{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,8]],"date-time":"2026-04-08T04:56:16Z","timestamp":1775624176649,"version":"3.50.1"},"reference-count":29,"publisher":"World Scientific Pub Co Pte Lt","issue":"10","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2010,10]]},"abstract":"<jats:p> The complex bifurcation structure in the parameter space of the general piecewise-linear scalar map with a single discontinuity \u2014 nowadays known as nested period adding structure \u2014 was completely studied analytically by N. N. Leonov already 50 years ago. He used an elegant and very efficient recursive technique, which allows the analytical calculation of the border-collision bifurcation curves, causing the nested period adding structure to occur. In this work, we have demonstrated that the application of Leonov's technique is not resticted to that particular bifurcation structure. On the contrary, the presented map replacement approach, which is an extension of Leonov's technique, allows the analytical calculation of border-collision bifurcation curves for periodic orbits with high periods and complex symbolic sequences using appropriate composite maps and the bifurcation curves for periodic orbits with much lower periods. <\/jats:p>","DOI":"10.1142\/s0218127410027581","type":"journal-article","created":{"date-parts":[[2010,11,10]],"date-time":"2010-11-10T21:10:06Z","timestamp":1289423406000},"page":"3105-3135","source":"Crossref","is-referenced-by-count":44,"title":["CALCULATION OF BIFURCATION CURVES BY MAP REPLACEMENT"],"prefix":"10.1142","volume":"20","author":[{"given":"VIKTOR","family":"AVRUTIN","sequence":"first","affiliation":[{"name":"Institute of Parallel and Distributed Systems, University of Stuttgart, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"MICHAEL","family":"SCHANZ","sequence":"additional","affiliation":[{"name":"Institute of Parallel and Distributed Systems, University of Stuttgart, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"LAURA","family":"GARDINI","sequence":"additional","affiliation":[{"name":"Department of Economics, University of Urbino, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/19\/8\/007"},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/21\/5\/010"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1142\/0674"},{"key":"rf5","first-page":"186","volume":"3","author":"Brocot A.","journal-title":"Revue Chronometrique"},{"key":"rf6","doi-asserted-by":"publisher","DOI":"10.1007\/s00285-005-0359-x"},{"key":"rf7","series-title":"Applied Mathematical Sciences","volume-title":"Piecewise-Smooth Dynamical Systems: Theory and Applications","volume":"163","author":"di Bernardo M.","year":"2007"},{"key":"rf8","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1080\/14786441608628487","volume":"47","author":"Farey J.","journal-title":"Philos. 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