{"id":"https://openalex.org/W4389620290","doi":"https://doi.org/10.1142/s021812742350178x","title":"Limit Cycles of a Planar Piecewise Linear System with an Improper Node","display_name":"Limit Cycles of a Planar Piecewise Linear System with an Improper Node","publication_year":2023,"publication_date":"2023-12-11","ids":{"openalex":"https://openalex.org/W4389620290","doi":"https://doi.org/10.1142/s021812742350178x"},"language":"en","primary_location":{"id":"doi:10.1142/s021812742350178x","is_oa":false,"landing_page_url":"http://dx.doi.org/10.1142/s021812742350178x","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/A5102495233","display_name":"Ning Xiao","orcid":"https://orcid.org/0009-0003-3814-7876"},"institutions":[{"id":"https://openalex.org/I178232147","display_name":"Guizhou University","ror":"https://ror.org/02wmsc916","country_code":"CN","type":"education","lineage":["https://openalex.org/I178232147"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Ning Xiao","raw_affiliation_strings":["School of Mathematics and Statistics, Guizhou University, Guiyang 550025, P. R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics, Guizhou University, Guiyang 550025, P. R. China","institution_ids":["https://openalex.org/I178232147"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5025837333","display_name":"Kuilin Wu","orcid":"https://orcid.org/0000-0001-5872-3808"},"institutions":[{"id":"https://openalex.org/I178232147","display_name":"Guizhou University","ror":"https://ror.org/02wmsc916","country_code":"CN","type":"education","lineage":["https://openalex.org/I178232147"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Kuilin Wu","raw_affiliation_strings":["School of Mathematics and Statistics, Guizhou University, Guiyang 550025, P. R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics, Guizhou University, Guiyang 550025, P. R. China","institution_ids":["https://openalex.org/I178232147"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I178232147"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.26667163,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"33","issue":"15","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12396","display_name":"Advanced Differential Equations and Dynamical Systems","score":0.9998999834060669,"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"}},"topics":[{"id":"https://openalex.org/T12396","display_name":"Advanced Differential Equations and Dynamical Systems","score":0.9998999834060669,"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":0.7449650168418884},{"id":"https://openalex.org/keywords/piecewise-linear-function","display_name":"Piecewise linear function","score":0.7093418836593628},{"id":"https://openalex.org/keywords/limit","display_name":"Limit (mathematics)","score":0.684588611125946},{"id":"https://openalex.org/keywords/node","display_name":"Node (physics)","score":0.6604921817779541},{"id":"https://openalex.org/keywords/piecewise","display_name":"Piecewise","score":0.6388988494873047},{"id":"https://openalex.org/keywords/planar","display_name":"Planar","score":0.5874862670898438},{"id":"https://openalex.org/keywords/focus","display_name":"Focus (optics)","score":0.5696576833724976},{"id":"https://openalex.org/keywords/boundary","display_name":"Boundary (topology)","score":0.4993572235107422},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.49241840839385986},{"id":"https://openalex.org/keywords/type","display_name":"Type (biology)","score":0.4807036519050598},{"id":"https://openalex.org/keywords/limit-cycle","display_name":"Limit cycle","score":0.47842103242874146},{"id":"https://openalex.org/keywords/displacement","display_name":"Displacement (psychology)","score":0.4608473479747772},{"id":"https://openalex.org/keywords/saddle","display_name":"Saddle","score":0.4477090537548065},{"id":"https://openalex.org/keywords/saddle-node-bifurcation","display_name":"Saddle-node bifurcation","score":0.4249630570411682},{"id":"https://openalex.org/keywords/line","display_name":"Line (geometry)","score":0.4115447700023651},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.25867825746536255},{"id":"https://openalex.org/keywords/bifurcation","display_name":"Bifurcation","score":0.2167968451976776},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.17277467250823975},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.16890877485275269},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.12071952223777771},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.09940236806869507}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7449650168418884},{"id":"https://openalex.org/C17095337","wikidata":"https://www.wikidata.org/wiki/Q2375229","display_name":"Piecewise linear function","level":2,"score":0.7093418836593628},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.684588611125946},{"id":"https://openalex.org/C62611344","wikidata":"https://www.wikidata.org/wiki/Q1062658","display_name":"Node (physics)","level":2,"score":0.6604921817779541},{"id":"https://openalex.org/C164660894","wikidata":"https://www.wikidata.org/wiki/Q2037833","display_name":"Piecewise","level":2,"score":0.6388988494873047},{"id":"https://openalex.org/C134786449","wikidata":"https://www.wikidata.org/wiki/Q3391255","display_name":"Planar","level":2,"score":0.5874862670898438},{"id":"https://openalex.org/C192209626","wikidata":"https://www.wikidata.org/wiki/Q190909","display_name":"Focus (optics)","level":2,"score":0.5696576833724976},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.4993572235107422},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.49241840839385986},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.4807036519050598},{"id":"https://openalex.org/C93357160","wikidata":"https://www.wikidata.org/wiki/Q1545916","display_name":"Limit cycle","level":3,"score":0.47842103242874146},{"id":"https://openalex.org/C107551265","wikidata":"https://www.wikidata.org/wiki/Q1458245","display_name":"Displacement (psychology)","level":2,"score":0.4608473479747772},{"id":"https://openalex.org/C2777127463","wikidata":"https://www.wikidata.org/wiki/Q10862618","display_name":"Saddle","level":2,"score":0.4477090537548065},{"id":"https://openalex.org/C101225446","wikidata":"https://www.wikidata.org/wiki/Q740190","display_name":"Saddle-node bifurcation","level":4,"score":0.4249630570411682},{"id":"https://openalex.org/C198352243","wikidata":"https://www.wikidata.org/wiki/Q37105","display_name":"Line (geometry)","level":2,"score":0.4115447700023651},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.25867825746536255},{"id":"https://openalex.org/C2781349735","wikidata":"https://www.wikidata.org/wiki/Q4904824","display_name":"Bifurcation","level":3,"score":0.2167968451976776},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.17277467250823975},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.16890877485275269},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.12071952223777771},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.09940236806869507},{"id":"https://openalex.org/C120665830","wikidata":"https://www.wikidata.org/wiki/Q14620","display_name":"Optics","level":1,"score":0.0},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C542102704","wikidata":"https://www.wikidata.org/wiki/Q183257","display_name":"Psychotherapist","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/C121684516","wikidata":"https://www.wikidata.org/wiki/Q7600677","display_name":"Computer graphics (images)","level":1,"score":0.0},{"id":"https://openalex.org/C15744967","wikidata":"https://www.wikidata.org/wiki/Q9418","display_name":"Psychology","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1142/s021812742350178x","is_oa":false,"landing_page_url":"http://dx.doi.org/10.1142/s021812742350178x","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":28,"referenced_works":["https://openalex.org/W27783439","https://openalex.org/W965103837","https://openalex.org/W1974272814","https://openalex.org/W1980228211","https://openalex.org/W1997262155","https://openalex.org/W2002282594","https://openalex.org/W2008393500","https://openalex.org/W2048083457","https://openalex.org/W2050710621","https://openalex.org/W2053658423","https://openalex.org/W2054652000","https://openalex.org/W2056794517","https://openalex.org/W2066824120","https://openalex.org/W2073072471","https://openalex.org/W2079569562","https://openalex.org/W2090366747","https://openalex.org/W2092046534","https://openalex.org/W2884876477","https://openalex.org/W2889668533","https://openalex.org/W2922368149","https://openalex.org/W2976778875","https://openalex.org/W3031244554","https://openalex.org/W3088941680","https://openalex.org/W3090096152","https://openalex.org/W3181518534","https://openalex.org/W4238307931","https://openalex.org/W4281672383","https://openalex.org/W4308438246"],"related_works":["https://openalex.org/W2765944944","https://openalex.org/W4226164715","https://openalex.org/W3096660584","https://openalex.org/W2133947431","https://openalex.org/W1596838019","https://openalex.org/W3182827720","https://openalex.org/W2490196280","https://openalex.org/W3208509670","https://openalex.org/W2100864966","https://openalex.org/W2109304297"],"abstract_inverted_index":{"This":[0],"paper":[1],"is":[2,60,98],"concerned":[3],"with":[4,17],"the":[5,29,32,58,75,88,92],"number":[6,40,76,94],"of":[7,28,31,41,77,79,95],"limit":[8,42,96],"cycles":[9,43,97],"for":[10,44,82,105],"a":[11,22,61],"planar":[12],"piecewise":[13],"linear":[14],"(PWL)":[15],"system":[16,34],"two":[18,101],"zones":[19],"separated":[20],"by":[21],"straight":[23],"line.":[24],"Assume":[25],"that":[26],"one":[27,99],"subsystems":[30],"PWL":[33],"has":[35],"an":[36],"improper":[37],"node.":[38],"The":[39],"saddle-improper":[45],"node":[46,49,53],"type,":[47],"focus-improper":[48],"type":[50,54],"and":[51,73],"center-improper":[52],"(the":[55],"focus":[56],"or":[57,63,100],"center":[59],"virtual":[62],"boundary":[64],"equilibrium)":[65],"are":[66],"studied.":[67],"First,":[68],"we":[69,86],"introduce":[70],"displacement":[71,80],"functions":[72,81],"study":[74],"zeros":[78],"different":[83,106],"types.":[84,107],"Then,":[85],"give":[87],"parameter":[89],"regions":[90],"where":[91],"exact":[93],"(at":[102],"least":[103],"two)":[104]},"counts_by_year":[],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}
