005-05-10T11:32:19Z","timestamp":1115724739000},"page":"395-423","source":"Crossref","is-referenced-by-count":8,"title":["BIFURCATIONS IN GLASS NETWORKS"],"prefix":"10.1142","volume":"15","author":[{"given":"D. B.","family":"KILLOUGH","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, University of Victoria, PO Box 3045, STN CSC, Victoria, BC, Canada V8W 3P4, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"R.","family":"EDWARDS","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, University of Victoria, PO Box 3045, STN CSC, Victoria, BC, Canada V8W 3P4, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/11\/4\/007"},{"key":"rf2","first-page":"1881","volume":"10","author":"di Bernardo M.","journal-title":"Chaos Solit. Fract."},{"key":"rf3","first-page":"171","volume":"154","author":"di Bernardo M.","journal-title":"Physica"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.86.2553"},{"key":"rf5","first-page":"175","volume":"170","author":"di Bernardo M.","journal-title":"Physica"},{"key":"rf6","first-page":"157","volume":"51","author":"Edwards R.","journal-title":"Bull. Math. Biol."},{"key":"rf7","first-page":"165","volume":"146","author":"Edwards R.","journal-title":"Physica"},{"key":"rf8","first-page":"187","volume":"9","author":"Edwards R.","journal-title":"Diff. Eqs. Dyn. Syst."},{"key":"rf9","doi-asserted-by":"publisher","DOI":"10.1063\/1.1336498"},{"key":"rf10","doi-asserted-by":"publisher","DOI":"10.1016\/0021-8928(95)00118-2"},{"key":"rf11","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-015-7793-9"},{"key":"rf12","doi-asserted-by":"publisher","DOI":"10.1063\/1.431518"},{"key":"rf13","doi-asserted-by":"publisher","DOI":"10.1007\/BF02463128"},{"key":"rf14","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1140-2"},{"key":"rf15","doi-asserted-by":"publisher","DOI":"10.1016\/S0165-1684(02)00479-6"},{"key":"rf16","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127491000373"},{"key":"rf17","doi-asserted-by":"publisher","DOI":"10.1162\/neco.1992.4.5.621"},{"key":"rf18","doi-asserted-by":"publisher","DOI":"10.1109\/81.873871"},{"key":"rf19","volume-title":"Dissipative Structures and Weak Turbulence","author":"Manneville P.","year":"1990"},{"key":"rf20","doi-asserted-by":"publisher","DOI":"10.1080\/02681119508806202"},{"key":"rf21","first-page":"33","volume":"98","author":"Mestl T.","journal-title":"Physica"},{"key":"rf22","first-page":"1073","volume":"49","author":"Nusse H. E.","journal-title":"Phys. Rev."},{"key":"rf23","volume-title":"Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering","author":"Strogatz S. H.","year":"1994"},{"key":"rf24","doi-asserted-by":"publisher","DOI":"10.1109\/81.703837"}],"container-title":["International Journal of Bifurcation and Chaos"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218127405012302","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,7]],"date-time":"2019-08-07T00:06:44Z","timestamp":1565136404000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218127405012302"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,2]]},"references-count":24,"journal-issue":{"issue":"02","published-online":{"date-parts":[[2011,11,20]]},"published-print":{"date-parts":[[2005,2]]}},"alternative-id":["10.1142\/S0218127405012302"],"URL":"https:\/\/doi.org\/10.1142\/s0218127405012302","relation":{},"ISSN":["0218-1274","1793-6551"],"issn-type":[{"value":"0218-1274","type":"print"},{"value":"1793-6551","type":"electronic"}],"subject":[],"published":{"date-parts":[[2005,2]]}}}