{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,3,27]],"date-time":"2025-03-27T10:33:45Z","timestamp":1743071625761},"reference-count":14,"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":[[2001,2]]},"abstract":"<jats:p> In this paper we use an exponential conformal mapping and a z-transform to \"translate\" the local activity criteria for continuous-time reaction\u2013diffusion cellular nonlinear networks (CNN) to those for difference-equation CNNs. A difference-equation CNN is modeled by a set of difference equations with a constant sampling interval \u03b4t&gt;0. Since a difference-equation CNN tends to a continuous-time CNN when \u03b4t\u21920, we can view the Laplace transform of a continuous-time CNN as the limit of the conformal-mapping z-transform of a corresponding difference-equation CNN. Based on the relation between Laplace transform and our conformal-mapping z-transform, we extend the local activity criteria from a continuous-time CNN to a difference-equation CNN. We have proved the rather surprising result that the class of all reaction\u2013diffusion difference-equation CNNs with two state variables and one diffusion coefficient is locally active everywhere, i.e. its local passive region is empty. In particular, as \u03b4t\u21920, the local-passive region of a continuous-time CNN cell transforms into the \"edge-of-chaos\" region of a corresponding difference-equation CNN cell with \u03b4t&gt;0. Remarkably, as \u03b4t\u21920 the locally active edge-of-chaos region degenerates into a locally passive region as the difference equation tends to a differential equation. These results highlight a fundamental difference between the qualitative properties of systems of nonlinear differential- and difference-equations. <\/jats:p>","DOI":"10.1142\/s0218127401002250","type":"journal-article","created":{"date-parts":[[2002,7,27]],"date-time":"2002-07-27T11:04:45Z","timestamp":1027767885000},"page":"311-419","source":"Crossref","is-referenced-by-count":8,"title":["THE LOCAL ACTIVITY CRITERIA FOR \"DIFFERENCE-EQUATION\" CNN"],"prefix":"10.1142","volume":"11","author":[{"given":"VALERY I.","family":"SBITNEV","sequence":"first","affiliation":[{"name":"Department of Electrical Engineering and Computer Sciences, University of California at Berkeley, Berkeley, California, CA 94720, USA"},{"name":"Department of Condensed State Research, B. P. Konstantinov Petersburg Nuclear Physics Institute, Russ. Ac. Sci., Gatchina, Leningrad district, 188350, Russia"}]},{"given":"TAO","family":"YANG","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering and Computer Sciences, University of California at Berkeley, Berkeley, California, CA 94720, USA"}]},{"given":"LEON O.","family":"CHUA","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering and Computer Sciences, University of California at Berkeley, Berkeley, California, CA 94720, USA"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"p_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.14.2338"},{"key":"p_3","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127497001618"},{"key":"p_5","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127498000152"},{"key":"p_6","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127498000899"},{"key":"p_7","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127498000899"},{"key":"p_8","doi-asserted-by":"publisher","DOI":"10.1109\/PROC.1987.13748"},{"key":"p_9","first-page":"87","volume":"132","author":"Ginzburg S. 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