{"id":"https://openalex.org/W2463076810","doi":"https://doi.org/10.1142/s0218127416501005","title":"Bifurcation Analysis for Neural Networks in Neutral Form","display_name":"Bifurcation Analysis for Neural Networks in Neutral Form","publication_year":2016,"publication_date":"2016-06-15","ids":{"openalex":"https://openalex.org/W2463076810","doi":"https://doi.org/10.1142/s0218127416501005","mag":"2463076810"},"language":"en","primary_location":{"id":"doi:10.1142/s0218127416501005","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127416501005","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/A5039736179","display_name":"Hongbing Chen","orcid":"https://orcid.org/0000-0003-2220-0487"},"institutions":[{"id":"https://openalex.org/I4210163834","display_name":"Tianshui Normal University","ror":"https://ror.org/05kc6dc21","country_code":"CN","type":"education","lineage":["https://openalex.org/I4210163834"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Hong-Bing Chen","raw_affiliation_strings":["School of Mathematics and Statistics, Tianshui Normal University, Tianshui, Gansu 741001, P. R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics, Tianshui Normal University, Tianshui, Gansu 741001, P. R. China","institution_ids":["https://openalex.org/I4210163834"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5102099834","display_name":"Xiao-Ke Sun","orcid":null},"institutions":[{"id":"https://openalex.org/I4210163834","display_name":"Tianshui Normal University","ror":"https://ror.org/05kc6dc21","country_code":"CN","type":"education","lineage":["https://openalex.org/I4210163834"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Xiao-Ke Sun","raw_affiliation_strings":["School of Mathematics and Statistics, Tianshui Normal University, Tianshui, Gansu 741001, P. R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics, Tianshui Normal University, Tianshui, Gansu 741001, P. R. China","institution_ids":["https://openalex.org/I4210163834"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I4210163834"],"apc_list":null,"apc_paid":null,"fwci":0.2833,"has_fulltext":false,"cited_by_count":7,"citation_normalized_percentile":{"value":0.62186852,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"26","issue":"06","first_page":"1650100","last_page":"1650100"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11347","display_name":"Neural Networks Stability and Synchronization","score":1.0,"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":1.0,"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/T11513","display_name":"stochastic dynamics and bifurcation","score":0.9991000294685364,"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/T10320","display_name":"Neural Networks and Applications","score":0.9984999895095825,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/hopf-bifurcation","display_name":"Hopf bifurcation","score":0.7930690050125122},{"id":"https://openalex.org/keywords/center-manifold","display_name":"Center manifold","score":0.786845862865448},{"id":"https://openalex.org/keywords/saddle-node-bifurcation","display_name":"Saddle-node bifurcation","score":0.7142344117164612},{"id":"https://openalex.org/keywords/biological-applications-of-bifurcation-theory","display_name":"Biological applications of bifurcation theory","score":0.6942036747932434},{"id":"https://openalex.org/keywords/bifurcation-diagram","display_name":"Bifurcation diagram","score":0.6287035346031189},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6095949411392212},{"id":"https://openalex.org/keywords/bifurcation","display_name":"Bifurcation","score":0.5911261439323425},{"id":"https://openalex.org/keywords/pitchfork-bifurcation","display_name":"Pitchfork bifurcation","score":0.5890865325927734},{"id":"https://openalex.org/keywords/transcritical-bifurcation","display_name":"Transcritical bifurcation","score":0.5568545460700989},{"id":"https://openalex.org/keywords/bogdanov\u2013takens-bifurcation","display_name":"Bogdanov\u2013Takens bifurcation","score":0.5260244011878967},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.5077399015426636},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.4812892973423004},{"id":"https://openalex.org/keywords/period-doubling-bifurcation","display_name":"Period-doubling bifurcation","score":0.4330224096775055},{"id":"https://openalex.org/keywords/infinite-period-bifurcation","display_name":"Infinite-period bifurcation","score":0.4268742501735687},{"id":"https://openalex.org/keywords/control-theory","display_name":"Control theory (sociology)","score":0.3819277882575989},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3538751006126404},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.33154210448265076},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.23800477385520935},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.19576460123062134},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.16991746425628662},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.062323302030563354}],"concepts":[{"id":"https://openalex.org/C172435161","wikidata":"https://www.wikidata.org/wiki/Q531126","display_name":"Hopf