{"id":"https://openalex.org/W2323999154","doi":"https://doi.org/10.1142/s0218127416500401","title":"Normal Form of Saddle-Node-Hopf Bifurcation in Retarded Functional Differential Equations and Applications","display_name":"Normal Form of Saddle-Node-Hopf Bifurcation in Retarded Functional Differential Equations and Applications","publication_year":2016,"publication_date":"2016-03-01","ids":{"openalex":"https://openalex.org/W2323999154","doi":"https://doi.org/10.1142/s0218127416500401","mag":"2323999154"},"language":"en","primary_location":{"id":"doi:10.1142/s0218127416500401","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127416500401","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/A5089559378","display_name":"Heping Jiang","orcid":null},"institutions":[{"id":"https://openalex.org/I116953780","display_name":"Tongji University","ror":"https://ror.org/03rc6as71","country_code":"CN","type":"education","lineage":["https://openalex.org/I116953780"]},{"id":"https://openalex.org/I3132008074","display_name":"Huangshan University","ror":"https://ror.org/05akhmy90","country_code":"CN","type":"education","lineage":["https://openalex.org/I3132008074"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Heping Jiang","raw_affiliation_strings":["Department of Mathematics, Tongji University, Shanghai 200092, P. R. China","School of Mathematics and Statistics, Huangshan University, Anhui 245041, P. R. China"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Tongji University, Shanghai 200092, P. R. China","institution_ids":["https://openalex.org/I116953780"]},{"raw_affiliation_string":"School of Mathematics and Statistics, Huangshan University, Anhui 245041, P. R. China","institution_ids":["https://openalex.org/I3132008074"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5042517297","display_name":"Jiao Jiang","orcid":"https://orcid.org/0000-0001-5273-9032"},"institutions":[{"id":"https://openalex.org/I96733725","display_name":"Shanghai Maritime University","ror":"https://ror.org/04z7qrj66","country_code":"CN","type":"education","lineage":["https://openalex.org/I96733725"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Jiao Jiang","raw_affiliation_strings":["Department of Mathematics, Shanghai Maritime University, Shanghai 201306, P. R. China"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Shanghai Maritime University, Shanghai 201306, P. R. China","institution_ids":["https://openalex.org/I96733725"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5026187828","display_name":"Yongli Song","orcid":"https://orcid.org/0000-0002-2209-2458"},"institutions":[{"id":"https://openalex.org/I116953780","display_name":"Tongji University","ror":"https://ror.org/03rc6as71","country_code":"CN","type":"education","lineage":["https://openalex.org/I116953780"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Yongli Song","raw_affiliation_strings":["Department of Mathematics, Tongji University, Shanghai 200092, P. R. China"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Tongji University, Shanghai 200092, P. R. China","institution_ids":["https://openalex.org/I116953780"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5026187828"],"corresponding_institution_ids":["https://openalex.org/I116953780"],"apc_list":null,"apc_paid":null,"fwci":1.1511,"has_fulltext":false,"cited_by_count":9,"citation_normalized_percentile":{"value":0.82111305,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"26","issue":"03","first_page":"1650040","last_page":"1650040"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10482","display_name":"Mathematical and Theoretical Epidemiology and Ecology Models","score":0.9997000098228455,"subfield":{"id":"https://openalex.org/subfields/2739","display_name":"Public Health, Environmental and Occupational Health"},"field":{"id":"https://openalex.org/fields/27","display_name":"Medicine"},"domain":{"id":"https://openalex.org/domains/4","display_name":"Health Sciences"}},"topics":[{"id":"https://openalex.org/T10482","display_name":"Mathematical and Theoretical Epidemiology and Ecology Models","score":0.9997000098228455,"subfield":{"id":"https://openalex.org/subfields/2739","display_name":"Public Health, Environmental and Occupational Health"},"field":{"id":"https://openalex.org/fields/27","display_name":"Medicine"},"domain":{"id":"https://openalex.org/domains/4","display_name":"Health Sciences"}},{"id":"https://openalex.org/T11187","display_name":"Nonlinear Dynamics and Pattern Formation","score":0.9959999918937683,"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/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9934999942779541,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"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/saddle-node-bifurcation","display_name":"Saddle-node bifurcation","score":0.7953012585639954},{"id":"https://openalex.org/keywords/center-manifold","display_name":"Center manifold","score":0.7943733930587769},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7529568076133728},{"id":"https://openalex.org/keywords/hopf-bifurcation","display_name":"Hopf bifurcation","score":0.7207118272781372},{"id":"https://openalex.org/keywords/pitchfork-bifurcation","display_name":"Pitchfork bifurcation","score":0.5537195205688477},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.5126134753227234},{"id":"https://openalex.org/keywords/biological-applications-of-bifurcation-theory","display_name":"Biological applications of bifurcation theory","score":0.49453040957450867},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4922098219394684},{"id":"https://openalex.org/keywords/saddle","display_name":"Saddle","score":0.4503331780433655},{"id":"https://openalex.org/keywords/dynamical-systems-theory","display_name":"Dynamical