{"id":"https://openalex.org/W3184757832","doi":"https://doi.org/10.1137/21m1436087","title":"Surprises in a Classic Boundary-Layer Problem","display_name":"Surprises in a Classic Boundary-Layer Problem","publication_year":2023,"publication_date":"2023-02-01","ids":{"openalex":"https://openalex.org/W3184757832","doi":"https://doi.org/10.1137/21m1436087","mag":"3184757832"},"language":"en","primary_location":{"id":"doi:10.1137/21m1436087","is_oa":false,"landing_page_url":"https://doi.org/10.1137/21m1436087","pdf_url":null,"source":{"id":"https://openalex.org/S160107561","display_name":"SIAM Review","issn_l":"0036-1445","issn":["0036-1445","1095-7200"],"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 Review","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2107.11624","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5000720807","display_name":"William Clark","orcid":"https://orcid.org/0000-0002-0817-0309"},"institutions":[{"id":"https://openalex.org/I205783295","display_name":"Cornell University","ror":"https://ror.org/05bnh6r87","country_code":"US","type":"education","lineage":["https://openalex.org/I205783295"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"William A. Clark","raw_affiliation_strings":["Department of Mathematics, Cornell University, Ithaca, New York 14853, USA"],"raw_orcid":"https://orcid.org/0000-0002-0817-0309","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Cornell University, Ithaca, New York 14853, USA","institution_ids":["https://openalex.org/I205783295"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5050482570","display_name":"Mario W. Gomes","orcid":"https://orcid.org/0009-0009-2253-4648"},"institutions":[{"id":"https://openalex.org/I155173764","display_name":"Rochester Institute of Technology","ror":"https://ror.org/00v4yb702","country_code":"US","type":"education","lineage":["https://openalex.org/I155173764"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Mario W. Gomes","raw_affiliation_strings":["Department of Mechanical Engineering, Rochester Institute of Technology, Rochester, NY 14623, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mechanical Engineering, Rochester Institute of Technology, Rochester, NY 14623, USA","institution_ids":["https://openalex.org/I155173764"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5010478698","display_name":"Arnaldo Rodriguez\u2010Gonzalez","orcid":"https://orcid.org/0000-0002-9200-4505"},"institutions":[{"id":"https://openalex.org/I205783295","display_name":"Cornell University","ror":"https://ror.org/05bnh6r87","country_code":"US","type":"education","lineage":["https://openalex.org/I205783295"]},{"id":"https://openalex.org/I368840534","display_name":"University of Mississippi","ror":"https://ror.org/02teq1165","country_code":"US","type":"education","lineage":["https://openalex.org/I368840534","https://openalex.org/I4210141039"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Arnaldo Rodriguez-Gonzalez","raw_affiliation_strings":["Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, New York 14853, USA","Department of Physics and Astronomy, The University of Mississippi, University, MS 38677, USA"],"raw_orcid":"https://orcid.org/0000-0002-9200-4505","affiliations":[{"raw_affiliation_string":"Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, New York 14853, USA","institution_ids":["https://openalex.org/I205783295"]},{"raw_affiliation_string":"Department of Physics and Astronomy, The University of Mississippi, University, MS 38677, USA","institution_ids":["https://openalex.org/I368840534"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5064122083","display_name":"Leo C. Stein","orcid":"https://orcid.org/0000-0001-7559-9597"},"institutions":[{"id":"https://openalex.org/I205783295","display_name":"Cornell University","ror":"https://ror.org/05bnh6r87","country_code":"US","type":"education","lineage":["https://openalex.org/I205783295"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Leo C. Stein","raw_affiliation_strings":["Department of Mathematics, Cornell University, Ithaca, New York 14853, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Cornell University, Ithaca, New York 14853, USA","institution_ids":["https://openalex.org/I205783295"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5029418921","display_name":"Steven H. Strogatz","orcid":"https://orcid.org/0000-0003-2923-3118"},"institutions":[{"id":"https://openalex.org/I205783295","display_name":"Cornell University","ror":"https://ror.org/05bnh6r87","country_code":"US","type":"education","lineage":["https://openalex.org/I205783295"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Steven H. Strogatz","raw_affiliation_strings":["Department