{"id":"https://openalex.org/W2113280598","doi":"https://doi.org/10.1142/s0218127409023494","title":"LOCAL HOPF BIFURCATION OF COMPLEX NONLINEAR SYSTEMS WITH TIME-DELAY","display_name":"LOCAL HOPF BIFURCATION OF COMPLEX NONLINEAR SYSTEMS WITH TIME-DELAY","publication_year":2009,"publication_date":"2009-03-01","ids":{"openalex":"https://openalex.org/W2113280598","doi":"https://doi.org/10.1142/s0218127409023494","mag":"2113280598"},"language":"en","primary_location":{"id":"doi:10.1142/s0218127409023494","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127409023494","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/A5005521971","display_name":"Jun Yu Li","orcid":null},"institutions":[{"id":"https://openalex.org/I9842412","display_name":"Nanjing University of Aeronautics and Astronautics","ror":"https://ror.org/01scyh794","country_code":"CN","type":"education","lineage":["https://openalex.org/I9842412"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"JUN YU LI","raw_affiliation_strings":["Institute of Vibration Engineering Research, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, P. R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Vibration Engineering Research, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, P. R. China","institution_ids":["https://openalex.org/I9842412"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5049923295","display_name":"Zaihua Wang","orcid":"https://orcid.org/0000-0003-4740-665X"},"institutions":[{"id":"https://openalex.org/I9842412","display_name":"Nanjing University of Aeronautics and Astronautics","ror":"https://ror.org/01scyh794","country_code":"CN","type":"education","lineage":["https://openalex.org/I9842412"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"ZAI HUA WANG","raw_affiliation_strings":["Institute of Vibration Engineering Research, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, P. R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Vibration Engineering Research, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, P. R. China","institution_ids":["https://openalex.org/I9842412"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I9842412"],"apc_list":null,"apc_paid":null,"fwci":0.7101,"has_fulltext":false,"cited_by_count":5,"citation_normalized_percentile":{"value":0.72404286,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"19","issue":"03","first_page":"1069","last_page":"1079"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10244","display_name":"Chaos control and synchronization","score":0.9965999722480774,"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/T10244","display_name":"Chaos control and synchronization","score":0.9965999722480774,"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/T11187","display_name":"Nonlinear Dynamics and Pattern Formation","score":0.9958999752998352,"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":0.9805999994277954,"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/hopf-bifurcation","display_name":"Hopf bifurcation","score":0.8081722259521484},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7167969942092896},{"id":"https://openalex.org/keywords/center-manifold","display_name":"Center manifold","score":0.7147769927978516},{"id":"https://openalex.org/keywords/period-doubling-bifurcation","display_name":"Period-doubling bifurcation","score":0.6072779297828674},{"id":"https://openalex.org/keywords/saddle-node-bifurcation","display_name":"Saddle-node bifurcation","score":0.5509454607963562},{"id":"https://openalex.org/keywords/pitchfork-bifurcation","display_name":"Pitchfork bifurcation","score":0.49064913392066956},{"id":"https://openalex.org/keywords/bifurcation-diagram","display_name":"Bifurcation diagram","score":0.4784976541996002},{"id":"https://openalex.org/keywords/scalar","display_name":"Scalar (mathematics)","score":0.4620508551597595},{"id":"https://openalex.org/keywords/transcritical-bifurcation","display_name":"Transcritical bifurcation","score":0.4506979286670685},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.44439759850502014},{"id":"https://openalex.org/keywords/control-theory","display_name":"Control theory (sociology)","score":0.4384461045265198},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.4377908706665039},{"id":"https://openalex.org/keywords/delay-differential-equation","display_name":"Delay differential equation","score":0.4210963249206543},{"id":"https://openalex.org/keywords/biological-applications-of-bifurcation-theory","display_name":"Biological applications of bifurcation theory","score":0.4139043986797333},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.38924089074134827},{"id":"https://openalex.org/keywords/bifurcation","display_name":"Bifurcation","score":0.37190306186676025},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.23020392656326294},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.09470546245574951},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.09389594197273254},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.08736202120780945}],"concepts":[{"id":"https://openalex.org/C172435161","wikidata":"https://www.wikidata.org/wiki/Q531126","display_name":"Hopf bifurcation","level":4,"score":0.8081722259521484},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7167969942092896},{"id":"https://openalex.org/C74774103","wikidata":"https://www.wikidata.org/wiki/Q4504202","display_name":"Center manifold","level":5,"score":0.7147769927978516},{"id":"https://openalex.org/C194271172","wikidata":"https://www.wikidata.org/wiki/Q2181373","display_name":"Period-doubling bifurcation","level":4,"score":0.6072779297828674},{"id":"https://openalex.org/C101225446","wikidata":"https://www.wikidata.org/wiki/Q740190","display_name":"Saddle-node bifurcation","level":4,"score":0.5509454607963562},{"id":"https://openalex.org/C155092403","wikidata":"https://www.wikidata.org/wiki/Q1785924","display_name":"Pitchfork