systems theory","score":0.4314131736755371},{"id":"https://openalex.org/keywords/bifurcation-theory","display_name":"Bifurcation theory","score":0.42228198051452637},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4144352078437805},{"id":"https://openalex.org/keywords/bifurcation","display_name":"Bifurcation","score":0.38384270668029785},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.13764935731887817},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.10677185654640198},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.10062220692634583}],"concepts":[{"id":"https://openalex.org/C101225446","wikidata":"https://www.wikidata.org/wiki/Q740190","display_name":"Saddle-node bifurcation","level":4,"score":0.7953012585639954},{"id":"https://openalex.org/C74774103","wikidata":"https://www.wikidata.org/wiki/Q4504202","display_name":"Center manifold","level":5,"score":0.7943733930587769},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7529568076133728},{"id":"https://openalex.org/C172435161","wikidata":"https://www.wikidata.org/wiki/Q531126","display_name":"Hopf bifurcation","level":4,"score":0.7207118272781372},{"id":"https://openalex.org/C155092403","wikidata":"https://www.wikidata.org/wiki/Q1785924","display_name":"Pitchfork bifurcation","level":5,"score":0.5537195205688477},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.5126134753227234},{"id":"https://openalex.org/C174914120","wikidata":"https://www.wikidata.org/wiki/Q4914973","display_name":"Biological applications of bifurcation theory","level":5,"score":0.49453040957450867},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4922098219394684},{"id":"https://openalex.org/C2777127463","wikidata":"https://www.wikidata.org/wiki/Q10862618","display_name":"Saddle","level":2,"score":0.4503331780433655},{"id":"https://openalex.org/C79379906","wikidata":"https://www.wikidata.org/wiki/Q3174497","display_name":"Dynamical systems theory","level":2,"score":0.4314131736755371},{"id":"https://openalex.org/C85075877","wikidata":"https://www.wikidata.org/wiki/Q1154357","display_name":"Bifurcation theory","level":4,"score":0.42228198051452637},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4144352078437805},{"id":"https://openalex.org/C2781349735","wikidata":"https://www.wikidata.org/wiki/Q4904824","display_name":"Bifurcation","level":3,"score":0.38384270668029785},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.13764935731887817},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.10677185654640198},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.10062220692634583},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1142/s0218127416500401","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127416500401","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":[{"id":"https://openalex.org/G1628596063","display_name":null,"funder_award_id":"11201294","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":31,"referenced_works":["https://openalex.org/W1966200641","https://openalex.org/W1968678806","https://openalex.org/W1970978128","https://openalex.org/W1984975374","https://openalex.org/W1993003400","https://openalex.org/W2001451047","https://openalex.org/W2008220409","https://openalex.org/W2009433941","https://openalex.org/W2010265353","https://openalex.org/W2010516800","https://openalex.org/W2014368710","https://openalex.org/W2018815721","https://openalex.org/W2020500274","https://openalex.org/W2030453979","https://openalex.org/W2033270072","https://openalex.org/W2040712046","https://openalex.org/W2049810622","https://openalex.org/W2050900164","https://openalex.org/W2055558450","https://openalex.org/W2059638741","https://openalex.org/W2061636103","https://openalex.org/W2074229944","https://openalex.org/W2086851860","https://openalex.org/W2089876099","https://openalex.org/W2092883264","https://openalex.org/W2099993067","https://openalex.org/W2125664084","https://openalex.org/W2152817052","https://openalex.org/W2161464354","https://openalex.org/W4211202937","https://openalex.org/W4294967943"],"related_works":["https://openalex.org/W2546487931","https://openalex.org/W2019027784","https://openalex.org/W2078078632","https://openalex.org/W4245057904","https://openalex.org/W2159151868","https://openalex.org/W1976081455","https://openalex.org/W4312595290","https://openalex.org/W4250493900","https://openalex.org/W4365441268","https://openalex.org/W2369348556"],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"we":[3],"firstly":[4],"employ":[5],"the":[6,21,28,35,53,57,65,69,74,81],"normal":[7,22,66],"form":[8,23,67],"theory":[9],"of":[10,24,38],"delayed":[11],"differential":[12,32],"equations":[13],"according":[14],"to":[15,19,79],"Faria":[16],"and":[17,46,68],"Magalh\u00e3es":[18],"derive":[20],"saddle-node-Hopf":[25,58],"bifurcation":[26,59],"for":[27],"general":[29],"retarded":[30],"functional":[31,48],"equations.":[33],"Then,":[34],"dynamical":[36,54],"behaviors":[37],"a":[39],"Leslie\u2013Gower":[40],"predator\u2013prey":[41],"model":[42],"with":[43],"time":[44],"delay":[45],"nonmonotonic":[47],"response":[49],"are":[50,77],"considered.":[51],"Specially,":[52],"classification":[55],"near":[56],"point":[60],"is":[61],"investigated":[62],"by":[63],"using":[64],"center":[70],"manifold":[71],"approaches.":[72],"Finally,":[73],"numerical":[75],"simulations":[76],"employed":[78],"support":[80],"theoretical":[82],"results.":[83]},"counts_by_year":[{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":1}],"updated_date":"2026-03-02T08:37:19.008085","created_date":"2025-10-10T00:00:00"}
