{"id":"https://openalex.org/W2084882300","doi":"https://doi.org/10.1137/07069972x","title":"Tangency Bifurcations of Global Poincar\u00e9 Maps","display_name":"Tangency Bifurcations of Global Poincar\u00e9 Maps","publication_year":2008,"publication_date":"2008-01-01","ids":{"openalex":"https://openalex.org/W2084882300","doi":"https://doi.org/10.1137/07069972x","mag":"2084882300"},"language":"en","primary_location":{"id":"doi:10.1137/07069972x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/07069972x","pdf_url":null,"source":{"id":"https://openalex.org/S161160597","display_name":"SIAM Journal on Applied Dynamical Systems","issn_l":"1536-0040","issn":["1536-0040"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Applied Dynamical Systems","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://ir.cwi.nl/pub/13314","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5053266944","display_name":"Clare M. Lee","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Clare M. Lee","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5019902087","display_name":"Pieter Collins","orcid":"https://orcid.org/0000-0002-8896-9603"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Pieter J. Collins","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5072817844","display_name":"Bernd Krauskopf","orcid":"https://orcid.org/0000-0002-8940-230X"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Bernd Krauskopf","raw_affiliation_strings":["Atoms, Molecules, Lasers"],"affiliations":[{"raw_affiliation_string":"Atoms, Molecules, Lasers","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5078314587","display_name":"Hinke M. Osinga","orcid":"https://orcid.org/0000-0003-2169-0883"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Hinke M. Osinga","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5053266944"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":3.6351,"has_fulltext":false,"cited_by_count":27,"citation_normalized_percentile":{"value":0.92722526,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"7","issue":"3","first_page":"712","last_page":"754"},"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":0.9990000128746033,"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":0.9990000128746033,"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/T10244","display_name":"Chaos control and synchronization","score":0.9977999925613403,"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/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.996399998664856,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/diffeomorphism","display_name":"Diffeomorphism","score":0.7287782430648804},{"id":"https://openalex.org/keywords/poincar\u00e9-map","display_name":"Poincar\u00e9 map","score":0.7265182733535767},{"id":"https://openalex.org/keywords/vector-field","display_name":"Vector field","score":0.7254562377929688},{"id":"https://openalex.org/keywords/section","display_name":"Section (typography)","score":0.6510063409805298},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.630892813205719},{"id":"https://openalex.org/keywords/tangent","display_name":"Tangent","score":0.5802173018455505},{"id":"https://openalex.org/keywords/bifurcation","display_name":"Bifurcation","score":0.5575215816497803},{"id":"https://openalex.org/keywords/orbit","display_name":"Orbit (dynamics)","score":0.4904932379722595},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.488724946975708},{"id":"https://openalex.org/keywords/codimension","display_name":"Codimension","score":0.4702780544757843},{"id":"https://openalex.org/keywords/manifold","display_name":"Manifold (fluid mechanics)","score":0.4565827250480652},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.4317173361778259},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.37506717443466187},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.22488805651664734},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.1331753432750702},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.08590537309646606}],"concepts":[{"id":"https://openalex.org/C47556283","wikidata":"https://www.wikidata.org/wiki/Q1058314","display_name":"Diffeomorphism","level":2,"score":0.7287782430648804},{"id":"https://openalex.org/C21955772","wikidata":"https://www.wikidata.org/wiki/Q339199","display_name":"Poincar\u00e9 