{"id":"https://openalex.org/W2047138290","doi":"https://doi.org/10.1142/s0218127412500605","title":"TWO-TO-ONE RESONANT HOPF BIFURCATIONS IN A QUADRATICALLY NONLINEAR OSCILLATOR INVOLVING TIME DELAY","display_name":"TWO-TO-ONE RESONANT HOPF BIFURCATIONS IN A QUADRATICALLY NONLINEAR OSCILLATOR INVOLVING TIME DELAY","publication_year":2012,"publication_date":"2012-02-15","ids":{"openalex":"https://openalex.org/W2047138290","doi":"https://doi.org/10.1142/s0218127412500605","mag":"2047138290"},"language":"en","primary_location":{"id":"doi:10.1142/s0218127412500605","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127412500605","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/A5084477919","display_name":"Jinchen Ji","orcid":"https://orcid.org/0000-0003-3280-5463"},"institutions":[{"id":"https://openalex.org/I114017466","display_name":"University of Technology Sydney","ror":"https://ror.org/03f0f6041","country_code":"AU","type":"education","lineage":["https://openalex.org/I114017466"]}],"countries":["AU"],"is_corresponding":true,"raw_author_name":"J. C. JI","raw_affiliation_strings":["School of Electrical, Mechanical and Mechatronic Systems, University of Technology, Sydney, PO Box 123, Broadway, NSW 2007, Australia"],"affiliations":[{"raw_affiliation_string":"School of Electrical, Mechanical and Mechatronic Systems, University of Technology, Sydney, PO Box 123, Broadway, NSW 2007, Australia","institution_ids":["https://openalex.org/I114017466"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5037582926","display_name":"Xiuying Li","orcid":"https://orcid.org/0000-0002-7025-6306"},"institutions":[{"id":"https://openalex.org/I184843921","display_name":"Hebei University of Technology","ror":"https://ror.org/018hded08","country_code":"CN","type":"education","lineage":["https://openalex.org/I184843921"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"X. Y. LI","raw_affiliation_strings":["School of Mechanical Engineering, Hebei University of Technology, Tianjin 300130, P. R. China"],"affiliations":[{"raw_affiliation_string":"School of Mechanical Engineering, Hebei University of Technology, Tianjin 300130, P. R. China","institution_ids":["https://openalex.org/I184843921"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5022276111","display_name":"Zhen Luo","orcid":"https://orcid.org/0000-0002-6953-3986"},"institutions":[{"id":"https://openalex.org/I114017466","display_name":"University of Technology Sydney","ror":"https://ror.org/03f0f6041","country_code":"AU","type":"education","lineage":["https://openalex.org/I114017466"]}],"countries":["AU"],"is_corresponding":false,"raw_author_name":"Z. LUO","raw_affiliation_strings":["School of Electrical, Mechanical and Mechatronic Systems, University of Technology, Sydney, PO Box 123, Broadway, NSW 2007, Australia"],"affiliations":[{"raw_affiliation_string":"School of Electrical, Mechanical and Mechatronic Systems, University of Technology, Sydney, PO Box 123, Broadway, NSW 2007, Australia","institution_ids":["https://openalex.org/I114017466"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5019610452","display_name":"Nong Zhang","orcid":"https://orcid.org/0000-0001-6408-0044"},"institutions":[{"id":"https://openalex.org/I114017466","display_name":"University of Technology Sydney","ror":"https://ror.org/03f0f6041","country_code":"AU","type":"education","lineage":["https://openalex.org/I114017466"]}],"countries":["AU"],"is_corresponding":false,"raw_author_name":"N. ZHANG","raw_affiliation_strings":["School of Electrical, Mechanical and Mechatronic Systems, University of Technology, Sydney, PO Box 123, Broadway, NSW 2007, Australia"],"affiliations":[{"raw_affiliation_string":"School of Electrical, Mechanical and Mechatronic Systems, University of Technology, Sydney, PO Box 123, Broadway, NSW 2007, Australia","institution_ids":["https://openalex.org/I114017466"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5084477919"],"corresponding_institution_ids":["https://openalex.org/I114017466"],"apc_list":null,"apc_paid":null,"fwci":0.7092,"has_fulltext":false,"cited_by_count":15,"citation_normalized_percentile":{"value":0.72435748,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"22","issue":"03","first_page":"1250060","last_page":"1250060"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11187","display_name":"Nonlinear Dynamics and Pattern Formation","score":0.9997000098228455,"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/T11187","display_name":"Nonlinear Dynamics and Pattern Formation","score":0.9997000098228455,"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":0.9984999895095825,"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/T11513","display_name":"stochastic dynamics and bifurcation","score":0.995199978351593,"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/hopf-bifurcation","display_name":"Hopf bifurcation","score":0.807835578918457},{"id":"https://openalex.org/keywords/center-manifold","display_name":"Center manifold","score":0.7543850541114807},{"id":"https://openalex.org/keywords/quadratic-growth","display_name":"Quadratic growth","score":0.6888182759284973},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6753939986228943},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.5981835722923279},{"id":"https://openalex.org/keywords/pitchfork-bifurcation","display_name":"Pitchfork