{"id":"https://openalex.org/W2152817052","doi":"https://doi.org/10.1142/s0218127412502860","title":"STEADY-STATE, HOPF AND STEADY-STATE-HOPF BIFURCATIONS IN DELAY DIFFERENTIAL EQUATIONS WITH APPLICATIONS TO A DAMPED HARMONIC OSCILLATOR WITH DELAY FEEDBACK","display_name":"STEADY-STATE, HOPF AND STEADY-STATE-HOPF BIFURCATIONS IN DELAY DIFFERENTIAL EQUATIONS WITH APPLICATIONS TO A DAMPED HARMONIC OSCILLATOR WITH DELAY FEEDBACK","publication_year":2012,"publication_date":"2012-12-01","ids":{"openalex":"https://openalex.org/W2152817052","doi":"https://doi.org/10.1142/s0218127412502860","mag":"2152817052"},"language":"en","primary_location":{"id":"doi:10.1142/s0218127412502860","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127412502860","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/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"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Tongji University, Shanghai 200092, P. R. China","institution_ids":["https://openalex.org/I116953780"]}]},{"author_position":"last","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"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Shanghai Maritime University, Shanghai 201306, P. R. China","institution_ids":["https://openalex.org/I96733725"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5026187828"],"corresponding_institution_ids":["https://openalex.org/I116953780"],"apc_list":null,"apc_paid":null,"fwci":1.4903,"has_fulltext":false,"cited_by_count":13,"citation_normalized_percentile":{"value":0.84628254,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":97},"biblio":{"volume":"22","issue":"12","first_page":"1250286","last_page":"1250286"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11347","display_name":"Neural Networks Stability and Synchronization","score":0.9988999962806702,"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"}},"topics":[{"id":"https://openalex.org/T11347","display_name":"Neural Networks Stability and Synchronization","score":0.9988999962806702,"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/T11187","display_name":"Nonlinear Dynamics and Pattern Formation","score":0.9986000061035156,"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/T10482","display_name":"Mathematical and Theoretical Epidemiology and Ecology Models","score":0.9955000281333923,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/hopf-bifurcation","display_name":"Hopf bifurcation","score":0.742035448551178},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7286702394485474},{"id":"https://openalex.org/keywords/center-manifold","display_name":"Center manifold","score":0.7194221019744873},{"id":"https://openalex.org/keywords/delay-differential-equation","display_name":"Delay differential equation","score":0.5689477324485779},{"id":"https://openalex.org/keywords/pitchfork-bifurcation","display_name":"Pitchfork bifurcation","score":0.5478413701057434},{"id":"https://openalex.org/keywords/steady-state","display_name":"Steady state (chemistry)","score":0.5111227631568909},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.49258118867874146},{"id":"https://openalex.org/keywords/singularity","display_name":"Singularity","score":0.470069944858551},{"id":"https://openalex.org/keywords/harmonic-oscillator","display_name":"Harmonic oscillator","score":0.4665714502334595},{"id":"https://openalex.org/keywords/flow","display_name":"Flow (mathematics)","score":0.43209755420684814},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.41454607248306274},{"id":"https://openalex.org/keywords/bifurcation","display_name":"Bifurcation","score":0.3017634153366089},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.1666237711906433},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.15084362030029297},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.11168035864830017}],"concepts":[{"id":"https://openalex.org/C172435161","wikidata":"https://www.wikidata.org/wiki/Q531126","display_name":"Hopf bifurcation","level":4,"score":0.742035448551178},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7286702394485474},{"id":"https://openalex.org/C74774103","wikidata":"https://www.wikidata.org/wiki/Q4504202","display_name":"Center manifold","level":5,"score":0.7194221019744873},{"id":"https://openalex.org/C147504518","wikidata":"https://www.wikidata.org/wiki/Q2146125","display_name":"Delay differential equation","level":3,"score":0.5689477324485779},{"id":"https://openalex.org/C155092403","wikidata":"https://www.wikidata.org/wiki/Q1785924","display_name":"Pitchfork bifurcation","level":5,"score":0.5478413701057434},{"id":"https://openalex.org/C8171440","wikidata":"https://www.wikidata.org/wiki/Q903414","display_name":"Steady state (chemistry)","level":2,"score":0.5111227631568909},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.49258118867874146},{"id":"https://openalex.org/C16171025","wikidata":"https://www.wikidata.org/wiki/Q863349","display_name":"Singularity","level":2,"score":0.470069944858551},{"id":"https://openalex.org/C40934718","wikidata":"https://www.wikidata.org/wiki/Q190070","display_name":"Harmonic