{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:57:07Z","timestamp":1760245027071},"reference-count":12,"publisher":"World Scientific Pub Co Pte Lt","issue":"08","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2006,8]]},"abstract":"<jats:p> In the sixties, \u0141ojasiewicz proved a fundamental inequality for vector fields defined by the gradient of an analytic function, which gives a lower bound on the norm of the gradient in a neighborhood of a (possibly) non-isolated critical point. The inequality involves a number belonging to (0, 1), which depends on the critical point, and is known as the \u0141ojasiewicz exponent. In this paper, a class of vector fields which are defined on a hypercube of \u211d<jats:sup>n<\/jats:sup>, is considered. Each vector field is the gradient of a quadratic function in the interior of the hypercube, however it is discontinuous on the boundary of the hypercube. An extended \u0141ojasiewicz inequality for this class of vector fields is proved, and it is also shown that the \u0141ojasiewicz exponent at each point where a vector field vanishes is equal to 1\/2. The considered fields include a class of vector fields which describe the dynamics of the output trajectories of a standard Cellular Neural Network (CNN) with a symmetric neuron interconnection matrix. By applying the extended \u0141ojasiewicz inequality, it is shown that each output trajectory of a symmetric CNN has finite length, and as a consequence it converges to an equilibrium point. Furthermore, since the \u0141ojasiewicz exponent at each equilibrium point of a symmetric CNN is equal to 1\/2, it follows that each (state) trajectory, and each output trajectory, is exponentially convergent toward an equilibrium point, and this is true even in the most general case where the CNN possesses infinitely many nonisolated equilibrium points. In essence, the obtained results mean that standard symmetric CNNs enjoy the property of absolute stability of exponential convergence. <\/jats:p>","DOI":"10.1142\/s0218127406016008","type":"journal-article","created":{"date-parts":[[2006,11,1]],"date-time":"2006-11-01T07:14:53Z","timestamp":1162365293000},"page":"2191-2205","source":"Crossref","is-referenced-by-count":18,"title":["THE \u0141OJASIEWICZ EXPONENT AT AN EQUILIBRIUM POINT OF A STANDARD CNN IS 1\/2"],"prefix":"10.1142","volume":"16","author":[{"given":"MAURO","family":"FORTI","sequence":"first","affiliation":[{"name":"Dipartimento di Ingegneria dell'Informazione, Universit\u00e0 di Siena, v. Roma 56, 53100 Siena, Italy"}]},{"given":"ALBERTO","family":"TESI","sequence":"additional","affiliation":[{"name":"Dipartimento di Sistemi e Informatica, Universit\u00e0 di Firenze, v. S. Marta 3, 50139 Firenze, Italy"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1109\/31.7600"},{"key":"rf2","volume-title":"Optimization and Non-Smooth Analysis","author":"Clarke F. H.","year":"1969"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1007\/s002220050304"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127401002328"},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1109\/TNN.2002.1031956"},{"key":"rf6","doi-asserted-by":"publisher","DOI":"10.1109\/TCSI.2004.834493"},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.1109\/TCSI.2004.838143"},{"key":"rf8","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-4426-4"},{"key":"rf9","volume-title":"The Theory of Matrices","author":"Lancaster P.","year":"1985"},{"key":"rf10","first-page":"87","author":"\u0141ojasiewicz S.","journal-title":"Studia Math."},{"key":"rf12","unstructured":"S.\u00a0\u0141ojasiewicz, Seminari di Geometria 1982\u20131983 (Universit\u00e0 di Bologna, Instituto di Geometria, Dipartimento di Matematica, 1984)\u00a0pp. 115\u2013117."},{"key":"rf13","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-5703-5"}],"container-title":["International Journal of Bifurcation and Chaos"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218127406016008","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T20:08:55Z","timestamp":1565122135000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218127406016008"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,8]]},"references-count":12,"journal-issue":{"issue":"08","published-online":{"date-parts":[[2011,11,20]]},"published-print":{"date-parts":[[2006,8]]}},"alternative-id":["10.1142\/S0218127406016008"],"URL":"https:\/\/doi.org\/10.1142\/s0218127406016008","relation":{},"ISSN":["0218-1274","1793-6551"],"issn-type":[{"value":"0218-1274","type":"print"},{"value":"1793-6551","type":"electronic"}],"subject":[],"published":{"date-parts":[[2006,8]]}}}