{"id":"https://openalex.org/W2071887423","doi":"https://doi.org/10.1142/s0218127412501052","title":"BIFURCATION AND CHAOS ANALYSIS FOR A DELAYED TWO-NEURAL NETWORK WITH A VARIATION SLOPE RATIO IN THE ACTIVATION FUNCTION","display_name":"BIFURCATION AND CHAOS ANALYSIS FOR A DELAYED TWO-NEURAL NETWORK WITH A VARIATION SLOPE RATIO IN THE ACTIVATION FUNCTION","publication_year":2012,"publication_date":"2012-03-29","ids":{"openalex":"https://openalex.org/W2071887423","doi":"https://doi.org/10.1142/s0218127412501052","mag":"2071887423"},"language":"en","primary_location":{"id":"doi:10.1142/s0218127412501052","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127412501052","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/A5102741445","display_name":"Zigen Song","orcid":"https://orcid.org/0000-0002-5790-7901"},"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":false,"raw_author_name":"ZIGEN SONG","raw_affiliation_strings":["School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, P. R. China","School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, P. R. China#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, P. R. China","institution_ids":["https://openalex.org/I116953780"]},{"raw_affiliation_string":"School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, P. R. China#TAB#","institution_ids":["https://openalex.org/I116953780"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5100743369","display_name":"Jian Xu","orcid":"https://orcid.org/0000-0002-3521-6843"},"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":false,"raw_author_name":"JIAN XU","raw_affiliation_strings":["School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, P. R. China","School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, P. R. China#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, P. R. China","institution_ids":["https://openalex.org/I116953780"]},{"raw_affiliation_string":"School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, P. R. China#TAB#","institution_ids":["https://openalex.org/I116953780"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I116953780"],"apc_list":null,"apc_paid":null,"fwci":3.2675,"has_fulltext":false,"cited_by_count":24,"citation_normalized_percentile":{"value":0.9253757,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"22","issue":"05","first_page":"1250105","last_page":"1250105"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11347","display_name":"Neural Networks Stability and Synchronization","score":0.9994999766349792,"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":0.9994999766349792,"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":0.9991999864578247,"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":0.9990000128746033,"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/sigmoid-function","display_name":"Sigmoid function","score":0.7077575325965881},{"id":"https://openalex.org/keywords/phase-portrait","display_name":"Phase portrait","score":0.6726377010345459},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6277890205383301},{"id":"https://openalex.org/keywords/bifurcation","display_name":"Bifurcation","score":0.6048380136489868},{"id":"https://openalex.org/keywords/saddle-node-bifurcation","display_name":"Saddle-node bifurcation","score":0.5694682598114014},{"id":"https://openalex.org/keywords/chaotic","display_name":"Chaotic","score":0.5166821479797363},{"id":"https://openalex.org/keywords/equilibrium-point","display_name":"Equilibrium point","score":0.4895502030849457},{"id":"https://openalex.org/keywords/torus","display_name":"Torus","score":0.47701048851013184},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.47079458832740784},{"id":"https://openalex.org/keywords/center-manifold","display_name":"Center manifold","score":0.4439448118209839},{"id":"https://openalex.org/keywords/hopf-bifurcation","display_name":"Hopf bifurcation","score":0.37466156482696533},{"id":"https://openalex.org/keywords/statistical-physics","display_name":"Statistical physics","score":0.35359275341033936},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.3463916778564453},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.2807157337665558},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.22883298993110657},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.16064012050628662},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.1156054437160492},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.11247342824935913}],"concepts":[{"id":"https://openalex.org/C81388566","wikidata":"https://www.wikidata.org/wiki/Q526668","display_name":"Sigmoid function","level":3,"score":0.7077575325965881},{"id":"https://openalex.org/C122283620","wikidata":"https://www.wikidata.org/wiki/Q2532943","display_name":"Phase portrait","level":4,"score":0.6726377010345459},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6277890205383301},{"id":"https://openalex.org/C2781349735","wikidata":"https://www.wikidata.org/wiki/Q4904824","display_name":"Bifurcation","level":3,"score":0.6048380136489868},{"id":"https://openalex.org/C101225446","wikidata":"https://www.wikidata.org/wiki/Q740190","display_name":"Saddle-node bifurcation","level":4,"score":0.5694682598114014},{"id":"https://openalex.org/C2777052490","wikidata":"https://www.wikidata.org/wiki/Q5072826","display_name":"Chaotic","level":2,"score":0.5166821479797363},{"id":"https://openalex.org/C94766913","wikidata":"https://www.wikidata.org/wiki/Q1530271","display_name":"Equilibrium