{"id":"https://openalex.org/W2063168848","doi":"https://doi.org/10.1142/s0218127409023706","title":"STABILITY AND BIFURCATION IN A LOGISTIC EQUATION WITH PIECEWISE CONSTANT ARGUMENTS","display_name":"STABILITY AND BIFURCATION IN A LOGISTIC EQUATION WITH PIECEWISE CONSTANT ARGUMENTS","publication_year":2009,"publication_date":"2009-04-01","ids":{"openalex":"https://openalex.org/W2063168848","doi":"https://doi.org/10.1142/s0218127409023706","mag":"2063168848"},"language":"en","primary_location":{"id":"doi:10.1142/s0218127409023706","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127409023706","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/A5046423503","display_name":"Chunrui Zhang","orcid":"https://orcid.org/0000-0002-9356-5064"},"institutions":[{"id":"https://openalex.org/I47689461","display_name":"Northeast Forestry University","ror":"https://ror.org/02yxnh564","country_code":"CN","type":"education","lineage":["https://openalex.org/I47689461"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"CHUNRUI ZHANG","raw_affiliation_strings":["Department of Mathematics, Northeast Forestry, University, Harbin 150040, P. R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Northeast Forestry, University, Harbin 150040, P. R. China","institution_ids":["https://openalex.org/I47689461"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5102204108","display_name":"Baodong Zheng","orcid":null},"institutions":[{"id":"https://openalex.org/I204983213","display_name":"Harbin Institute of Technology","ror":"https://ror.org/01yqg2h08","country_code":"CN","type":"education","lineage":["https://openalex.org/I204983213"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"BAODONG ZHENG","raw_affiliation_strings":["Department of Mathematics, Harbin Institute of Technology, Harbin 150001, P. R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Harbin Institute of Technology, Harbin 150001, P. R. China","institution_ids":["https://openalex.org/I204983213"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5100711411","display_name":"Yazhuo Zhang","orcid":"https://orcid.org/0000-0002-8583-2580"},"institutions":[{"id":"https://openalex.org/I204983213","display_name":"Harbin Institute of Technology","ror":"https://ror.org/01yqg2h08","country_code":"CN","type":"education","lineage":["https://openalex.org/I204983213"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"YAZHUO ZHANG","raw_affiliation_strings":["Department of Mathematics, Harbin Institute of Technology, Harbin 150001, P. R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Harbin Institute of Technology, Harbin 150001, P. R. China","institution_ids":["https://openalex.org/I204983213"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.3412,"has_fulltext":false,"cited_by_count":5,"citation_normalized_percentile":{"value":0.71269807,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":95},"biblio":{"volume":"19","issue":"04","first_page":"1373","last_page":"1379"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10482","display_name":"Mathematical and Theoretical Epidemiology and Ecology Models","score":0.9994000196456909,"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"}},"topics":[{"id":"https://openalex.org/T10482","display_name":"Mathematical and Theoretical Epidemiology and Ecology Models","score":0.9994000196456909,"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"}},{"id":"https://openalex.org/T10541","display_name":"Nonlinear Differential Equations Analysis","score":0.987500011920929,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12396","display_name":"Advanced Differential Equations and Dynamical Systems","score":0.9835000038146973,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/mathematics","display_name":"Mathematics","score":0.8019343018531799},{"id":"https://openalex.org/keywords/center-manifold","display_name":"Center manifold","score":0.7846484780311584},{"id":"https://openalex.org/keywords/constant","display_name":"Constant (computer programming)","score":0.7153719663619995},{"id":"https://openalex.org/keywords/hopf-bifurcation","display_name":"Hopf bifurcation","score":0.6712919473648071},{"id":"https://openalex.org/keywords/saddle-node-bifurcation","display_name":"Saddle-node bifurcation","score":0.6698139905929565},{"id":"https://openalex.org/keywords/bifurcation-diagram","display_name":"Bifurcation diagram","score":0.64619380235672},{"id":"https://openalex.org/keywords/transcritical-bifurcation","display_name":"Transcritical bifurcation","score":0.6104933619499207},{"id":"https://openalex.org/keywords/piecewise","display_name":"Piecewise","score":0.584170401096344},{"id":"https://openalex.org/keywords/bifurcation","display_name":"Bifurcation","score":0.5739307403564453},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.5662323832511902},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.5636125206947327},{"id":"https://openalex.org/keywords/period-doubling-bifurcation","display_name":"Period-doubling