oscillator","level":2,"score":0.4665714502334595},{"id":"https://openalex.org/C38349280","wikidata":"https://www.wikidata.org/wiki/Q1434290","display_name":"Flow (mathematics)","level":2,"score":0.43209755420684814},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.41454607248306274},{"id":"https://openalex.org/C2781349735","wikidata":"https://www.wikidata.org/wiki/Q4904824","display_name":"Bifurcation","level":3,"score":0.3017634153366089},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.1666237711906433},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.15084362030029297},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.11168035864830017},{"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/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C147789679","wikidata":"https://www.wikidata.org/wiki/Q11372","display_name":"Physical chemistry","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1142/s0218127412502860","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127412502860","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":[{"id":"https://metadata.un.org/sdg/11","score":0.4099999964237213,"display_name":"Sustainable cities and communities"}],"awards":[],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320334924","display_name":"Program for New Century Excellent Talents in University","ror":"https://ror.org/01mv9t934"},{"id":"https://openalex.org/F4320335787","display_name":"Fundamental Research Funds for the Central Universities","ror":null}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":41,"referenced_works":["https://openalex.org/W590413840","https://openalex.org/W592068574","https://openalex.org/W610586754","https://openalex.org/W1494118823","https://openalex.org/W1561509108","https://openalex.org/W1586658033","https://openalex.org/W1596321532","https://openalex.org/W1958809597","https://openalex.org/W1970964221","https://openalex.org/W1988583185","https://openalex.org/W2000260888","https://openalex.org/W2003181422","https://openalex.org/W2006648678","https://openalex.org/W2009433941","https://openalex.org/W2018815721","https://openalex.org/W2019027784","https://openalex.org/W2020975304","https://openalex.org/W2030519113","https://openalex.org/W2033270072","https://openalex.org/W2034615008","https://openalex.org/W2035033442","https://openalex.org/W2037037095","https://openalex.org/W2041854671","https://openalex.org/W2048676671","https://openalex.org/W2050514524","https://openalex.org/W2057388390","https://openalex.org/W2062210472","https://openalex.org/W2065513785","https://openalex.org/W2074212690","https://openalex.org/W2078089188","https://openalex.org/W2082256594","https://openalex.org/W2083072945","https://openalex.org/W2089876099","https://openalex.org/W2090584918","https://openalex.org/W2096290885","https://openalex.org/W2104323710","https://openalex.org/W2163284349","https://openalex.org/W2167797372","https://openalex.org/W2474479551","https://openalex.org/W4211202937","https://openalex.org/W4213264637"],"related_works":["https://openalex.org/W1987790493","https://openalex.org/W1986426368","https://openalex.org/W2363522146","https://openalex.org/W2393612219","https://openalex.org/W2064215520","https://openalex.org/W2889598458","https://openalex.org/W2352502327","https://openalex.org/W2737309493","https://openalex.org/W2033747895","https://openalex.org/W2884051366"],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"employing":[3],"the":[4,22,30,37,45,57,66,75,94,106,114],"normal":[5,26,79],"form":[6,27,80],"theory":[7],"of":[8,21,24,47,70,77,102,105],"delay":[9,41,111],"differential":[10,42],"equations":[11,43],"due":[12],"to":[13,88,122,136],"Faria":[14],"and":[15,50,59,68,73,127],"Magalh\u00e3es,":[16],"we":[17,97],"present":[18],"explicit":[19,54],"formulas":[20],"coefficients":[23,76],"a":[25,33,78,99,134],"associated":[28],"with":[29,36,81,110],"flow":[31],"on":[32],"center":[34],"manifold":[35],"unfolding":[38,83],"for":[39,62,84],"general":[40],"under":[44],"cases":[46],"steady-state,":[48],"Hopf":[49,71],"steady-state-Hopf":[51,85],"singularities.":[52],"The":[53],"conditions":[55],"determining":[56,65,74],"transcritical":[58],"pitchfork":[60],"bifurcations":[61],"steady-state":[63],"singularity,":[64],"direction":[67],"stability":[69],"bifurcations,":[72],"universal":[82],"singularity":[86],"up":[87],"third":[89],"order":[90],"are":[91,120,131],"obtained.":[92],"Using":[93],"obtained":[95,132],"results,":[96],"give":[98],"complete":[100],"description":[101],"bifurcation":[103],"scenario":[104],"damped":[107],"harmonic":[108],"oscillator":[109],"feedback":[112],"near":[113],"zero":[115],"equilibrium.":[116],"Finally,":[117],"numerical":[118,129],"simulations":[119],"given":[121],"illustrate":[123],"our":[124,137],"theoretical":[125,138],"results":[126],"some":[128],"extensions":[130],"as":[133],"supplement":[135],"analysis.":[139]},"counts_by_year":[{"year":2022,"cited_by_count":2},{"year":2019,"cited_by_count":3},{"year":2018,"cited_by_count":2},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":3},{"year":2013,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}