point","level":3,"score":0.4895502030849457},{"id":"https://openalex.org/C9767117","wikidata":"https://www.wikidata.org/wiki/Q12510","display_name":"Torus","level":2,"score":0.47701048851013184},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.47079458832740784},{"id":"https://openalex.org/C74774103","wikidata":"https://www.wikidata.org/wiki/Q4504202","display_name":"Center manifold","level":5,"score":0.4439448118209839},{"id":"https://openalex.org/C172435161","wikidata":"https://www.wikidata.org/wiki/Q531126","display_name":"Hopf bifurcation","level":4,"score":0.37466156482696533},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.35359275341033936},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.3463916778564453},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.2807157337665558},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.22883298993110657},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.16064012050628662},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.1156054437160492},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.11247342824935913},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","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/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1142/s0218127412501052","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127412501052","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":0.5899999737739563,"display_name":"Sustainable cities and communities"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":49,"referenced_works":["https://openalex.org/W1531715726","https://openalex.org/W1574436885","https://openalex.org/W1586335931","https://openalex.org/W1592322637","https://openalex.org/W1596321532","https://openalex.org/W1963913942","https://openalex.org/W1970175853","https://openalex.org/W1970680382","https://openalex.org/W1970851593","https://openalex.org/W1971411983","https://openalex.org/W1978212693","https://openalex.org/W1988447968","https://openalex.org/W1995341919","https://openalex.org/W1997118891","https://openalex.org/W2000162766","https://openalex.org/W2003404212","https://openalex.org/W2003875765","https://openalex.org/W2005414790","https://openalex.org/W2006557790","https://openalex.org/W2010586249","https://openalex.org/W2014352001","https://openalex.org/W2016354087","https://openalex.org/W2022242081","https://openalex.org/W2025574280","https://openalex.org/W2030594480","https://openalex.org/W2035364483","https://openalex.org/W2038294548","https://openalex.org/W2038970958","https://openalex.org/W2061271581","https://openalex.org/W2062585003","https://openalex.org/W2065644906","https://openalex.org/W2067428061","https://openalex.org/W2067607110","https://openalex.org/W2079431495","https://openalex.org/W2079807479","https://openalex.org/W2084822120","https://openalex.org/W2088804152","https://openalex.org/W2089876099","https://openalex.org/W2090194301","https://openalex.org/W2090606621","https://openalex.org/W2095192539","https://openalex.org/W2112246162","https://openalex.org/W2122764273","https://openalex.org/W2151497088","https://openalex.org/W2171264374","https://openalex.org/W3025560754","https://openalex.org/W3104865783","https://openalex.org/W4214513274","https://openalex.org/W4234284487"],"related_works":["https://openalex.org/W2752952648","https://openalex.org/W1987790493","https://openalex.org/W2153388945","https://openalex.org/W1975783007","https://openalex.org/W2038701038","https://openalex.org/W2382752082","https://openalex.org/W1988489950","https://openalex.org/W2352502327","https://openalex.org/W2884051366","https://openalex.org/W2737309493"],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"some":[3],"complex":[4],"phenomena":[5],"of":[6,35,51,89],"dynamical":[7,132],"bifurcations":[8],"are":[9,44,66,82,98,128],"shown":[10],"for":[11],"a":[12,25],"two-neural":[13],"network":[14],"with":[15,22],"delay":[16],"coupling.":[17],"The":[18,41,58,75],"sigmoid":[19,37],"activation":[20],"function":[21],"slope":[23,90],"ratio,":[24],"monotonically":[26],"increasing":[27],"function,":[28],"is":[29],"proposed":[30],"to":[31,68,116,130],"consider":[32],"the":[33,36,52,101,119,124,144],"relations":[34],"and":[38,55,62,72,79,92,110,118,142],"Hardlim":[39],"functions.":[40],"equilibrium":[42,77,136],"points":[43,78],"studied":[45],"analytically":[46],"in":[47,49,84],"detail":[48],"terms":[50],"characteristic":[53],"equation":[54],"static":[56],"bifurcation.":[57],"central":[59],"manifold":[60],"reduction":[61],"normal":[63],"form":[64],"method":[65],"employed":[67],"determine":[69],"Hopf":[70],"bifurcation":[71],"its":[73],"stability.":[74],"stable":[76,135],"periodic":[80,138],"motions":[81],"observed":[83],"different":[85],"parameter":[86,126],"regions.":[87],"Effects":[88],"ratio":[91],"delayed":[93],"coupling":[94],"on":[95,123],"dynamic":[96],"behaviors":[97,133],"investigated":[99],"by":[100],"numerical":[102],"simulation,":[103],"such":[104],"as":[105],"Poincare":[106],"map,":[107],"phase":[108],"portraits,":[109],"power":[111],"spectrum.":[112],"Various":[113],"active":[114],"transitions":[115],"chaos":[117],"corresponding":[120],"critical":[121],"boundaries":[122],"focused":[125],"regions":[127],"obtained":[129],"classify":[131],"including":[134],"point,":[137],"solution,":[139],"2-torus,":[140],"3-torus,":[141],"then":[143],"chaotic":[145],"motions.":[146]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":3},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":4},{"year":2015,"cited_by_count":5},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":2},{"year":2012,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}