bifurcation","score":0.49589285254478455},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.47613266110420227},{"id":"https://openalex.org/keywords/bifurcation","display_name":"Bifurcation","score":0.2927939295768738},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.26191091537475586},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.15344035625457764}],"concepts":[{"id":"https://openalex.org/C172435161","wikidata":"https://www.wikidata.org/wiki/Q531126","display_name":"Hopf bifurcation","level":4,"score":0.807835578918457},{"id":"https://openalex.org/C74774103","wikidata":"https://www.wikidata.org/wiki/Q4504202","display_name":"Center manifold","level":5,"score":0.7543850541114807},{"id":"https://openalex.org/C195956108","wikidata":"https://www.wikidata.org/wiki/Q7268362","display_name":"Quadratic growth","level":2,"score":0.6888182759284973},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6753939986228943},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5981835722923279},{"id":"https://openalex.org/C155092403","wikidata":"https://www.wikidata.org/wiki/Q1785924","display_name":"Pitchfork bifurcation","level":5,"score":0.49589285254478455},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.47613266110420227},{"id":"https://openalex.org/C2781349735","wikidata":"https://www.wikidata.org/wiki/Q4904824","display_name":"Bifurcation","level":3,"score":0.2927939295768738},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.26191091537475586},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.15344035625457764}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1142/s0218127412500605","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127412500605","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":[{"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":19,"referenced_works":["https://openalex.org/W17144807","https://openalex.org/W21567595","https://openalex.org/W1531715726","https://openalex.org/W1607633954","https://openalex.org/W1995715179","https://openalex.org/W2022116410","https://openalex.org/W2022444921","https://openalex.org/W2030865530","https://openalex.org/W2060697038","https://openalex.org/W2063818274","https://openalex.org/W2079871064","https://openalex.org/W2084119948","https://openalex.org/W2089876099","https://openalex.org/W2099883907","https://openalex.org/W2113280598","https://openalex.org/W2141901926","https://openalex.org/W4211202937","https://openalex.org/W4231409469","https://openalex.org/W4237837031"],"related_works":["https://openalex.org/W1987790493","https://openalex.org/W2153388945","https://openalex.org/W1975783007","https://openalex.org/W2363522146","https://openalex.org/W2393612219","https://openalex.org/W2064215520","https://openalex.org/W2352502327","https://openalex.org/W2737309493","https://openalex.org/W2033747895","https://openalex.org/W2884051366"],"abstract_inverted_index":{"The":[0,115,133],"trivial":[1,181],"equilibrium":[2,206],"of":[3,41,47,73,86,95,106,109,112,117,120,158,173,198,231],"a":[4,12,21,96,156,208],"weakly":[5],"nonlinear":[6,98,201,233],"oscillator":[7,99,202,234],"having":[8,211],"quadratic":[9],"nonlinearities":[10],"under":[11,163,235],"delayed":[13],"feedback":[14],"control":[15],"can":[16,138],"change":[17],"its":[18],"stability":[19],"via":[20],"single":[22],"Hopf":[23,31,64,122,136],"bifurcation":[24,32],"as":[25,81,83],"the":[26,35,55,62,67,71,84,89,93,104,107,118,145,171,174,180,187,191,195,199,204,220,232,236],"time":[27,58,101],"delay":[28,102],"increases.":[29],"Double":[30],"occurs":[33],"when":[34,54],"characteristic":[36],"equation":[37],"has":[38],"two":[39,56,63,121,134,140,151,175,212],"pairs":[40],"purely":[42],"imaginary":[43],"solutions.":[44,143],"An":[45],"interaction":[46],"resonant":[48,135,238],"Hopf\u2013Hopf":[49,113,239],"bifurcations":[50,65,123,137],"may":[51],"be":[52,128],"possible":[53],"critical":[57],"delays":[59],"corresponding":[60,148],"to":[61,127,131,149,218],"have":[66],"same":[68],"value.":[69],"With":[70],"aid":[72],"normal":[74],"form":[75],"theory":[76],"and":[77,183,207,227],"centre":[78,146,192],"manifold":[79,147],"theorem":[80],"well":[82],"method":[85],"multiple":[87],"scales,":[88],"present":[90],"paper":[91],"studies":[92],"dynamics":[94],"quadratically":[97],"involving":[100],"in":[103],"vicinity":[105],"point":[108],"two-to-one":[110,164,237],"resonances":[111],"bifurcations.":[114],"ratio":[116],"frequencies":[119],"is":[124,153,168],"numerically":[125],"found":[126],"nearly":[129],"equal":[130],"two.":[132],"generate":[139],"respective":[141],"periodic":[142,177],"Consequently,":[144],"these":[150],"solutions":[152,178,185],"determined":[154],"by":[155],"set":[157],"four":[159],"first-order":[160],"differential":[161],"equations":[162,189],"internal":[165],"resonances.":[166],"It":[167],"shown":[169],"that":[170],"amplitudes":[172],"bifurcating":[176],"admit":[179],"solution":[182,230],"two-mode":[184,229],"for":[186],"averaged":[188],"on":[190],"manifolds.":[193],"Correspondingly,":[194],"cumulative":[196],"behavior":[197],"original":[200],"exhibits":[203],"initial":[205],"quasi-periodic":[209],"motion":[210],"frequencies.":[213],"Illustrative":[214],"examples":[215],"are":[216],"given":[217],"show":[219],"unstable":[221],"zero":[222,225],"solution,":[223,226],"stable":[224,228],"interactions.":[240]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":3},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}