of Mathematics, Cornell University, Ithaca, New York 14853, USA"],"raw_orcid":"https://orcid.org/0000-0003-2923-3118","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Cornell University, Ithaca, New York 14853, USA","institution_ids":["https://openalex.org/I205783295"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":5,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.6213,"has_fulltext":false,"cited_by_count":4,"citation_normalized_percentile":{"value":0.62327599,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":98},"biblio":{"volume":"65","issue":"1","first_page":"291","last_page":"315"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T13126","display_name":"Scientific Research and Discoveries","score":0.9932000041007996,"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/T13126","display_name":"Scientific Research and Discoveries","score":0.9932000041007996,"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/T12727","display_name":"Differential Equations and Numerical Methods","score":0.9901000261306763,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"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/T11416","display_name":"Numerical methods for differential equations","score":0.9811999797821045,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"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/bifurcation","display_name":"Bifurcation","score":0.6678799986839294},{"id":"https://openalex.org/keywords/perturbation","display_name":"Perturbation (astronomy)","score":0.6657474040985107},{"id":"https://openalex.org/keywords/boundary-value-problem","display_name":"Boundary value problem","score":0.5859153866767883},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.5730541348457336},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5679618716239929},{"id":"https://openalex.org/keywords/boundary-layer","display_name":"Boundary layer","score":0.5570514798164368},{"id":"https://openalex.org/keywords/pitchfork-bifurcation","display_name":"Pitchfork bifurcation","score":0.5487263798713684},{"id":"https://openalex.org/keywords/singular-perturbation","display_name":"Singular perturbation","score":0.4891379177570343},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.42636755108833313},{"id":"https://openalex.org/keywords/boundary","display_name":"Boundary (topology)","score":0.4108172357082367},{"id":"https://openalex.org/keywords/calculus","display_name":"Calculus (dental)","score":0.40833622217178345},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.33878475427627563},{"id":"https://openalex.org/keywords/bifurcation-theory","display_name":"Bifurcation theory","score":0.2461976408958435},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.18573680520057678},{"id":"https://openalex.org/keywords/mechanics","display_name":"Mechanics","score":0.1129327118396759}],"concepts":[{"id":"https://openalex.org/C2781349735","wikidata":"https://www.wikidata.org/wiki/Q4904824","display_name":"Bifurcation","level":3,"score":0.6678799986839294},{"id":"https://openalex.org/C177918212","wikidata":"https://www.wikidata.org/wiki/Q803623","display_name":"Perturbation (astronomy)","level":2,"score":0.6657474040985107},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.5859153866767883},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.5730541348457336},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5679618716239929},{"id":"https://openalex.org/C111603439","wikidata":"https://www.wikidata.org/wiki/Q752193","display_name":"Boundary layer","level":2,"score":0.5570514798164368},{"id":"https://openalex.org/C155092403","wikidata":"https://www.wikidata.org/wiki/Q1785924","display_name":"Pitchfork bifurcation","level":5,"score":0.5487263798713684},{"id":"https://openalex.org/C165160513","wikidata":"https://www.wikidata.org/wiki/Q7524249","display_name":"Singular perturbation","level":2,"score":0.4891379177570343},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.42636755108833313},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.4108172357082367},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.40833622217178345},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.33878475427627563},{"id":"https://openalex.org/C85075877","wikidata":"https://www.wikidata.org/wiki/Q1154357","display_name":"Bifurcation