{"id":"https://openalex.org/W2159650417","doi":"https://doi.org/10.1142/s0218127405012314","title":"PLANE MAPS WITH DENOMINATOR. PART III: NONSIMPLE FOCAL POINTS AND RELATED BIFURCATIONS","display_name":"PLANE MAPS WITH DENOMINATOR. PART III: NONSIMPLE FOCAL POINTS AND RELATED BIFURCATIONS","publication_year":2005,"publication_date":"2005-02-01","ids":{"openalex":"https://openalex.org/W2159650417","doi":"https://doi.org/10.1142/s0218127405012314","mag":"2159650417"},"language":"en","primary_location":{"id":"doi:10.1142/s0218127405012314","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127405012314","pdf_url":null,"source":{"id":"https://openalex.org/S81011612","display_name":"International Journal of Bifurcation and Chaos","issn_l":"0218-1274","issn":["0218-1274","1793-6551"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319815","host_organization_name":"World Scientific","host_organization_lineage":["https://openalex.org/P4310319815"],"host_organization_lineage_names":["World Scientific"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Bifurcation and Chaos","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5109096698","display_name":"Gian-Italo Bischi","orcid":null},"institutions":[{"id":"https://openalex.org/I190397597","display_name":"University of Urbino","ror":"https://ror.org/04q4kt073","country_code":"IT","type":"education","lineage":["https://openalex.org/I190397597"]}],"countries":["IT"],"is_corresponding":true,"raw_author_name":"GIAN-ITALO BISCHI","raw_affiliation_strings":["Istituto di Scienze Economiche, University of Urbino, 61029 Urbino, Italy"],"affiliations":[{"raw_affiliation_string":"Istituto di Scienze Economiche, University of Urbino, 61029 Urbino, Italy","institution_ids":["https://openalex.org/I190397597"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5045586765","display_name":"Laura Gardini","orcid":"https://orcid.org/0000-0002-3386-6543"},"institutions":[{"id":"https://openalex.org/I190397597","display_name":"University of Urbino","ror":"https://ror.org/04q4kt073","country_code":"IT","type":"education","lineage":["https://openalex.org/I190397597"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"LAURA GARDINI","raw_affiliation_strings":["Istituto di Scienze Economiche, University of Urbino, 61029 Urbino, Italy"],"affiliations":[{"raw_affiliation_string":"Istituto di Scienze Economiche, University of Urbino, 61029 Urbino, Italy","institution_ids":["https://openalex.org/I190397597"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5108428626","display_name":"Christian Mira","orcid":null},"institutions":[{"id":"https://openalex.org/I190397597","display_name":"University of Urbino","ror":"https://ror.org/04q4kt073","country_code":"IT","type":"education","lineage":["https://openalex.org/I190397597"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"CHRISTIAN MIRA","raw_affiliation_strings":["19 rue d'Occitanie, Fonsegrives 31130 Quint, and Istituto di Scienze Economiche, University of Urbino, 61029 Urbino, Italy"],"affiliations":[{"raw_affiliation_string":"19 rue d'Occitanie, Fonsegrives 31130 Quint, and Istituto di Scienze Economiche, University of Urbino, 61029 Urbino, Italy","institution_ids":["https://openalex.org/I190397597"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5109096698"],"corresponding_institution_ids":["https://openalex.org/I190397597"],"apc_list":null,"apc_paid":null,"fwci":2.5239,"has_fulltext":false,"cited_by_count":28,"citation_normalized_percentile":{"value":0.89312812,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"15","issue":"02","first_page":"451","last_page":"496"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12396","display_name":"Advanced Differential Equations and Dynamical Systems","score":0.9987999796867371,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12396","display_name":"Advanced Differential Equations and Dynamical Systems","score":0.9987999796867371,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.9947999715805054,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.9868000149726868,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7813115119934082},{"id":"https://openalex.org/keywords/tangent","display_name":"Tangent","score":0.6771127581596375},{"id":"https://openalex.org/keywords/cardinal-point","display_name":"Cardinal