{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,19]],"date-time":"2025-10-19T15:41:53Z","timestamp":1760888513757},"reference-count":21,"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":[[2011,2]]},"abstract":"<jats:p> We show that, for a certain class of systems, the problem of establishing the existence of periodic orbits can be successfully studied by a symbolic dynamics approach combined with interval methods. Symbolic dynamics is used to find approximate positions of periodic points, and the existence of periodic orbits in a neighborhood of these approximations is proved using an interval operator. As an example, the Lorenz system is studied; a theoretical argument is used to prove that each periodic orbit has a distinct symbol sequence. All periodic orbits with the period p \u2264 16 of the Poincar\u00e9 map associated with the Lorenz system are found. Estimates of the topological entropy of the Poincar\u00e9 map and the flow, based on the number and flow-times of short periodic orbits, are calculated. Finally, we establish the existence of several long periodic orbits with specific symbol sequences. <\/jats:p>","DOI":"10.1142\/s021812741102857x","type":"journal-article","created":{"date-parts":[[2011,3,10]],"date-time":"2011-03-10T05:07:52Z","timestamp":1299733672000},"page":"551-563","source":"Crossref","is-referenced-by-count":10,"title":["VALIDATED STUDY OF THE EXISTENCE OF SHORT CYCLES FOR CHAOTIC SYSTEMS USING SYMBOLIC DYNAMICS AND INTERVAL TOOLS"],"prefix":"10.1142","volume":"21","author":[{"given":"ZBIGNIEW","family":"GALIAS","sequence":"first","affiliation":[{"name":"Department of Electrical Engineering, AGH University of Science and Technology, Mickiewicza 30, 30-059 Krak\u00f3w, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"WARWICK","family":"TUCKER","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Uppsala University, Box 480, 751 06 Uppsala, Sweden"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.2307\/2944326"},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.49.R971"},{"key":"rf3","first-page":"377","volume":"154","author":"Bowen R.","journal-title":"Trans. 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