{"id":"https://openalex.org/W3167837829","doi":"https://doi.org/10.1155/2021/9978320","title":"Stability, Multistability, and Complexity Analysis in a Dynamic Duopoly Game with Exponential Demand Function","display_name":"Stability, Multistability, and Complexity Analysis in a Dynamic Duopoly Game with Exponential Demand Function","publication_year":2021,"publication_date":"2021-01-01","ids":{"openalex":"https://openalex.org/W3167837829","doi":"https://doi.org/10.1155/2021/9978320","mag":"3167837829"},"language":"en","primary_location":{"id":"doi:10.1155/2021/9978320","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2021/9978320","pdf_url":"https://downloads.hindawi.com/journals/complexity/2021/9978320.pdf","source":{"id":"https://openalex.org/S207319443","display_name":"Complexity","issn_l":"1076-2787","issn":["1076-2787","1099-0526"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Complexity","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://downloads.hindawi.com/journals/complexity/2021/9978320.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5103101642","display_name":"Hui Li","orcid":"https://orcid.org/0000-0002-4929-4949"},"institutions":[{"id":"https://openalex.org/I3133134087","display_name":"Lanzhou Jiaotong University","ror":"https://ror.org/03144pv92","country_code":"CN","type":"education","lineage":["https://openalex.org/I3133134087"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Hui Li","raw_affiliation_strings":["Research Centre of Game eory and Economic Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, Gansu, China","School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, Gansu, China","Research Centre of Game Theory and Economic Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, Gansu, China"],"raw_orcid":"https://orcid.org/0000-0002-4929-4949","affiliations":[{"raw_affiliation_string":"Research Centre of Game eory and Economic Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, Gansu, China","institution_ids":["https://openalex.org/I3133134087"]},{"raw_affiliation_string":"School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, Gansu, China","institution_ids":["https://openalex.org/I3133134087"]},{"raw_affiliation_string":"Research Centre of Game Theory and Economic Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, Gansu, China","institution_ids":["https://openalex.org/I3133134087"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5041367076","display_name":"Wei Zhou","orcid":"https://orcid.org/0000-0002-4477-3940"},"institutions":[{"id":"https://openalex.org/I3133134087","display_name":"Lanzhou Jiaotong University","ror":"https://ror.org/03144pv92","country_code":"CN","type":"education","lineage":["https://openalex.org/I3133134087"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Wei Zhou","raw_affiliation_strings":["Research Centre of Game eory and Economic Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, Gansu, China","School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, Gansu, China","Research Centre of Game Theory and Economic Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, Gansu, China"],"raw_orcid":"https://orcid.org/0000-0002-4477-3940","affiliations":[{"raw_affiliation_string":"Research Centre of Game eory and Economic Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, Gansu, China","institution_ids":["https://openalex.org/I3133134087"]},{"raw_affiliation_string":"School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, Gansu, China","institution_ids":["https://openalex.org/I3133134087"]},{"raw_affiliation_string":"Research Centre of Game Theory and Economic Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, Gansu, China","institution_ids":["https://openalex.org/I3133134087"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5077917968","display_name":"Tong Chu","orcid":"https://orcid.org/0000-0002-0200-772X"},"institutions":[{"id":"https://openalex.org/I90727586","display_name":"Zhejiang University of Finance and Economics","ror":"https://ror.org/055vj5234","country_code":"CN","type":"education","lineage":["https://openalex.org/I90727586"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Tong Chu","raw_affiliation_strings":["School of Law, Zhejiang University of Finance and Economics, Hangzhou 310018, Zhejiang, China"],"raw_orcid":"https://orcid.org/0000-0002-0200-772X","affiliations":[{"raw_affiliation_string":"School of Law, Zhejiang University of Finance and Economics, Hangzhou 