point","score":0.6509788036346436},{"id":"https://openalex.org/keywords/gravitational-singularity","display_name":"Gravitational singularity","score":0.577267050743103},{"id":"https://openalex.org/keywords/plane","display_name":"Plane (geometry)","score":0.5519775152206421},{"id":"https://openalex.org/keywords/foliation","display_name":"Foliation (geology)","score":0.4883115291595459},{"id":"https://openalex.org/keywords/submanifold","display_name":"Submanifold","score":0.480476051568985},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4696749150753021},{"id":"https://openalex.org/keywords/point","display_name":"Point (geometry)","score":0.46662983298301697},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.45431140065193176},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.4418456256389618},{"id":"https://openalex.org/keywords/tangent-space","display_name":"Tangent space","score":0.43879976868629456},{"id":"https://openalex.org/keywords/relation","display_name":"Relation (database)","score":0.41267579793930054},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.08778107166290283}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7813115119934082},{"id":"https://openalex.org/C138187205","wikidata":"https://www.wikidata.org/wiki/Q131251","display_name":"Tangent","level":2,"score":0.6771127581596375},{"id":"https://openalex.org/C138395690","wikidata":"https://www.wikidata.org/wiki/Q376733","display_name":"Cardinal point","level":2,"score":0.6509788036346436},{"id":"https://openalex.org/C12843","wikidata":"https://www.wikidata.org/wiki/Q201721","display_name":"Gravitational singularity","level":2,"score":0.577267050743103},{"id":"https://openalex.org/C17825722","wikidata":"https://www.wikidata.org/wiki/Q17285","display_name":"Plane (geometry)","level":2,"score":0.5519775152206421},{"id":"https://openalex.org/C168425004","wikidata":"https://www.wikidata.org/wiki/Q525802","display_name":"Foliation (geology)","level":3,"score":0.4883115291595459},{"id":"https://openalex.org/C151300846","wikidata":"https://www.wikidata.org/wiki/Q125539","display_name":"Submanifold","level":2,"score":0.480476051568985},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4696749150753021},{"id":"https://openalex.org/C28719098","wikidata":"https://www.wikidata.org/wiki/Q44946","display_name":"Point (geometry)","level":2,"score":0.46662983298301697},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.45431140065193176},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4418456256389618},{"id":"https://openalex.org/C157157409","wikidata":"https://www.wikidata.org/wiki/Q909601","display_name":"Tangent space","level":2,"score":0.43879976868629456},{"id":"https://openalex.org/C25343380","wikidata":"https://www.wikidata.org/wiki/Q277521","display_name":"Relation (database)","level":2,"score":0.41267579793930054},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.08778107166290283},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0},{"id":"https://openalex.org/C127313418","wikidata":"https://www.wikidata.org/wiki/Q1069","display_name":"Geology","level":0,"score":0.0},{"id":"https://openalex.org/C120665830","wikidata":"https://www.wikidata.org/wiki/Q14620","display_name":"Optics","level":1,"score":0.0},{"id":"https://openalex.org/C77088390","wikidata":"https://www.wikidata.org/wiki/Q8513","display_name":"Database","level":1,"score":0.0},{"id":"https://openalex.org/C26687426","wikidata":"https://www.wikidata.org/wiki/Q47069","display_name":"Metamorphic rock","level":2,"score":0.0},{"id":"https://openalex.org/C17409809","wikidata":"https://www.wikidata.org/wiki/Q161764","display_name":"Geochemistry","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1142/s0218127405012314","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127405012314","pdf_url":null,"source":{"id":"https://openalex.org/S81011612","display_name":"International Journal of Bifurcation and Chaos","issn_l":"0218-1274","issn":["0218-1274","1793-6551"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319815","host_organization_name":"World Scientific","host_organization_lineage":["https://openalex.org/P4310319815"],"host_organization_lineage_names":["World Scientific"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Bifurcation and