{"id":"https://openalex.org/W2015243509","doi":"https://doi.org/10.1142/s0218127410026927","title":"DEGENERATE BIFURCATIONS AND BORDER COLLISIONS IN PIECEWISE SMOOTH 1D AND 2D MAPS","display_name":"DEGENERATE BIFURCATIONS AND BORDER COLLISIONS IN PIECEWISE SMOOTH 1D AND 2D MAPS","publication_year":2010,"publication_date":"2010-07-01","ids":{"openalex":"https://openalex.org/W2015243509","doi":"https://doi.org/10.1142/s0218127410026927","mag":"2015243509"},"language":"en","primary_location":{"id":"doi:10.1142/s0218127410026927","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127410026927","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/A5085576419","display_name":"Iryna Sushko","orcid":"https://orcid.org/0000-0001-5879-0699"},"institutions":[{"id":"https://openalex.org/I108970525","display_name":"Kyiv School of Economics","ror":"https://ror.org/006kf9d11","country_code":"UA","type":"education","lineage":["https://openalex.org/I108970525"]},{"id":"https://openalex.org/I149851306","display_name":"National Academy of Sciences of Ukraine","ror":"https://ror.org/00je4t102","country_code":"UA","type":"government","lineage":["https://openalex.org/I149851306"]},{"id":"https://openalex.org/I190444043","display_name":"Institute of Mathematics","ror":"https://ror.org/04q3mp532","country_code":"UA","type":"facility","lineage":["https://openalex.org/I149851306","https://openalex.org/I190444043"]}],"countries":["UA"],"is_corresponding":false,"raw_author_name":"IRYNA SUSHKO","raw_affiliation_strings":["Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv School of Economics, Kyiv, Ukraine"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv School of Economics, Kyiv, Ukraine","institution_ids":["https://openalex.org/I108970525","https://openalex.org/I190444043","https://openalex.org/I149851306"]}]},{"author_position":"last","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":["Department of Economics and Quantitative Methods, University of Urbino, Italy","Department of Economics and Quantitative Methods, University of Urbino, Italy#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Economics and Quantitative Methods, University of Urbino, Italy","institution_ids":["https://openalex.org/I190397597"]},{"raw_affiliation_string":"Department of Economics and Quantitative Methods, University of Urbino, Italy#TAB#","institution_ids":["https://openalex.org/I190397597"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":7.5367,"has_fulltext":false,"cited_by_count":122,"citation_normalized_percentile":{"value":0.97942339,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":100},"biblio":{"volume":"20","issue":"07","first_page":"2045","last_page":"2070"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11261","display_name":"Quantum chaos and dynamical systems","score":0.9926000237464905,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11261","display_name":"Quantum chaos and dynamical systems","score":0.9926000237464905,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11187","display_name":"Nonlinear Dynamics and Pattern Formation","score":0.9854000210762024,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10244","display_name":"Chaos control and synchronization","score":0.9746999740600586,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/degenerate-energy-levels","display_name":"Degenerate energy levels","score":0.7885738015174866},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7173136472702026},{"id":"https://openalex.org/keywords/codimension","display_name":"Codimension","score":0.6886608600616455},{"id":"https://openalex.org/keywords/bifurcation","display_name":"Bifurcation","score":0.6080195903778076},{"id":"https://openalex.org/keywords/piecewise","display_name":"Piecewise","score":0.5969557166099548},{"id":"https://openalex.org/keywords/parameter-space","display_name":"Parameter space","score":0.49715808033943176},{"id":"https://openalex.org/keywords/piecewise-linear-function","display_name":"Piecewise linear function","score":0.4709133803844452},{"id":"https://openalex.org/keywords/piecewise-linear-manifold","display_name":"Piecewise linear manifold","score":0.45808425545692444},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.45196232199668884},{"id":"https://openalex.org/keywords/degeneracy","display_name":"Degeneracy (biology)","score":0.4465904235839844},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.443820059299469},{"id":"https://openalex.org/keywords/saddle-node-bifurcation","display_name":"Saddle-node bifurcation","score":0.44240429997444153},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.42906466126441956},{"id":"https://openalex.org/keywords/subspace-topology","display_name":"Subspace topology","score":0.41821256279945374},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.13735339045524597},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.12592115998268127},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.12343275547027588}],"concepts":[{"id":"https://openalex.org/C72319582","wikidata":"https://www.wikidata.org/wiki/Q584304","display_name":"Degenerate energy levels","level":2,"score":0.7885738015174866},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7173136472702026},{"id":"https://openalex.org/C83979697","wikidata":"https://www.wikidata.org/wiki/Q1778247","display_name":"Codimension","level":2,"score":0.6886608600616455},{"id":"https://openalex.org/C2781349735","wikidata":"https://www.wikidata.org/wiki/Q4904824","display_name":"Bifurcation","level":3,"score":0.6080195903778076},{"id":"https://openalex.org/C164660894","wikidata":"https://www.wikidata.org/wiki/Q2037833","display_name":"Piecewise","level":2,"score":0.5969557166099548},{"id":"https://openalex.org/C73586568","wikidata":"https://www.wikidata.org/wiki/Q2600211","display_name":"Parameter