map","level":4,"score":0.7265182733535767},{"id":"https://openalex.org/C91188154","wikidata":"https://www.wikidata.org/wiki/Q186247","display_name":"Vector field","level":2,"score":0.7254562377929688},{"id":"https://openalex.org/C2780129039","wikidata":"https://www.wikidata.org/wiki/Q1931107","display_name":"Section (typography)","level":2,"score":0.6510063409805298},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.630892813205719},{"id":"https://openalex.org/C138187205","wikidata":"https://www.wikidata.org/wiki/Q131251","display_name":"Tangent","level":2,"score":0.5802173018455505},{"id":"https://openalex.org/C2781349735","wikidata":"https://www.wikidata.org/wiki/Q4904824","display_name":"Bifurcation","level":3,"score":0.5575215816497803},{"id":"https://openalex.org/C196644772","wikidata":"https://www.wikidata.org/wiki/Q3884964","display_name":"Orbit (dynamics)","level":2,"score":0.4904932379722595},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.488724946975708},{"id":"https://openalex.org/C83979697","wikidata":"https://www.wikidata.org/wiki/Q1778247","display_name":"Codimension","level":2,"score":0.4702780544757843},{"id":"https://openalex.org/C529865628","wikidata":"https://www.wikidata.org/wiki/Q1790740","display_name":"Manifold (fluid mechanics)","level":2,"score":0.4565827250480652},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4317173361778259},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.37506717443466187},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.22488805651664734},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.1331753432750702},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.08590537309646606},{"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/C78519656","wikidata":"https://www.wikidata.org/wiki/Q101333","display_name":"Mechanical engineering","level":1,"score":0.0},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0},{"id":"https://openalex.org/C146978453","wikidata":"https://www.wikidata.org/wiki/Q3798668","display_name":"Aerospace engineering","level":1,"score":0.0},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0}],"mesh":[],"locations_count":8,"locations":[{"id":"doi:10.1137/07069972x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/07069972x","pdf_url":null,"source":{"id":"https://openalex.org/S161160597","display_name":"SIAM Journal on Applied Dynamical Systems","issn_l":"1536-0040","issn":["1536-0040"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Applied Dynamical Systems","raw_type":"journal-article"},{"id":"pmh:cwi:oai:cwi.nl:13314","is_oa":true,"landing_page_url":"https://ir.cwi.nl/pub/13314","pdf_url":null,"source":{"id":"https://openalex.org/S4306401843","display_name":"Data Archiving and Networked Services (DANS)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1322597698","host_organization_name":"Royal Netherlands Academy of Arts and Sciences","host_organization_lineage":["https://openalex.org/I1322597698"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"SIAM Journal on Applied Dynamical Systems, 7(3), 712 - 754","raw_type":"info:eu-repo/semantics/article"},{"id":"pmh:hdl:10068/963068","is_oa":false,"landing_page_url":"http://hdl.handle.net/10068/963068","pdf_url":null,"source":{"id":"https://openalex.org/S4377196900","display_name":"OpenGrey (Institut de l'Information Scientifique et Technique)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1295345811","host_organization_name":"Institut de l'Information Scientifique et Technique","host_organization_lineage":["https://openalex.org/I1295345811"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"U - Thesis"},{"id":"pmh:oai:research-information.bris.ac.uk:openaire_cris_publications/dfe06ff4-4efc-422b-b103-11c848669e2e","is_oa":false,"landing_page_url":"https://research-information.bris.ac.uk/en/publications/dfe06ff4-4efc-422b-b103-11c848669e2e","pdf_url":null,"source":{"id":"https://openalex.org/S4306400895","display_name":"Bristol Research (University of