bifurcation","level":4,"score":0.7930690050125122},{"id":"https://openalex.org/C74774103","wikidata":"https://www.wikidata.org/wiki/Q4504202","display_name":"Center manifold","level":5,"score":0.786845862865448},{"id":"https://openalex.org/C101225446","wikidata":"https://www.wikidata.org/wiki/Q740190","display_name":"Saddle-node bifurcation","level":4,"score":0.7142344117164612},{"id":"https://openalex.org/C174914120","wikidata":"https://www.wikidata.org/wiki/Q4914973","display_name":"Biological applications of bifurcation theory","level":5,"score":0.6942036747932434},{"id":"https://openalex.org/C86243350","wikidata":"https://www.wikidata.org/wiki/Q2595915","display_name":"Bifurcation diagram","level":4,"score":0.6287035346031189},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6095949411392212},{"id":"https://openalex.org/C2781349735","wikidata":"https://www.wikidata.org/wiki/Q4904824","display_name":"Bifurcation","level":3,"score":0.5911261439323425},{"id":"https://openalex.org/C155092403","wikidata":"https://www.wikidata.org/wiki/Q1785924","display_name":"Pitchfork bifurcation","level":5,"score":0.5890865325927734},{"id":"https://openalex.org/C55610538","wikidata":"https://www.wikidata.org/wiki/Q1502581","display_name":"Transcritical bifurcation","level":5,"score":0.5568545460700989},{"id":"https://openalex.org/C22257346","wikidata":"https://www.wikidata.org/wiki/Q4937510","display_name":"Bogdanov\u2013Takens bifurcation","level":5,"score":0.5260244011878967},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.5077399015426636},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.4812892973423004},{"id":"https://openalex.org/C194271172","wikidata":"https://www.wikidata.org/wiki/Q2181373","display_name":"Period-doubling bifurcation","level":4,"score":0.4330224096775055},{"id":"https://openalex.org/C165150564","wikidata":"https://www.wikidata.org/wiki/Q1154357","display_name":"Infinite-period bifurcation","level":5,"score":0.4268742501735687},{"id":"https://openalex.org/C47446073","wikidata":"https://www.wikidata.org/wiki/Q5165890","display_name":"Control theory (sociology)","level":3,"score":0.3819277882575989},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3538751006126404},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.33154210448265076},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.23800477385520935},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.19576460123062134},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.16991746425628662},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.062323302030563354},{"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},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1142/s0218127416501005","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127416501005","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":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":16,"referenced_works":["https://openalex.org/W1531715726","https://openalex.org/W1967859092","https://openalex.org/W1968375879","https://openalex.org/W1973084963","https://openalex.org/W1976002650","https://openalex.org/W1977202197","https://openalex.org/W1981514016","https://openalex.org/W1985957601","https://openalex.org/W1990517958","https://openalex.org/W1997592679","https://openalex.org/W2047453114","https://openalex.org/W2084318376","https://openalex.org/W2091280897","https://openalex.org/W2095296030","https://openalex.org/W2112246162","https://openalex.org/W2132570681"],"related_works":["https://openalex.org/W2546487931","https://openalex.org/W2019027784","https://openalex.org/W2775656422","https://openalex.org/W3020506472","https://openalex.org/W4245057904","https://openalex.org/W2025138898","https://openalex.org/W4281733204","https://openalex.org/W1976081455","https://openalex.org/W2347648614","https://openalex.org/W2078078632"],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"a":[3,24,74],"system":[4],"of":[5,45,67],"neural":[6],"networks":[7],"in":[8],"neutral":[9],"form":[10,57],"with":[11],"time":[12],"delay":[13,19],"is":[14,28,38,70,82],"investigated.":[15],"Further,":[16],"by":[17,54,72],"introducing":[18],"[Formula:":[20,35],"see":[21,36],"text]":[22,37],"as":[23],"bifurcation":[25,32,77],"parameter,":[26],"it":[27],"found":[29],"that":[30],"Hopf":[31,47,76],"occurs":[33],"when":[34],"across":[39],"some":[40],"critical":[41],"values.":[42],"The":[43],"direction":[44],"the":[46,50,64,86],"bifurcations":[48],"and":[49,59],"stability":[51],"are":[52],"determined":[53],"using":[55,73],"normal":[56],"method":[58],"center":[60],"manifold":[61],"theory.":[62],"Next,":[63],"global":[65,75],"existence":[66],"periodic":[68],"solution":[69],"established":[71],"result.":[78],"Finally,":[79],"an":[80],"example":[81],"given":[83],"to":[84],"support":[85],"theoretical":[87],"predictions.":[88]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2017,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}