space","level":2,"score":0.49715808033943176},{"id":"https://openalex.org/C17095337","wikidata":"https://www.wikidata.org/wiki/Q2375229","display_name":"Piecewise linear function","level":2,"score":0.4709133803844452},{"id":"https://openalex.org/C184175843","wikidata":"https://www.wikidata.org/wiki/Q7191425","display_name":"Piecewise linear manifold","level":3,"score":0.45808425545692444},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.45196232199668884},{"id":"https://openalex.org/C2777727622","wikidata":"https://www.wikidata.org/wiki/Q5251772","display_name":"Degeneracy (biology)","level":2,"score":0.4465904235839844},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.443820059299469},{"id":"https://openalex.org/C101225446","wikidata":"https://www.wikidata.org/wiki/Q740190","display_name":"Saddle-node bifurcation","level":4,"score":0.44240429997444153},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.42906466126441956},{"id":"https://openalex.org/C32834561","wikidata":"https://www.wikidata.org/wiki/Q660730","display_name":"Subspace topology","level":2,"score":0.41821256279945374},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.13735339045524597},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.12592115998268127},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.12343275547027588},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C60644358","wikidata":"https://www.wikidata.org/wiki/Q128570","display_name":"Bioinformatics","level":1,"score":0.0},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1142/s0218127410026927","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127410026927","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"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":45,"referenced_works":["https://openalex.org/W207809827","https://openalex.org/W250423424","https://openalex.org/W571130116","https://openalex.org/W650925811","https://openalex.org/W658567699","https://openalex.org/W1552277675","https://openalex.org/W1596227557","https://openalex.org/W1596321532","https://openalex.org/W1597461812","https://openalex.org/W1608917902","https://openalex.org/W1970189657","https://openalex.org/W1971772978","https://openalex.org/W1974520525","https://openalex.org/W1996762115","https://openalex.org/W2006894653","https://openalex.org/W2009744004","https://openalex.org/W2024819864","https://openalex.org/W2028276121","https://openalex.org/W2032017885","https://openalex.org/W2040131740","https://openalex.org/W2043672595","https://openalex.org/W2049375413","https://openalex.org/W2049793915","https://openalex.org/W2061194688","https://openalex.org/W2062124659","https://openalex.org/W2069486963","https://openalex.org/W2080761408","https://openalex.org/W2083121060","https://openalex.org/W2089876099","https://openalex.org/W2094656677","https://openalex.org/W2102615391","https://openalex.org/W2116922934","https://openalex.org/W2117446133","https://openalex.org/W2118395206","https://openalex.org/W2127666064","https://openalex.org/W2159251357","https://openalex.org/W2172290680","https://openalex.org/W2487413971","https://openalex.org/W2500622636","https://openalex.org/W4206651839","https://openalex.org/W4232709242","https://openalex.org/W4234277200","https://openalex.org/W4240452402","https://openalex.org/W4251915023","https://openalex.org/W4285719527"],"related_works":["https://openalex.org/W2619539882","https://openalex.org/W2043586353","https://openalex.org/W57070652","https://openalex.org/W4256576314","https://openalex.org/W1989138816","https://openalex.org/W2886522954","https://openalex.org/W2774123355","https://openalex.org/W2260494922","https://openalex.org/W4206180239","https://openalex.org/W2123941794"],"abstract_inverted_index":{"We":[0,114],"recall":[1],"three":[2],"well-known":[3],"theorems":[4,39],"related":[5],"to":[6,62],"the":[7,19,36,66,69,76,80,84,98,117],"simplest":[8],"codimension-one":[9],"bifurcations":[10,28,48,86,121],"occurring":[11],"in":[12,29,52,64,89,94],"discrete":[13],"time":[14],"dynamical":[15],"systems,":[16],"such":[17,46,112],"as":[18,71],"fold,":[20],"flip":[21],"and":[22,25,122],"Neimark\u2013Sacker":[23],"bifurcations,":[24],"analyze":[26],"these":[27],"presence":[30],"of":[31,45,68,79,97],"certain":[32],"degeneracy":[33],"conditions,":[34],"when":[35],"above":[37],"mentioned":[38],"are":[40],"not":[41,60,108],"applied.":[42],"The":[43],"occurrence":[44],"degenerate":[47,85,120],"is":[49,59],"particularly":[50],"important":[51],"piecewise":[53,90],"smooth":[54,91],"maps,":[55],"for":[56],"which":[57],"it":[58,72],"possible":[61],"specify":[63],"general":[65],"result":[67],"bifurcation,":[70],"strongly":[73],"depends":[74],"on":[75],"global":[77],"properties":[78],"map.":[81],"In":[82],"fact,":[83],"mainly":[87],"occur":[88],"maps":[92],"defined":[93],"some":[95],"subspace":[96],"phase":[99],"space":[100],"by":[101,111],"a":[102],"linear":[103],"or":[104],"linear-fractional":[105],"function,":[106],"although":[107],"necessarily":[109],"only":[110],"functions.":[113],"also":[115],"discuss":[116],"relation":[118],"between":[119],"border-collision":[123],"bifurcations.":[124]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":4},{"year":2024,"cited_by_count":11},{"year":2023,"cited_by_count":7},{"year":2022,"cited_by_count":9},{"year":2021,"cited_by_count":6},{"year":2020,"cited_by_count":9},{"year":2019,"cited_by_count":5},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":7},{"year":2016,"cited_by_count":6},{"year":2015,"cited_by_count":14},{"year":2014,"cited_by_count":10},{"year":2013,"cited_by_count":8},{"year":2012,"cited_by_count":9}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}