Chaos","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.661.4510","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.661.4510","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.mdef.it/uploads/tx_sibibtex/BiGaMi_IJBC2005.pdf","raw_type":"text"},{"id":"pmh:oai:ora.uniurb.it:11576/1879750","is_oa":false,"landing_page_url":"http://hdl.handle.net/11576/1879750","pdf_url":null,"source":{"id":"https://openalex.org/S4377196377","display_name":"CINECA IRIS Institutional Research information system (University of Urbino)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I190397597","host_organization_name":"University of Urbino","host_organization_lineage":["https://openalex.org/I190397597"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":null,"sustainable_development_goals":[{"display_name":"Sustainable cities and communities","score":0.5899999737739563,"id":"https://metadata.un.org/sdg/11"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":22,"referenced_works":["https://openalex.org/W103253098","https://openalex.org/W259857763","https://openalex.org/W567049363","https://openalex.org/W1597461812","https://openalex.org/W1980137179","https://openalex.org/W1981128100","https://openalex.org/W1981553896","https://openalex.org/W1991156357","https://openalex.org/W2009764090","https://openalex.org/W2014537461","https://openalex.org/W2026265058","https://openalex.org/W2070830204","https://openalex.org/W2078010300","https://openalex.org/W2087840616","https://openalex.org/W2087934695","https://openalex.org/W2088126776","https://openalex.org/W2106428635","https://openalex.org/W2145930759","https://openalex.org/W2168581605","https://openalex.org/W4205905703","https://openalex.org/W4250146611","https://openalex.org/W4292510609"],"related_works":["https://openalex.org/W1422058139","https://openalex.org/W1800171802","https://openalex.org/W4302414700","https://openalex.org/W2020468792","https://openalex.org/W3104630249","https://openalex.org/W2361271799","https://openalex.org/W2294316159","https://openalex.org/W47544861","https://openalex.org/W2577818139","https://openalex.org/W2183615287"],"abstract_inverted_index":{"This":[0],"paper":[1,75],"continues":[2],"the":[3,6,14,18,34,38,56,60,63,68,79,86,93,100,109,114,121,131,135,144,147,159,176,195,200,203,207],"study":[4],"of":[5,13,20,40,43,59,70,81,95,102,123,137,170,178,187,194,202,206],"global":[7,64],"dynamic":[8],"properties":[9],"specific":[10],"to":[11,67,78],"maps":[12],"plane":[15],"characterized":[16],"by":[17,33],"presence":[19,69,177],"a":[21,26,96,103,128,140],"denominator":[22],"that":[23,151],"vanishes":[24],"in":[25,158,175],"one-dimensional":[27],"submanifold.":[28],"After":[29],"two":[30],"previous":[31,160],"papers":[32],"same":[35,115],"authors,":[36],"where":[37],"definitions":[39],"new":[41],"kinds":[42,169],"singularities,":[44,72],"called":[45],"focal":[46,83,110,125,141,180],"points":[47,101,145,181],"and":[48,62,85,106,143,155,185],"prefocal":[49,104,149,183],"sets,":[50,197],"are":[51,189],"given,":[52],"as":[53,55],"well":[54],"particular":[57],"structures":[58],"basins":[61],"bifurcations":[65,87],"related":[66],"such":[71,127],"this":[73],"third":[74],"is":[76],"devoted":[77],"analysis":[80],"nonsimple":[82,124,179],"points,":[84,126],"associated":[88,148,190],"with":[89,165,191,199],"them.":[90],"We":[91],"prove":[92],"existence":[94],"one-to-one":[97,132],"relation":[98,129],"between":[99,134],"curve":[105,150],"arcs":[107,138],"through":[108,139],"point":[111,142],"having":[112],"all":[113],"tangent":[116],"but":[117],"different":[118],"curvatures.":[119],"In":[120],"case":[122],"replaces":[130],"correspondence":[133],"slopes":[136],"along":[146],"have":[152],"been":[153],"proved":[154],"extensively":[156],"discussed":[157],"papers.":[161],"Moreover,":[162],"when":[163],"dealing":[164],"noninvertible":[166],"maps,":[167],"other":[168],"relations":[171],"can":[172],"be":[173],"obtained":[174],"or":[182],"curves,":[184],"some":[186],"them":[188],"qualitative":[192],"changes":[193],"critical":[196],"i.e.":[198],"structure":[201],"Riemann":[204],"foliation":[205],"plane.":[208]},"counts_by_year":[{"year":2024,"cited_by_count":3},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":4},{"year":2019,"cited_by_count":4},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":1}],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2025-10-10T00:00:00"}