bifurcation","score":0.5398676991462708},{"id":"https://openalex.org/keywords/pitchfork-bifurcation","display_name":"Pitchfork bifurcation","score":0.5151585340499878},{"id":"https://openalex.org/keywords/biological-applications-of-bifurcation-theory","display_name":"Biological applications of bifurcation theory","score":0.464956134557724},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.386918842792511},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.1627393662929535},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.08530494570732117},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.060389578342437744}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8019343018531799},{"id":"https://openalex.org/C74774103","wikidata":"https://www.wikidata.org/wiki/Q4504202","display_name":"Center manifold","level":5,"score":0.7846484780311584},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.7153719663619995},{"id":"https://openalex.org/C172435161","wikidata":"https://www.wikidata.org/wiki/Q531126","display_name":"Hopf bifurcation","level":4,"score":0.6712919473648071},{"id":"https://openalex.org/C101225446","wikidata":"https://www.wikidata.org/wiki/Q740190","display_name":"Saddle-node bifurcation","level":4,"score":0.6698139905929565},{"id":"https://openalex.org/C86243350","wikidata":"https://www.wikidata.org/wiki/Q2595915","display_name":"Bifurcation diagram","level":4,"score":0.64619380235672},{"id":"https://openalex.org/C55610538","wikidata":"https://www.wikidata.org/wiki/Q1502581","display_name":"Transcritical bifurcation","level":5,"score":0.6104933619499207},{"id":"https://openalex.org/C164660894","wikidata":"https://www.wikidata.org/wiki/Q2037833","display_name":"Piecewise","level":2,"score":0.584170401096344},{"id":"https://openalex.org/C2781349735","wikidata":"https://www.wikidata.org/wiki/Q4904824","display_name":"Bifurcation","level":3,"score":0.5739307403564453},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5662323832511902},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.5636125206947327},{"id":"https://openalex.org/C194271172","wikidata":"https://www.wikidata.org/wiki/Q2181373","display_name":"Period-doubling bifurcation","level":4,"score":0.5398676991462708},{"id":"https://openalex.org/C155092403","wikidata":"https://www.wikidata.org/wiki/Q1785924","display_name":"Pitchfork bifurcation","level":5,"score":0.5151585340499878},{"id":"https://openalex.org/C174914120","wikidata":"https://www.wikidata.org/wiki/Q4914973","display_name":"Biological applications of bifurcation theory","level":5,"score":0.464956134557724},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.386918842792511},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.1627393662929535},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.08530494570732117},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.060389578342437744},{"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/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","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}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1142/s0218127409023706","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127409023706","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":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":10,"referenced_works":["https://openalex.org/W1963688261","https://openalex.org/W1965136568","https://openalex.org/W1970655894","https://openalex.org/W1976002650","https://openalex.org/W2019449051","https://openalex.org/W2024723232","https://openalex.org/W2043523019","https://openalex.org/W2065732144","https://openalex.org/W2077512839","https://openalex.org/W2112910821"],"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/W2273563257"],"abstract_inverted_index":{"A":[0],"logistic":[1],"equation":[2],"with":[3],"piecewise":[4],"constant":[5],"arguments":[6],"is":[7,16,19,54],"investigated.":[8],"Firstly,":[9],"the":[10,14,28,35,40,43,47,50,58,73],"linear":[11],"stability":[12,48],"of":[13,42,49],"model":[15],"studied.":[17],"It":[18],"found":[20],"that":[21],"there":[22],"exists":[23],"a":[24,31],"Hopf":[25,44],"bifurcation":[26,45],"when":[27],"parameter":[29],"passes":[30],"critical":[32],"value.":[33],"Then":[34],"explicit":[36],"algorithm":[37],"for":[38],"determining":[39],"direction":[41],"and":[46,62],"bifurcating":[51],"periodic":[52],"solution":[53],"derived":[55],"by":[56],"using":[57],"normal":[59],"form":[60],"method":[61],"center":[63],"manifold":[64],"theorem.":[65],"Finally,":[66],"computer":[67],"simulations":[68],"are":[69],"performed":[70],"to":[71],"illustrate":[72],"analytical":[74],"results":[75],"found.":[76]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2018,"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"}
