{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,3,21]],"date-time":"2023-03-21T13:51:44Z","timestamp":1679406704577},"reference-count":8,"publisher":"World Scientific Pub Co Pte Lt","issue":"05","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2004,5]]},"abstract":"<jats:p> The present work describes a family of polynomial noninvertible maps of the plane shared within two open regions: (i) (denoted by Z<jats:sub>0<\/jats:sub>) each point having no real preimage, and (ii) (denoted by Z<jats:sub>2<\/jats:sub>) each point having two real preimages. The regions Z<jats:sub>0<\/jats:sub>, Z<jats:sub>2<\/jats:sub> are separated by the critical curve LC, locus of points having two coincident preimages. Z<jats:sub>2<\/jats:sub> is made up of two sheets (giving rise to a plane foliation) joining on LC, each being associated with a well-defined inverse of the map. The considered maps family is structurally unstable. For a wide choice of the parameter space it generates a singular foliation in the sense that the region Z<jats:sub>2<\/jats:sub> is separated into two zones, Z'<jats:sub>2<\/jats:sub> and [Formula: see text], inside which the two preimages do not have the same qualitative behavior. Moreover, the boundary between Z'<jats:sub>2<\/jats:sub> and [Formula: see text] is made up of points having only one real preimage at finite distance, the second one being at infinity. This situation gives rise to a nonclassical homoclinic bifurcation. The maps of the family have another important feature: their inverses present a denominator which vanish along a line of the plane. This has a great consequence on a chaotic attractor structure, when it exists. The imbedding of the map into a wider structurally stable family, generating regions Z<jats:sub>k<\/jats:sub> (k=1,3 being the number of real preimages), permits to understand the foliation nature when the imbedding parameter cancels leading to the structurally unstable map. <\/jats:p>","DOI":"10.1142\/s0218127404010254","type":"journal-article","created":{"date-parts":[[2004,6,18]],"date-time":"2004-06-18T05:34:05Z","timestamp":1087536845000},"page":"1601-1624","source":"Crossref","is-referenced-by-count":1,"title":["ON A PARTICULAR FOLIATION ASSOCIATED WITH A POLYNOMIAL FAMILY OF NONINVERTIBLE MAPS OF THE PLANE"],"prefix":"10.1142","volume":"14","author":[{"given":"LAURA","family":"GARDINI","sequence":"first","affiliation":[{"name":"Istituto di Scienze Economiche, University of Urbino, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"ILARIA","family":"FORONI","sequence":"additional","affiliation":[{"name":"University Bicocca, Milano, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"CHRISTIAN","family":"MIRA","sequence":"additional","affiliation":[{"name":"Istituto di Scienze Economiche, University of Urbino, Italy"},{"name":"19 rue d'Occitanie, 31130 Quint Fonsegrives, France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127499000079"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1142\/S021812740300793X"},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1016\/S0960-0779(02)00176-5"},{"key":"rf6","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127497000972"},{"key":"rf7","volume-title":"Dynamique Chaotique","author":"Gumowski I.","year":"1980"},{"key":"rf8","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127494000241"},{"key":"rf9","doi-asserted-by":"publisher","DOI":"10.1142\/2252"},{"key":"rf10","first-page":"1439","volume":"6","author":"Mira C.","journal-title":"Int. J. Bifurcation and Chaos"}],"container-title":["International Journal of Bifurcation and Chaos"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218127404010254","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,7]],"date-time":"2019-08-07T11:02:14Z","timestamp":1565175734000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218127404010254"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,5]]},"references-count":8,"journal-issue":{"issue":"05","published-online":{"date-parts":[[2011,11,20]]},"published-print":{"date-parts":[[2004,5]]}},"alternative-id":["10.1142\/S0218127404010254"],"URL":"https:\/\/doi.org\/10.1142\/s0218127404010254","relation":{},"ISSN":["0218-1274","1793-6551"],"issn-type":[{"value":"0218-1274","type":"print"},{"value":"1793-6551","type":"electronic"}],"subject":[],"published":{"date-parts":[[2004,5]]}}}