{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,10]],"date-time":"2026-01-10T03:29:06Z","timestamp":1768015746656,"version":"3.49.0"},"reference-count":15,"publisher":"World Scientific Pub Co Pte Ltd","issue":"07","funder":[{"DOI":"10.13039\/501100011033","name":"Agencia Estatal de Investigaci\u00f3n","doi-asserted-by":"publisher","award":["PID2021-125535NB-I00"],"award-info":[{"award-number":["PID2021-125535NB-I00"]}],"id":[{"id":"10.13039\/501100011033","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2024,6,15]]},"abstract":"<jats:p> We study a family of one-dimensional quasi-periodically forced maps [Formula: see text], where [Formula: see text] is real, [Formula: see text] is an angle, and [Formula: see text] is an irrational frequency, such that [Formula: see text] is a real piecewise-linear map with respect to [Formula: see text] of certain kind. The family depends on two real parameters, [Formula: see text] and [Formula: see text]. For this family, we prove the existence of nonsmooth pitchfork bifurcations. For [Formula: see text] and any [Formula: see text] there is only one continuous invariant curve. For [Formula: see text] there exists a smooth map [Formula: see text] such that: (a) For [Formula: see text], [Formula: see text] has two continuous attracting invariant curves and one continuous repelling curve; (b) For [Formula: see text] it has one continuous repelling invariant curve and two semi-continuous (noncontinuous) attracting invariant curves that intersect the unstable one in a zero-Lebesgue measure set of angles; (c) For [Formula: see text] it has one continuous attracting invariant curve. The case [Formula: see text] is a degenerate case that is also discussed in the paper. It is interesting to note that this family is a simplified version of the smooth family [Formula: see text] for which there is numerical evidence of a nonsmooth pitchfork bifurcation. Finally, we also discuss the limit case when [Formula: see text]. <\/jats:p>","DOI":"10.1142\/s0218127424500846","type":"journal-article","created":{"date-parts":[[2024,5,25]],"date-time":"2024-05-25T04:14:06Z","timestamp":1716610446000},"source":"Crossref","is-referenced-by-count":3,"title":["Nonsmooth Pitchfork Bifurcations in a Quasi-Periodically Forced Piecewise-Linear Map"],"prefix":"10.1142","volume":"34","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2210-6589","authenticated-orcid":false,"given":"\u00c0ngel","family":"Jorba","sequence":"first","affiliation":[{"name":"Departament de Matem\u00e0tiques i Inform\u00e0tica, Universitat de Barcelona, Spain"},{"name":"Centre de Recerca Matem\u00e0tica, Edifici C, Campus Bellaterra, 08193 Bellaterra, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8309-3940","authenticated-orcid":false,"given":"Joan Carles","family":"Tatjer","sequence":"additional","affiliation":[{"name":"Departament de Matem\u00e0tiques i Inform\u00e0tica, Universitat de Barcelona, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0009-0000-5501-9647","authenticated-orcid":false,"given":"Yuan","family":"Zhang","sequence":"additional","affiliation":[{"name":"Complex Systems Research Center, Shanxi University, China"}]}],"member":"219","published-online":{"date-parts":[[2024,5,23]]},"reference":[{"key":"S0218127424500846BIB001","doi-asserted-by":"publisher","DOI":"10.4064\/fm206-0-2"},{"key":"S0218127424500846BIB002","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2023.02.051"},{"key":"S0218127424500846BIB003","doi-asserted-by":"publisher","DOI":"10.1017\/etds.2014.92"},{"key":"S0218127424500846BIB004","doi-asserted-by":"publisher","DOI":"10.1017\/etds.2017.4"},{"key":"S0218127424500846BIB005","first-page":"457","volume":"4","author":"Glendinning P.","year":"2004","journal-title":"Discrete Contin. 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