Bristol)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I36234482","host_organization_name":"University of Bristol","host_organization_lineage":["https://openalex.org/I36234482"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Lee, CM, Collins, PJ, Krauskopf, B & Osinga, HM 2008, 'Tangency bifurcations of global Poincar\u00e9 maps', SIAM Journal on Applied Dynamical Systems, vol. 7(3), pp. 712 - 754. https://doi.org/10.1137/07069972X","raw_type":"info:eu-repo/semantics/publishedVersion"},{"id":"pmh:oai:research-information.bris.ac.uk:publications/dfe06ff4-4efc-422b-b103-11c848669e2e","is_oa":false,"landing_page_url":"https://hdl.handle.net/1983/dfe06ff4-4efc-422b-b103-11c848669e2e","pdf_url":null,"source":{"id":"https://openalex.org/S7407055359","display_name":"Explore Bristol Research","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Lee, CM, Collins, PJ, Krauskopf, B & Osinga, HM 2008, 'Tangency bifurcations of global Poincar\u00e9 maps', SIAM Journal on Applied Dynamical Systems, vol. 7(3), pp. 712 - 754. https://doi.org/10.1137/07069972X","raw_type":"info:eu-repo/semantics/publishedVersion"},{"id":"pmh:oai:research.vu.nl:publications/38d7bd91-fade-4b13-9000-1be18ada59c2","is_oa":false,"landing_page_url":"https://hdl.handle.net/1871.1/38d7bd91-fade-4b13-9000-1be18ada59c2","pdf_url":null,"source":{"id":"https://openalex.org/S4306401107","display_name":"VU Research Portal","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I865915315","host_organization_name":"Vrije Universiteit Amsterdam","host_organization_lineage":["https://openalex.org/I865915315"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Lee, C M, Collins, P J, Krauskopf, B & Osinga, H M 2008, 'Tangency Bifurcations of Global Poincare Maps', SIAM Journal on Applied Dynamical Systems, vol. 7, no. 3, pp. 712-754. https://doi.org/10.1137/07069972X","raw_type":"article"},{"id":"pmh:vu:oai:research.vu.nl:publications/38d7bd91-fade-4b13-9000-1be18ada59c2","is_oa":false,"landing_page_url":"https://research.vu.nl/en/publications/38d7bd91-fade-4b13-9000-1be18ada59c2","pdf_url":null,"source":{"id":"https://openalex.org/S4306401843","display_name":"Data Archiving and Networked Services (DANS)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1322597698","host_organization_name":"Royal Netherlands Academy of Arts and Sciences","host_organization_lineage":["https://openalex.org/I1322597698"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"SIAM Journal on Applied Dynamical Systems, 7(3), 712 - 754. Society of Industrial and Applied Mathematics","raw_type":"info:eu-repo/semantics/article"},{"id":"pmh:vu:oai:research.vu.nl:publications/7d31157f-d759-49c8-9964-5c27e9288a71","is_oa":false,"landing_page_url":"https://research.vu.nl/en/publications/7d31157f-d759-49c8-9964-5c27e9288a71","pdf_url":null,"source":{"id":"https://openalex.org/S4306401843","display_name":"Data Archiving and Networked Services (DANS)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1322597698","host_organization_name":"Royal Netherlands Academy of Arts and Sciences","host_organization_lineage":["https://openalex.org/I1322597698"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"SIAM Journal on Applied Dynamical Systems, 7(3), 712 - 754. Society of Industrial and Applied Mathematics","raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":{"id":"pmh:cwi:oai:cwi.nl:13314","is_oa":true,"landing_page_url":"https://ir.cwi.nl/pub/13314","pdf_url":null,"source":{"id":"https://openalex.org/S4306401843","display_name":"Data Archiving and Networked Services (DANS)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1322597698","host_organization_name":"Royal Netherlands Academy of Arts and Sciences","host_organization_lineage":["https://openalex.org/I1322597698"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"SIAM Journal on Applied Dynamical Systems, 7(3), 712 - 