{"id":"https://openalex.org/W4415731352","doi":"https://doi.org/10.1142/s0218127426500148","title":"A Geometric Method for Studying the Stability Switching Criteria with Two Discrete Delays","display_name":"A Geometric Method for Studying the Stability Switching Criteria with Two Discrete Delays","publication_year":2025,"publication_date":"2025-10-31","ids":{"openalex":"https://openalex.org/W4415731352","doi":"https://doi.org/10.1142/s0218127426500148"},"language":"en","primary_location":{"id":"doi:10.1142/s0218127426500148","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127426500148","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/A5102708789","display_name":"Jianzhi Cao","orcid":"https://orcid.org/0009-0005-5840-7617"},"institutions":[{"id":"https://openalex.org/I43337087","display_name":"Hebei University","ror":"https://ror.org/01p884a79","country_code":"CN","type":"education","lineage":["https://openalex.org/I43337087"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Jianzhi Cao","raw_affiliation_strings":["Hebei Key Laboratory of Machine Learning and Computational Intelligence, College of Mathematics and Information Science, Hebei University, Baoding, Hebei 071002, P. R. China"],"affiliations":[{"raw_affiliation_string":"Hebei Key Laboratory of Machine Learning and Computational Intelligence, College of Mathematics and Information Science, Hebei University, Baoding, Hebei 071002, P. R. China","institution_ids":["https://openalex.org/I43337087"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5034674230","display_name":"Qi Liu","orcid":"https://orcid.org/0009-0009-1762-5051"},"institutions":[{"id":"https://openalex.org/I43337087","display_name":"Hebei University","ror":"https://ror.org/01p884a79","country_code":"CN","type":"education","lineage":["https://openalex.org/I43337087"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Qi Liu","raw_affiliation_strings":["Hebei Key Laboratory of Machine Learning and Computational Intelligence, College of Mathematics and Information Science, Hebei University, Baoding, Hebei 071002, P. R. China"],"affiliations":[{"raw_affiliation_string":"Hebei Key Laboratory of Machine Learning and Computational Intelligence, College of Mathematics and Information Science, Hebei University, Baoding, Hebei 071002, P. R. China","institution_ids":["https://openalex.org/I43337087"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5004222079","display_name":"Pengmiao Hao","orcid":"https://orcid.org/0000-0001-8605-1906"},"institutions":[{"id":"https://openalex.org/I43337087","display_name":"Hebei University","ror":"https://ror.org/01p884a79","country_code":"CN","type":"education","lineage":["https://openalex.org/I43337087"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Pengmiao Hao","raw_affiliation_strings":["Hebei Key Laboratory of Machine Learning and Computational Intelligence, College of Mathematics and Information Science, Hebei University, Baoding, Hebei 071002, P. R. China"],"affiliations":[{"raw_affiliation_string":"Hebei Key Laboratory of Machine Learning and Computational Intelligence, College of Mathematics and Information Science, Hebei University, Baoding, Hebei 071002, P. R. China","institution_ids":["https://openalex.org/I43337087"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5001869791","display_name":"Zhichao Jiang","orcid":"https://orcid.org/0000-0002-1077-2261"},"institutions":[{"id":"https://openalex.org/I4210119087","display_name":"North China Institute of Aerospace Engineering","ror":"https://ror.org/02m7msy24","country_code":"CN","type":"education","lineage":["https://openalex.org/I4210119087"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Zhichao Jiang","raw_affiliation_strings":["School of Liberal Arts and Sciences, North China Institute of Aerospace Engineering, Langfang, Hebei 065000, P. R. China"],"affiliations":[{"raw_affiliation_string":"School of Liberal Arts and Sciences, North China Institute of Aerospace Engineering, Langfang, Hebei 065000, P. R. China","institution_ids":["https://openalex.org/I4210119087"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5102708789"],"corresponding_institution_ids":["https://openalex.org/I43337087"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.36374293,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"36","issue":"02","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11347","display_name":"Neural Networks Stability and Synchronization","score":0.9793000221252441,"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":0.9793000221252441,"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/T12772","display_name":"Control and Stability of Dynamical Systems","score":0.005799999926239252,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11110","display_name":"Stability and Controllability of Differential Equations","score":0.0010999999940395355,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/parallelogram","display_name":"Parallelogram","score":0.7904000282287598},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.6031000018119812},{"id":"https://openalex.org/keywords/control-theory","display_name":"Control theory (sociology)","score":0.48420000076293945},{"id":"https://openalex.org/keywords/geometric-distribution","display_name":"Geometric distribution","score":0.4794999957084656},{"id":"https://openalex.org/keywords/type","display_name":"Type (biology)","score":0.453900009393692},{"id":"https://openalex.org/keywords/distribution","display_name":"Distribution (mathematics)","score":0.42309999465942383},{"id":"https://openalex.org/keywords/class","display_name":"Class (philosophy)","score":0.41909998655319214},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.4059000015258789}],"concepts":[{"id":"https://openalex.org/C53118041","wikidata":"https://www.wikidata.org/wiki/Q45867","display_name":"Parallelogram","level":3,"score":0.7904000282287598},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7523000240325928},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.6031000018119812},{"id":"https://openalex.org/C47446073","wikidata":"https://www.wikidata.org/wiki/Q5165890","display_name":"Control theory (sociology)","level":3,"score":0.48420000076293945},{"id":"https://openalex.org/C39947850","wikidata":"https://www.wikidata.org/wiki/Q729523","display_name":"Geometric distribution","level":3,"score":0.4794999957084656},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.453900009393692},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4275999963283539},{"id":"https://openalex.org/C110121322","wikidata":"https://www.wikidata.org/wiki/Q865811","display_name":"Distribution (mathematics)","level":2,"score":0.42309999465942383},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.41909998655319214},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4138000011444092},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.4059000015258789},{"id":"https://openalex.org/C93226319","wikidata":"https://www.wikidata.org/wiki/Q193137","display_name":"Differential (mechanical device)","level":2,"score":0.38830000162124634},{"id":"https://openalex.org/C55689738","wikidata":"https://www.wikidata.org/wiki/Q15963867","display_name":"Discrete time and continuous time","level":2,"score":0.3650999963283539},{"id":"https://openalex.org/C53744967","wikidata":"https://www.wikidata.org/wiki/Q1062971","display_name":"Characteristic equation","level":3,"score":0.3479999899864197},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.34619998931884766},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.3434999883174896},{"id":"https://openalex.org/C147504518","wikidata":"https://www.wikidata.org/wiki/Q2146125","display_name":"Delay differential equation","level":3,"score":0.3138999938964844},{"id":"https://openalex.org/C154968394","wikidata":"https://www.wikidata.org/wiki/Q5535474","display_name":"Geometric analysis","level":5,"score":0.31029999256134033},{"id":"https://openalex.org/C143170015","wikidata":"https://www.wikidata.org/wiki/Q17007850","display_name":"Stability conditions","level":3,"score":0.3019999861717224},{"id":"https://openalex.org/C7305733","wikidata":"https://www.wikidata.org/wiki/Q207961","display_name":"Geometric shape","level":2,"score":0.29820001125335693},{"id":"https://openalex.org/C148043351","wikidata":"https://www.wikidata.org/wiki/Q4456944","display_name":"Current (fluid)","level":2,"score":0.2648000121116638}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1142/s0218127426500148","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127426500148","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":[{"id":"https://openalex.org/G5194430032","display_name":null,"funder_award_id":"A2024409003","funder_id":"https://openalex.org/F4320322163","funder_display_name":"Natural Science Foundation of Hebei Province"},{"id":"https://openalex.org/G6343325761","display_name":null,"funder_award_id":"12201274","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G6843204641","display_name":null,"funder_award_id":"12171135","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320322163","display_name":"Natural Science Foundation of Hebei Province","ror":"https://ror.org/01h0zpd94"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":28,"referenced_works":["https://openalex.org/W1972385922","https://openalex.org/W1991319287","https://openalex.org/W2019823335","https://openalex.org/W2030228294","https://openalex.org/W2062264317","https://openalex.org/W2064340892","https://openalex.org/W2074178424","https://openalex.org/W2088786472","https://openalex.org/W2120026810","https://openalex.org/W2129165621","https://openalex.org/W2149971042","https://openalex.org/W2155267437","https://openalex.org/W2158630456","https://openalex.org/W2281360737","https://openalex.org/W2344955137","https://openalex.org/W2900959448","https://openalex.org/W3041834562","https://openalex.org/W3097069147","https://openalex.org/W3097680321","https://openalex.org/W4213116370","https://openalex.org/W4312255936","https://openalex.org/W4315701374","https://openalex.org/W4320020519","https://openalex.org/W4320891058","https://openalex.org/W4323064929","https://openalex.org/W4366980293","https://openalex.org/W4386314075","https://openalex.org/W4390730605"],"related_works":[],"abstract_inverted_index":{"This":[0],"paper":[1],"studies":[2],"the":[3,22,32,45,51,63,72,75,82,91],"characteristic":[4,23],"equations":[5,12],"for":[6,50],"a":[7,18,28,68,98],"class":[8],"of":[9,21,37,53,71,77,97],"functional":[10],"differential":[11],"with":[13,102],"two":[14,103],"discrete":[15,104],"delays,":[16],"and":[17,47,57,81],"complete":[19],"distribution":[20],"roots":[24,56],"is":[25,41,106],"derived":[26],"by":[27],"geometric":[29],"method.":[30],"With":[31],"method,":[33],"one":[34],"more":[35],"type":[36],"stable":[38,78],"switching":[39,79],"curves":[40,80],"obtained.":[42],"At":[43],"first,":[44],"necessary":[46],"sufficient":[48],"conditions":[49],"existence":[52],"pure":[54],"imaginary":[55],"crossing":[58,73,83],"sets":[59],"are":[60,85],"established":[61],"through":[62],"parallelogram":[64],"rule.":[65],"Then,":[66],"after":[67],"careful":[69],"analysis":[70],"set,":[74],"types":[76],"direction":[84],"revealed.":[86],"In":[87],"addition,":[88],"to":[89],"validate":[90],"theoretical":[92],"results":[93],"obtained,":[94],"an":[95],"example":[96],"neural":[99],"network":[100],"model":[101],"delays":[105],"presented.":[107]},"counts_by_year":[],"updated_date":"2026-04-09T08:11:56.329763","created_date":"2025-10-31T00:00:00"}