theory","level":4,"score":0.2461976408958435},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.18573680520057678},{"id":"https://openalex.org/C57879066","wikidata":"https://www.wikidata.org/wiki/Q41217","display_name":"Mechanics","level":1,"score":0.1129327118396759},{"id":"https://openalex.org/C71924100","wikidata":"https://www.wikidata.org/wiki/Q11190","display_name":"Medicine","level":0,"score":0.0},{"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/C199343813","wikidata":"https://www.wikidata.org/wiki/Q12128","display_name":"Dentistry","level":1,"score":0.0}],"mesh":[],"locations_count":7,"locations":[{"id":"doi:10.1137/21m1436087","is_oa":false,"landing_page_url":"https://doi.org/10.1137/21m1436087","pdf_url":null,"source":{"id":"https://openalex.org/S160107561","display_name":"SIAM Review","issn_l":"0036-1445","issn":["0036-1445","1095-7200"],"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 Review","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:2107.11624","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2107.11624","pdf_url":"https://arxiv.org/pdf/2107.11624","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:zenodo.org:5261382","is_oa":true,"landing_page_url":"https://zenodo.org/record/5261382","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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":"","raw_type":"info:eu-repo/semantics/other"},{"id":"pmh:oai:zenodo.org:5261679","is_oa":true,"landing_page_url":"https://zenodo.org/record/5261679","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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":"","raw_type":"info:eu-repo/semantics/other"},{"id":"doi:10.5281/zenodo.5261381","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.5261381","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"},{"id":"doi:10.5281/zenodo.5261382","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.5261382","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"},{"id":"doi:10.5281/zenodo.5261679","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.5261679","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2107.11624","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2107.11624","pdf_url":"https://arxiv.org/pdf/2107.11624","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/4","score":0.7699999809265137,"display_name":"Quality Education"}],"awards":[{"id":"https://openalex.org/G6287446353","display_name":null,"funder_award_id":"DMS-1645643","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":26,"referenced_works":["https://openalex.org/W80544570","https://openalex.org/W420509224","https://openalex.org/W609655943","https://openalex.org/W1547666981","https://openalex.org/W1599623497","https://openalex.org/W1876456565","https://openalex.org/W1986739294","https://openalex.org/W2008187346","https://openalex.org/W2011301426","https://openalex.org/W2068728923","https://openalex.org/W2075621462","https://openalex.org/W2145281112","https://openalex.org/W2432517183","https://openalex.org/W2552207324","https://openalex.org/W2766325652","https://openalex.org/W2899598218","https://openalex.org/W2925488132","https://openalex.org/W3003257820","https://openalex.org/W3035965352","https://openalex.org/W3099878876","https://openalex.org/W3103885388","https://openalex.org/W3105632539","https://openalex.org/W4210832181","https://openalex.org/W4237351959","https://openalex.org/W4252247111","https://openalex.org/W4298313869"],"related_works":["https://openalex.org/W2246258018","https://openalex.org/W4296238711","https://openalex.org/W3105759999","https://openalex.org/W4327639758","https://openalex.org/W1998082046","https://openalex.org/W2116787190","https://openalex.org/W2513783521","https://openalex.org/W2154304710","https://openalex.org/W2019676381","https://openalex.org/W2544244289"],"abstract_inverted_index":{"We":[0],"revisit":[1],"a":[2,6,15,27],"textbook":[3],"example":[4],"of":[5,17,33],"singularly":[7],"perturbed":[8],"nonlinear":[9],"boundary-value":[10],"problem.":[11],"Unexpectedly,":[12],"it":[13],"shows":[14],"wealth":[16],"phenomena":[18],"that":[19,49],"seem":[20],"to":[21],"have":[22],"been":[23],"overlooked":[24],"previously,":[25],"including":[26],"pitchfork":[28],"bifurcation":[29],"in":[30,45,73],"the":[31,38,46],"number":[32],"solutions":[34],"as":[35],"one":[36],"varies":[37],"small":[39,43],"parameter,":[40],"and":[41],"transcendentally":[42],"terms":[44],"initial":[47],"conditions":[48],"can":[50],"be":[51],"calculated":[52],"by":[53],"elementary":[54],"means.":[55],"Based":[56],"on":[57,75],"our":[58],"own":[59],"classroom":[60],"experience,":[61],"we":[62],"believe":[63],"this":[64],"problem":[65],"could":[66],"provide":[67],"an":[68],"enjoyable":[69],"workout":[70],"for":[71],"students":[72],"courses":[74],"perturbation":[76],"methods,":[77],"applied":[78],"dynamical":[79],"systems,":[80],"or":[81],"numerical":[82],"analysis.":[83]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":2},{"year":2023,"cited_by_count":1}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2021-08-02T00:00:00"}