754","raw_type":"info:eu-repo/semantics/article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":27,"referenced_works":["https://openalex.org/W1966523973","https://openalex.org/W1969135339","https://openalex.org/W1977907719","https://openalex.org/W1978423500","https://openalex.org/W1986103149","https://openalex.org/W1991658172","https://openalex.org/W1997693480","https://openalex.org/W1999849889","https://openalex.org/W2008232178","https://openalex.org/W2009568675","https://openalex.org/W2031464967","https://openalex.org/W2038036340","https://openalex.org/W2043695568","https://openalex.org/W2047494798","https://openalex.org/W2056494071","https://openalex.org/W2071248198","https://openalex.org/W2072758819","https://openalex.org/W2077011938","https://openalex.org/W2077933648","https://openalex.org/W2088432334","https://openalex.org/W2119779417","https://openalex.org/W2132909330","https://openalex.org/W2133311306","https://openalex.org/W2142118721","https://openalex.org/W2143054589","https://openalex.org/W2156443431","https://openalex.org/W3098768354"],"related_works":["https://openalex.org/W4308167938","https://openalex.org/W2949992241","https://openalex.org/W2073018304","https://openalex.org/W4381664001","https://openalex.org/W2095285628","https://openalex.org/W2963743559","https://openalex.org/W2006798731","https://openalex.org/W2022079858","https://openalex.org/W2022211455","https://openalex.org/W1246246827"],"abstract_inverted_index":{"One":[0],"tool":[1],"to":[2,18,24,74,119,126,143,188,275],"analyze":[3],"the":[4,20,31,43,50,53,57,66,83,92,115,120,129,134,144,149,184,189,198,202,221,226,229,244,247,269],"qualitative":[5,195],"behavior":[6],"of":[7,11,34,52,128,136,164,167,173,197,220,228,235,246,271],"a":[8,12,35,71,75,79,101,182,194,209,254,257,266],"periodic":[9,44,76,84],"orbit":[10,45,77],"vector":[13,54,174],"field":[14,55],"in":[15,87,146,176,193,225],"$\\mathbb{R}^n$":[16],"is":[17,46,62,78,117,152,238],"consider":[19],"Poincar\u00e9":[21,32,67,93,130,150,168,199],"return":[22,142],"map":[23,33,68,94,151,200],"an":[25],"$(n-1)$-dimensional":[26],"section.":[27,108],"The":[28,233],"image":[29],"under":[30],"point":[36],"on":[37,70,100,170],"this":[38,158],"section":[39,72],"that":[40,65],"lies":[41],"near":[42,82],"obtained":[47],"by":[48,240],"following":[49],"flow":[51,116,204],"until":[56],"next":[58],"(local)":[59],"intersection.":[60],"It":[61],"well":[63],"known":[64],"defined":[69,154],"transverse":[73],"diffeomorphism":[80],"locally":[81,97],"orbit.":[85],"However,":[86],"practice":[88],"one":[89],"often":[90],"considers":[91],"not":[95,140,153,207],"only":[96],"but":[98],"also":[99],"much":[102],"larger":[103],"global":[104,171],"and":[105,122,178,241,253,278],"typically":[106],"unbounded":[107],"Generically,":[109],"there":[110],"are":[111],"then":[112],"points":[113,138],"where":[114],"tangent":[118,187],"section,":[121,145,190],"these":[123,236],"give":[124],"rise":[125],"discontinuities":[127],"map.":[131],"In":[132,157],"fact,":[133],"orbits":[135],"some":[137],"may":[139],"even":[141],"which":[147,191],"case":[148],"at":[155],"all.":[156],"paper":[159],"we":[160,216,264],"study":[161,234],"tangency":[162,223,231],"bifurcations":[163,224,237],"invariant":[165],"manifolds":[166],"maps":[169],"sections":[172],"fields":[175],"$\\mathbb{R}^2$":[177],"$\\mathbb{R}^3$.":[179],"At":[180],"such":[181],"bifurcation":[183],"manifold":[185],"becomes":[186],"results":[192,274],"change":[196],"while":[201],"underlying":[203],"itself":[205],"does":[206],"undergo":[208],"bifurcation.":[210],"Using":[211],"tools":[212],"from":[213],"singularity":[214],"theory,":[215],"present":[217,265],"normal":[218],"forms":[219],"codimension-one":[222],"neighborhood":[227],"respective":[230],"point.":[232],"motivated":[239],"illustrated":[242],"with":[243,260],"examples":[245],"(unforced)":[248],"Van":[249],"der":[250],"Pol":[251],"oscillator":[252],"system":[255],"modeling":[256],"semiconductor":[258],"laser":[259],"optical":[261],"injection.":[262],"Finally,":[263],"framework":[267],"for":[268],"generalization":[270],"our":[272],"normal-form":[273],"arbitrary":[276],"dimension":[277],"codimension.":[279]},"counts_by_year":[{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":2},{"year":2020,"cited_by_count":3},{"year":2019,"cited_by_count":2},{"year":2017,"cited_by_count":3},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":1},{"year":2012,"cited_by_count":1}],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2025-10-10T00:00:00"}