{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,1]],"date-time":"2022-04-01T14:01:42Z","timestamp":1648821702391},"reference-count":15,"publisher":"World Scientific Pub Co Pte Lt","issue":"05","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2017,5]]},"abstract":"<jats:p> This paper introduces the notion of second minimal [Formula: see text]-periodic orbits of continuous maps on the interval according to whether [Formula: see text] is a successor of the minimal period of the map in the Sharkovski ordering. We pursue the classification of second minimal [Formula: see text]-orbits in terms of cyclic permutations and digraphs. It is proven that there are nine types of second minimal 7-orbits with accuracy up to inverses. The result is applied to the problem of the distribution of periodic windows within the chaotic regime of the bifurcation diagram of the one-parameter family of unimodal maps. It is revealed that by fixing the maximum number of appearances of periodic windows, there is a universal pattern of distribution. In particular, the first appearance of all the orbits is always a minimal orbit, while the second appearance is a second minimal orbit. It is observed that the second appearance of the 7-orbit is a second minimal 7-orbit with a Type 1 digraph. The reason for the relevance of the Type 1 second minimal orbit is the fact that the topological structure of the unimodal map with a single maximum, is equivalent to the structure of the Type 1 piecewise monotonic endomorphism associated with the second minimal 7-orbit. Yet another important report of this paper is the revelation of universal pattern dynamics with respect to an increased number of appearances. <\/jats:p>","DOI":"10.1142\/s021812741730018x","type":"journal-article","created":{"date-parts":[[2017,6,9]],"date-time":"2017-06-09T08:58:50Z","timestamp":1496998730000},"page":"1730018","source":"Crossref","is-referenced-by-count":1,"title":["Second Minimal Orbits, Sharkovski Ordering and Universality in Chaos"],"prefix":"10.1142","volume":"27","author":[{"given":"Ugur G.","family":"Abdulla","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, Florida Institute of Technology, 150 W University Blvd, Melbourne, Florida 32901, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rashad U.","family":"Abdulla","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Florida Institute of Technology, 150 W University Blvd, Melbourne, Florida 32901, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Muhammad U.","family":"Abdulla","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Florida Institute of Technology, 150 W University Blvd, Melbourne, Florida 32901, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Naveed H.","family":"Iqbal","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Florida Institute of Technology, 150 W University Blvd, Melbourne, Florida 32901, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2017,6,9]]},"reference":[{"key":"S021812741730018XBIB001","doi-asserted-by":"publisher","DOI":"10.1080\/10236198.2012.752468"},{"key":"S021812741730018XBIB003","doi-asserted-by":"publisher","DOI":"10.2307\/1999811"},{"key":"S021812741730018XBIB004","doi-asserted-by":"publisher","DOI":"10.1142\/4205"},{"key":"S021812741730018XBIB006","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1986-0854086-8"},{"key":"S021812741730018XBIB007","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0084762"},{"key":"S021812741730018XBIB008","doi-asserted-by":"publisher","DOI":"10.1007\/BF01942063"},{"key":"S021812741730018XBIB009","volume-title":"Iterated Maps on the Interval as Dynamical Systems","author":"Collet P.","year":"1980"},{"key":"S021812741730018XBIB010","doi-asserted-by":"publisher","DOI":"10.1007\/BF02193555"},{"key":"S021812741730018XBIB011","doi-asserted-by":"publisher","DOI":"10.1007\/BF01020332"},{"key":"S021812741730018XBIB012","doi-asserted-by":"publisher","DOI":"10.1007\/BF01107909"},{"key":"S021812741730018XBIB013","doi-asserted-by":"publisher","DOI":"10.1016\/0167-2789(83)90112-4"},{"key":"S021812741730018XBIB015","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(73)90033-2"},{"key":"S021812741730018XBIB016","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0082847"},{"key":"S021812741730018XBIB017","first-page":"61","volume":"16","author":"Sharkovski A.","year":"1964","journal-title":"Ukrains\u2019kii Mathematical Zhurnal"},{"key":"S021812741730018XBIB018","doi-asserted-by":"publisher","DOI":"10.1007\/BF01614086"}],"container-title":["International Journal of Bifurcation and Chaos"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S021812741730018X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,7]],"date-time":"2019-08-07T03:52:08Z","timestamp":1565149928000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S021812741730018X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,5]]},"references-count":15,"journal-issue":{"issue":"05","published-online":{"date-parts":[[2017,6,9]]},"published-print":{"date-parts":[[2017,5]]}},"alternative-id":["10.1142\/S021812741730018X"],"URL":"https:\/\/doi.org\/10.1142\/s021812741730018x","relation":{},"ISSN":["0218-1274","1793-6551"],"issn-type":[{"value":"0218-1274","type":"print"},{"value":"1793-6551","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,5]]}}}