{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,8]],"date-time":"2025-09-08T05:43:52Z","timestamp":1757310232900},"reference-count":23,"publisher":"World Scientific Pub Co Pte Lt","issue":"12","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2002,12]]},"abstract":"<jats:p> In computer simulations of smooth dynamical systems, the original phase space is replaced by machine arithmetic, which is a finite set. The resulting spatially discretized dynamical systems do not inherit all functional properties of the original systems, such as surjectivity and existence of absolutely continuous invariant measures. This can lead to computational collapse to fixed points or short cycles. The paper studies loss of such properties in spatial discretizations of dynamical systems induced by unimodal mappings of the unit interval. The problem reduces to studying set-valued negative semitrajectories of the discretized system. As the grid is refined, the asymptotic behavior of the cardinality structure of the semitrajectories follows probabilistic laws corresponding to a branching process. The transition probabilities of this process are explicitly calculated. These results are illustrated by the example of the discretized logistic mapping. <\/jats:p>","DOI":"10.1142\/s0218127402006229","type":"journal-article","created":{"date-parts":[[2003,3,12]],"date-time":"2003-03-12T07:12:14Z","timestamp":1047453134000},"page":"2847-2867","source":"Crossref","is-referenced-by-count":2,"title":["BRANCHING PROCESSES AND COMPUTATIONAL COLLAPSE OF DISCRETIZED UNIMODAL MAPPINGS"],"prefix":"10.1142","volume":"12","author":[{"given":"P.","family":"DIAMOND","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Queensland, Brisbane, QLD 4072, Australia"}]},{"given":"I.","family":"VLADIMIROV","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Queensland, Brisbane, QLD 4072, Australia"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","volume-title":"Convergence of Probability Measures","author":"Billingsley P.","year":"1968"},{"key":"rf2","doi-asserted-by":"crossref","DOI":"10.1090\/mmono\/161","volume-title":"Discreteness and Continuity in Problems of Chaotic Dynamics","author":"Blank M.","year":"1997"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4615-6927-5"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-78043-1"},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1016\/0167-2789(95)00188-A"},{"key":"rf6","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127496001533"},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.1007\/BF02179655"},{"key":"rf8","doi-asserted-by":"publisher","DOI":"10.1016\/0898-1221(96)00065-X"},{"key":"rf9","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-98-04277-4"},{"key":"rf10","doi-asserted-by":"publisher","DOI":"10.1002\/mana.19981910104"},{"key":"rf11","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127498001133"},{"key":"rf12","doi-asserted-by":"publisher","DOI":"10.1007\/s00332-001-0450-4"},{"key":"rf13","author":"Diamond P.","journal-title":"Automat. i Telemakh."},{"key":"rf14","first-page":"3688","volume":"34","author":"Grebogi E.","journal-title":"Phys. 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