{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,18]],"date-time":"2025-10-18T10:26:24Z","timestamp":1760783184432},"reference-count":20,"publisher":"World Scientific Pub Co Pte Lt","issue":"06","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2009,6]]},"abstract":"<jats:p> The 11th part of our tour through one-dimensional binary Cellular Automata concerns period-2 rules, which form the second group in our classification of the 88 globally-independent CA rules according to the properties of their periodic orbits. In this article, we display the basin tree diagrams of all period-2 rules along with their time-2 characteristic functions, and then we prove that all rules belonging to group 2 have robust period-2 \u03c9-limit orbits for any finite, and infinite, bit string length. This rigorous result, which pairs with the one about period-1 rules given in the tenth installment of our chronicle, confirms what we stated about period-2 rules on the basis of empirical evidence. In the second part of this tutorial, we introduce the notion of quasi global-equivalence and prove that there are only 82 quasi globally-independent CA rules. For the first time, we show that the space-time patterns of globally-independent local rules can depend on each other, and we present an example of quasi-global transformation. We also define the super string \ud834\udd1e, and its unique decimal representation x<jats:sub>\ud834\udd1e<\/jats:sub>, dubbed the super decimal, which provides a completely transparent yet rigorous proof that rule [Formula: see text] is chaotic when L \u2192 \u221e. Moreover, we present the basin tree generation formulas, which uncover the analytical relationships between basin trees of globally-equivalent rules. Last but not least, for pedagogical and epistemological reasons, we conclude this paper with the selection of rule[Formula: see text], instead of rule [Formula: see text], as the prototypic universal Turing machine for our future discourse. <\/jats:p>","DOI":"10.1142\/s0218127409023974","type":"journal-article","created":{"date-parts":[[2009,9,1]],"date-time":"2009-09-01T11:33:49Z","timestamp":1251804829000},"page":"1751-1930","source":"Crossref","is-referenced-by-count":8,"title":["A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM'S NEW KIND OF SCIENCE PART XI: PERIOD-2 RULES"],"prefix":"10.1142","volume":"19","author":[{"given":"LEON O.","family":"CHUA","sequence":"first","affiliation":[{"name":"Department of Electrical Engineering and Computer Sciences, University of California at Berkeley, Berkeley, CA 94720, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"GIOVANNI E.","family":"PAZIENZA","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering and Computer Sciences, University of California at Berkeley, Berkeley, CA 94720, USA"},{"name":"Cellular Sensory and Wave Computing Laboratory, Computer and Automation Research Institute, MTA-SZTAKI, Budapest, Hungary"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"JINWOOK","family":"SHIN","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering and Computer Sciences, University of California at Berkeley, Berkeley, CA 94720, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","volume-title":"The General Topology of Dynamical Systems","author":"Akin E.","year":"1993"},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.2307\/2324899"},{"key":"rf3","volume-title":"Dynamical Systems","author":"Birkhoff G. 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