{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:02:53Z","timestamp":1760058173584,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2025,3,17]],"date-time":"2025-03-17T00:00:00Z","timestamp":1742169600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>We prove a conjecture on the second minimal odd periodic orbits with respect to Sharkovski ordering for the continuous endomorphisms on the real line. A (2k+1)-periodic orbit {\u03b21&lt;\u03b22&lt;\u22ef&lt;\u03b22k+1}, (k\u22653) is called second minimal for the map f, if 2k\u22121 is a minimal period of f|[\u03b21,\u03b22k+1] in the Sharkovski ordering. Full classification of second minimal orbits is presented in terms of cyclic permutations and directed graphs of transitions. It is proved that second minimal odd orbits either have a Stefan-type structure like minimal odd orbits or one of the 4k\u22123 types, each characterized with unique cyclic permutations and directed graphs of transitions with an accuracy up to the inverses. The new concept of second minimal orbits and its classification have an important application towards an understanding of the universal structure of the distribution of the periodic windows in the bifurcation diagram generated by the chaotic dynamics of nonlinear maps on the interval.<\/jats:p>","DOI":"10.3390\/axioms14030222","type":"journal-article","created":{"date-parts":[[2025,3,17]],"date-time":"2025-03-17T11:04:22Z","timestamp":1742209462000},"page":"222","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Classification of the Second Minimal Orbits in the Sharkovski Ordering"],"prefix":"10.3390","volume":"14","author":[{"given":"Ugur G.","family":"Abdulla","sequence":"first","affiliation":[{"name":"Analysis & PDE Unit, Okinawa Institute of Science and Technology, Onna 904-0497, Okinawa, Japan"}]},{"given":"Naveed H.","family":"Iqbal","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA"}]},{"given":"Muhammad U.","family":"Abdulla","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, Stanford University, Stanford, CA 94305, USA"}]},{"given":"Rashad U.","family":"Abdulla","sequence":"additional","affiliation":[{"name":"HyAxiom Inc., East Hartford, CT 06108, USA"}]}],"member":"1968","published-online":{"date-parts":[[2025,3,17]]},"reference":[{"key":"ref_1","first-page":"61","article-title":"Coexistence of cycles of a continuous transofrmation of a line into itself","volume":"16","author":"Sharkovsky","year":"1964","journal-title":"Ukr. 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