bifurcation","level":5,"score":0.49064913392066956},{"id":"https://openalex.org/C86243350","wikidata":"https://www.wikidata.org/wiki/Q2595915","display_name":"Bifurcation diagram","level":4,"score":0.4784976541996002},{"id":"https://openalex.org/C57691317","wikidata":"https://www.wikidata.org/wiki/Q1289248","display_name":"Scalar (mathematics)","level":2,"score":0.4620508551597595},{"id":"https://openalex.org/C55610538","wikidata":"https://www.wikidata.org/wiki/Q1502581","display_name":"Transcritical bifurcation","level":5,"score":0.4506979286670685},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.44439759850502014},{"id":"https://openalex.org/C47446073","wikidata":"https://www.wikidata.org/wiki/Q5165890","display_name":"Control theory (sociology)","level":3,"score":0.4384461045265198},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.4377908706665039},{"id":"https://openalex.org/C147504518","wikidata":"https://www.wikidata.org/wiki/Q2146125","display_name":"Delay differential equation","level":3,"score":0.4210963249206543},{"id":"https://openalex.org/C174914120","wikidata":"https://www.wikidata.org/wiki/Q4914973","display_name":"Biological applications of bifurcation theory","level":5,"score":0.4139043986797333},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.38924089074134827},{"id":"https://openalex.org/C2781349735","wikidata":"https://www.wikidata.org/wiki/Q4904824","display_name":"Bifurcation","level":3,"score":0.37190306186676025},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.23020392656326294},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.09470546245574951},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.09389594197273254},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.08736202120780945},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"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/C2775924081","wikidata":"https://www.wikidata.org/wiki/Q55608371","display_name":"Control (management)","level":2,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1142/s0218127409023494","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127409023494","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":[{"score":0.6000000238418579,"id":"https://metadata.un.org/sdg/6","display_name":"Clean water and sanitation"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":32,"referenced_works":["https://openalex.org/W323081086","https://openalex.org/W1484776651","https://openalex.org/W1557499183","https://openalex.org/W1589786325","https://openalex.org/W1710243769","https://openalex.org/W1965880522","https://openalex.org/W1966652527","https://openalex.org/W1977590383","https://openalex.org/W1980654598","https://openalex.org/W1993217257","https://openalex.org/W1998340996","https://openalex.org/W2013472564","https://openalex.org/W2027832948","https://openalex.org/W2029040258","https://openalex.org/W2033942534","https://openalex.org/W2038197536","https://openalex.org/W2038454880","https://openalex.org/W2046177894","https://openalex.org/W2046356110","https://openalex.org/W2057661282","https://openalex.org/W2062210472","https://openalex.org/W2063534197","https://openalex.org/W2064017741","https://openalex.org/W2066545822","https://openalex.org/W2067858249","https://openalex.org/W2068728923","https://openalex.org/W2074217687","https://openalex.org/W2085967357","https://openalex.org/W2088912702","https://openalex.org/W2106689163","https://openalex.org/W2134651339","https://openalex.org/W3021623482"],"related_works":["https://openalex.org/W2546487931","https://openalex.org/W2019027784","https://openalex.org/W2078078632","https://openalex.org/W2347648614","https://openalex.org/W2978098199","https://openalex.org/W2054978396","https://openalex.org/W4293768108","https://openalex.org/W2388294899","https://openalex.org/W2271873789","https://openalex.org/W3015096850"],"abstract_inverted_index":{"A":[0],"Hopf":[1,28,57,77,115,168],"bifurcation":[2,29,58,78,116,169],"can":[3,117,137],"occur":[4],"commonly":[5],"in":[6,17,128,184],"the":[7,24,40,56,70,76,95,105,110,114,121,125,130,133,141,158,165,167],"time-delay":[8],"systems":[9],"described":[10],"by":[11,120,140],"delay":[12],"differential":[13],"equations":[14],"(DDEs),":[15],"even":[16],"first":[18,172],"order":[19,173,179],"autonomous":[20],"DDEs.":[21,90],"Compared":[22],"with":[23,32,61,104,187],"intensive":[25],"study":[26],"of":[27,59,69,79,94,124,132,164,170],"for":[30,55,75,86],"DDEs":[31,60,83],"real":[33,81],"coefficients,":[34],"which":[35],"was":[36],"mainly":[37],"done":[38],"using":[39],"center":[41],"manifold":[42],"reduction,":[43],"perturbation":[44],"methods":[45],"and":[46,152,176],"so":[47,108],"on,":[48],"little":[49],"effort":[50],"has":[51],"been":[52],"made":[53],"directly":[54],"complex":[62,88,174,180],"coefficients.":[63],"In":[64],"this":[65],"paper,":[66],"a":[67,101,171,177],"generalization":[68],"newly":[71],"developed":[72],"\"pseudo-oscillator":[73],"analysis\"":[74],"scalar":[80,87],"nonlinear":[82,89],"is":[84,98,154,182],"presented":[85],"The":[91,145],"main":[92],"idea":[93],"pseudo-oscillator":[96,102,146],"analysis":[97,147],"to":[99],"construct":[100],"associated":[103],"original":[106],"system":[107],"that":[109],"local":[111],"dynamics":[112],"near":[113],"be":[118,138],"justified":[119],"pseudo-power":[122],"function":[123],"generated":[126],"oscillator,":[127],"particular,":[129],"stability":[131],"bifurcated":[134],"periodic":[135],"solution":[136],"determined":[139],"averaged":[142],"power":[143],"function.":[144],"involves":[148],"easy":[149],"computation":[150],"only":[151],"it":[153],"more":[155],"tractable":[156],"than":[157],"current":[159],"methods.":[160],"As":[161],"an":[162],"application":[163],"method,":[166],"DDE":[175,181],"second":[178],"investigated":[183],"detail,":[185],"illustrated":[186],"numerical":[188],"simulation.":[189]},"counts_by_year":[{"year":2020,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2012,"cited_by_count":2}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}