{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T14:32:41Z","timestamp":1787236361561,"version":"build-2736575974"},"reference-count":26,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Appl. Dyn. Syst."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>We analyze three codimension-two bifurcations occurring in nonsmooth systems, when a nonhyperbolic cycle (fold, flip, and Neimark\u2013Sacker cases, in both continuous and discrete time) interacts with one of the discontinuity boundaries characterizing the system's dynamics. Rather than aiming at a complete unfolding of the three cases, which would require specific assumptions on both the class of nonsmooth system and the geometry of the involved boundary, we concentrate on the geometric features that are common to all scenarios. We show that, at a generic intersection between the smooth and discontinuity induced bifurcation curves, a third curve generically emanates tangentially to the former. This is the discontinuity induced bifurcation curve of the secondary invariant set (the other cycle, the double-period cycle, or the torus, respectively) involved in the smooth bifurcation. The result can be explained intuitively, but its validity is proved here rigorously under very general conditions. Three examples from different fields of science and engineering are also reported.<\/jats:p>","DOI":"10.1137\/080732377","type":"journal-article","created":{"date-parts":[[2010,1,8]],"date-time":"2010-01-08T19:17:05Z","timestamp":1262978225000},"page":"62-83","source":"Crossref","is-referenced-by-count":35,"title":["Discontinuity Induced Bifurcations of Nonhyperbolic Cycles in Nonsmooth Systems"],"prefix":"10.1137","volume":"9","author":[{"given":"Alessandro","family":"Colombo","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Fabio","family":"Dercole","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,1,8]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1109\/9.654885"},{"key":"R2","doi-asserted-by":"crossref","unstructured":"B. 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Oldeman,\n                      Numerical bifurcation analysis\n                      , in Encyclopedia of Complexity and Systems Science, Springer-Verlag, New York, Berlin, 2009, pp. 6329\u20136352.","DOI":"10.1007\/978-0-387-30440-3_373"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/19\/1\/001"},{"key":"R21","doi-asserted-by":"crossref","unstructured":"D. J. W. Simpson and J. D. Meiss,\n                      Unfolding a codimension-two, discontinuous, Andronov-Hopf bifurcation\n                      , Chaos, 18 (2008), paper 033125.","DOI":"10.1063\/1.2976165"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/22\/5\/009"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1137\/070703028"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1115\/1.2338658"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1076\/vesd.38.2.149.5617"},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/19\/2\/009"}],"container-title":["SIAM Journal on Applied Dynamical Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/080732377","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T13:27:27Z","timestamp":1787232447000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/080732377"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,1]]},"references-count":26,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2010,1]]}},"alternative-id":["10.1137\/080732377"],"URL":"https:\/\/doi.org\/10.1137\/080732377","relation":{},"ISSN":["1536-0040"],"issn-type":[{"value":"1536-0040","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,1]]}}}