{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,31]],"date-time":"2025-12-31T04:23:58Z","timestamp":1767155038150},"reference-count":17,"publisher":"World Scientific Pub Co Pte Lt","issue":"10","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2006,10]]},"abstract":"<jats:p>One purpose of this paper is to document the fact that, in dynamical systems described by ordinary differential equations, the trajectories can be organized not only around fixed points (steady states), but also around lines. In 2D, these lines are the nullclines themselves, in 3D, the intersections of the nullclines two by two, etc.<\/jats:p><jats:p>We precise the concepts of \"partial steady states\" (i.e. steady states in a subsystem that consists of sections of phase space by planes normal to one of the axes) and of \"partial multistationarity\" (multistationarity in such a subsystem).<\/jats:p><jats:p>Steady states, nullclines or their intersections are revisited in terms of circuits, defined from nonzero elements of the Jacobian matrix. It is shown how the mere examination of the Jacobian matrix and the sign patterns of its circuits can help interpreting (and often predicting) aspects of the dynamics of systems.<\/jats:p><jats:p>The results reinforce the idea that chaotic dynamics requires both a positive circuit, to provide (if only partial) multistationarity, and a negative circuit, to provide sustained oscillations. As shown elsewhere, a single circuit may suffice if it is ambiguous (i.e. positive or negative depending on the location in phase space).<\/jats:p><jats:p>The description in terms of circuits is by no means exclusive of the classical description. In many cases, a fruitful approach involves repeated feedback between the two viewpoints.<\/jats:p>","DOI":"10.1142\/s0218127406016628","type":"journal-article","created":{"date-parts":[[2006,12,8]],"date-time":"2006-12-08T11:26:15Z","timestamp":1165577175000},"page":"3023-3033","source":"Crossref","is-referenced-by-count":4,"title":["NULLCLINES AND NULLCLINE INTERSECTIONS"],"prefix":"10.1142","volume":"16","author":[{"given":"REN\u00c9","family":"THOMAS","sequence":"first","affiliation":[{"name":"Center for Nonlinear Phenomena and Complex Systems, Universit\u00e9 de Bruxelles, Bd. du Triomphe, B-1050 Bruxelles, Belgium"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1006\/jtbi.2002.2544"},{"key":"rf2","unstructured":"J.\u00a0Eisenfeld and C.\u00a0De Lisi, Mathematics and Computers in Biomedical Applications, eds. 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