310018, Zhejiang, China","institution_ids":["https://openalex.org/I90727586"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5041367076"],"corresponding_institution_ids":["https://openalex.org/I3133134087"],"apc_list":{"value":2550,"currency":"USD","value_usd":2550},"apc_paid":{"value":2550,"currency":"USD","value_usd":2550},"fwci":0.0,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.08735928,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":91,"max":95},"biblio":{"volume":"2021","issue":"1","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11043","display_name":"Merger and Competition Analysis","score":0.9886000156402588,"subfield":{"id":"https://openalex.org/subfields/2002","display_name":"Economics and Econometrics"},"field":{"id":"https://openalex.org/fields/20","display_name":"Economics, Econometrics and Finance"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},"topics":[{"id":"https://openalex.org/T11043","display_name":"Merger and Competition Analysis","score":0.9886000156402588,"subfield":{"id":"https://openalex.org/subfields/2002","display_name":"Economics and Econometrics"},"field":{"id":"https://openalex.org/fields/20","display_name":"Economics, Econometrics and Finance"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T11187","display_name":"Nonlinear Dynamics and Pattern Formation","score":0.9754999876022339,"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":0.9606999754905701,"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/multistability","display_name":"Multistability","score":0.9626466035842896},{"id":"https://openalex.org/keywords/duopoly","display_name":"Duopoly","score":0.676591694355011},{"id":"https://openalex.org/keywords/attractor","display_name":"Attractor","score":0.6725659370422363},{"id":"https://openalex.org/keywords/bifurcation","display_name":"Bifurcation","score":0.6050273776054382},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.5228427052497864},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.46835628151893616},{"id":"https://openalex.org/keywords/boundary","display_name":"Boundary (topology)","score":0.45915523171424866},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.44032707810401917},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.36061614751815796},{"id":"https://openalex.org/keywords/control-theory","display_name":"Control theory (sociology)","score":0.3548923432826996},{"id":"https://openalex.org/keywords/mathematical-economics","display_name":"Mathematical economics","score":0.3279162645339966},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.31695908308029175},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.29444533586502075},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.22534850239753723},{"id":"https://openalex.org/keywords/cournot-competition","display_name":"Cournot competition","score":0.0950319766998291}],"concepts":[{"id":"https://openalex.org/C2776966350","wikidata":"https://www.wikidata.org/wiki/Q6935096","display_name":"Multistability","level":3,"score":0.9626466035842896},{"id":"https://openalex.org/C180456769","wikidata":"https://www.wikidata.org/wiki/Q750207","display_name":"Duopoly","level":3,"score":0.676591694355011},{"id":"https://openalex.org/C164380108","wikidata":"https://www.wikidata.org/wiki/Q507187","display_name":"Attractor","level":2,"score":0.6725659370422363},{"id":"https://openalex.org/C2781349735","wikidata":"https://www.wikidata.org/wiki/Q4904824","display_name":"Bifurcation","level":3,"score":0.6050273776054382},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.5228427052497864},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.46835628151893616},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.45915523171424866},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.44032707810401917},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.36061614751815796},{"id":"https://openalex.org/C47446073","wikidata":"https://www.wikidata.org/wiki/Q5165890","display_name":"Control theory (sociology)","level":3,"score":0.3548923432826996},{"id":"https://openalex.org/C144237770","wikidata":"https://www.wikidata.org/wiki/Q747534","display_name":"Mathematical economics","level":1,"score":0.3279162645339966},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.31695908308029175},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.29444533586502075},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.22534850239753723},{"id":"https://openalex.org/C16520705","wikidata":"https://www.wikidata.org/wiki/Q601799","display_name":"Cournot competition","level":2,"score":0.0950319766998291},{"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/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0},{"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}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1155/2021/9978320","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2021/9978320","pdf_url":"https://downloads.hindawi.com/journals/complexity/2021/9978320.pdf","source":{"id":"https://openalex.org/S207319443","display_name":"Complexity","issn_l":"1076-2787","issn":["1076-2787","1099-0526"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Complexity","raw_type":"journal-article"},{"id":"pmh:oai:RePEc:hin:complx:9978320","is_oa":false,"landing_page_url":"http://downloads.hindawi.com/journals/complexity/2021/9978320.xml","pdf_url":null,"source":{"id":"https://openalex.org/S4306401271","display_name":"RePEc: Research Papers in Economics","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I77793887","host_organization_name":"Federal Reserve Bank of St. Louis","host_organization_lineage":["https://openalex.org/I77793887"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},{"id":"pmh:oai:doaj.org/article:bf1db8b031864edabb1e153b050abea5","is_oa":true,"landing_page_url":"https://doaj.org/article/bf1db8b031864edabb1e153b050abea5","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Complexity, Vol 2021 (2021)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.1155/2021/9978320","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2021/9978320","pdf_url":"https://downloads.hindawi.com/journals/complexity/2021/9978320.pdf","source":{"id":"https://openalex.org/S207319443","display_name":"Complexity","issn_l":"1076-2787","issn":["1076-2787","1099-0526"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Complexity","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320325566","display_name":"Lanzhou Jiaotong University","ror":"https://ror.org/03144pv92"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3167837829.pdf","grobid_xml":"https://content.openalex.org/works/W3167837829.grobid-xml"},"referenced_works_count":26,"referenced_works":["https://openalex.org/W163026172","https://openalex.org/W845833244","https://openalex.org/W2003908419","https://openalex.org/W2045817086","https://openalex.org/W2065977186","https://openalex.org/W2127669871","https://openalex.org/W2128527177","https://openalex.org/W2145217563","https://openalex.org/W2150778261","https://openalex.org/W2161137744","https://openalex.org/W2196316838","https://openalex.org/W2283374429","https://openalex.org/W2524056771","https://openalex.org/W2544246901","https://openalex.org/W2605783304","https://openalex.org/W2748278404","https://openalex.org/W2767249390","https://openalex.org/W2894079548","https://openalex.org/W2894107962","https://openalex.org/W2964412063","https://openalex.org/W2971969138","https://openalex.org/W2980564742","https://openalex.org/W3048785002","https://openalex.org/W3082766877","https://openalex.org/W3098617939","https://openalex.org/W3152663402"],"related_works":["https://openalex.org/W4226329279","https://openalex.org/W1527571384","https://openalex.org/W2383189914","https://openalex.org/W4301445084","https://openalex.org/W1464403884","https://openalex.org/W2108117187","https://openalex.org/W2015173963","https://openalex.org/W3015125480","https://openalex.org/W2140187929","https://openalex.org/W4362633503"],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"a":[3,119],"discrete\u2010time":[4],"dynamic":[5],"duopoly":[6],"model,":[7],"with":[8,37],"nonlinear":[9],"demand":[10],"and":[11,20,29,56,63,75,91,110,142],"cost":[12],"functions,":[13],"is":[14,133],"established.":[15],"The":[16,31,84],"properties":[17],"of":[18,23,40,46,49,79,81,106,108,113,123,139],"existence":[19],"local":[21],"stability":[22,32,62],"equilibrium":[24],"points":[25],"have":[26],"been":[27],"verified":[28],"analyzed.":[30,117],"conditions":[33],"are":[34,116],"also":[35],"given":[36],"the":[38,41,47,51,89,100,103,111,121,137],"help":[39],"Jury":[42],"criterion.":[43],"With":[44],"changing":[45],"values":[48],"parameters,":[50],"system":[52],"shows":[53],"some":[54],"new":[55],"interesting":[57],"phenomena":[58],"in":[59,102],"terms":[60],"to":[61],"multistability,":[64],"such":[65,126],"as":[66,127],"V\u2010shaped":[67],"stable":[68],"structures":[69,86],"(also":[70],"called":[71],"Isoperiodic":[72],"Stable":[73],"Structures)":[74],"different":[76],"shape":[77],"basins":[78],"attraction":[80,109],"coexisting":[82],"attractors.":[83],"eye\u2010shaped":[85],"appear":[87],"where":[88],"period\u2010doubling":[90],"period\u2010halving":[92],"bifurcations":[93],"occur.":[94],"Finally,":[95],"by":[96,136],"utilizing":[97],"critical":[98,140],"curves,":[99],"changes":[101],"topological":[104],"structure":[105],"basin":[107],"reason":[112],"\u201choles\u201d":[114],"formation":[115],"As":[118],"result,":[120],"generation":[122],"global":[124],"bifurcation,":[125,132],"contact":[128,138],"bifurcation":[129],"or":[130],"final":[131],"usually":[134],"accompanied":[135],"curves":[141],"boundary.":[143]},"counts_by_year":[{"year":2025,"cited_by_count":1}],"updated_date":"2026-07-23T08:03:31.855105","created_date":"2025-10-10T00:00:00"}
