{"id":"https://openalex.org/W2885969990","doi":"https://doi.org/10.1142/s0218127418501018","title":"Loop Numbers for the Stability of Homoclinic Loops of Planar Vector Fields","display_name":"Loop Numbers for the Stability of Homoclinic Loops of Planar Vector Fields","publication_year":2018,"publication_date":"2018-07-01","ids":{"openalex":"https://openalex.org/W2885969990","doi":"https://doi.org/10.1142/s0218127418501018","mag":"2885969990"},"language":"en","primary_location":{"id":"doi:10.1142/s0218127418501018","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127418501018","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/A5023278785","display_name":"Xingbo Liu","orcid":"https://orcid.org/0000-0002-7706-9044"},"institutions":[{"id":"https://openalex.org/I66867065","display_name":"East China Normal University","ror":"https://ror.org/02n96ep67","country_code":"CN","type":"education","lineage":["https://openalex.org/I66867065"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Xingbo Liu","raw_affiliation_strings":["Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, P. R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, P. R. China","institution_ids":["https://openalex.org/I66867065"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5101764662","display_name":"Xiao-Biao Lin","orcid":"https://orcid.org/0000-0001-6572-5545"},"institutions":[{"id":"https://openalex.org/I137902535","display_name":"North Carolina State University","ror":"https://ror.org/04tj63d06","country_code":"US","type":"education","lineage":["https://openalex.org/I137902535"]},{"id":"https://openalex.org/I2801313472","display_name":"North Central State College","ror":"https://ror.org/04b6b6f76","country_code":"US","type":"education","lineage":["https://openalex.org/I2801313472"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Xiao-Biao Lin","raw_affiliation_strings":["Department of Mathematics, Box 8205, NC State University, Raleigh, NC 27695-8205, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Box 8205, NC State University, Raleigh, NC 27695-8205, USA","institution_ids":["https://openalex.org/I137902535","https://openalex.org/I2801313472"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5062067924","display_name":"Yuzhen Bai","orcid":"https://orcid.org/0000-0002-0370-9395"},"institutions":[{"id":"https://openalex.org/I202126657","display_name":"Qufu Normal University","ror":"https://ror.org/03ceheh96","country_code":"CN","type":"education","lineage":["https://openalex.org/I202126657"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Yuzhen Bai","raw_affiliation_strings":["School of Mathematical Sciences, Qufu Normal University, Qufu 273165, P. R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences, Qufu Normal University, Qufu 273165, P. R. China","institution_ids":["https://openalex.org/I202126657"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5079872641","display_name":"Deming Zhu","orcid":"https://orcid.org/0000-0002-6317-4329"},"institutions":[{"id":"https://openalex.org/I66867065","display_name":"East China Normal University","ror":"https://ror.org/02n96ep67","country_code":"CN","type":"education","lineage":["https://openalex.org/I66867065"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Deming Zhu","raw_affiliation_strings":["Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, P. R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, P. R. China","institution_ids":["https://openalex.org/I66867065"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.17729311,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"28","issue":"08","first_page":"1850101","last_page":"1850101"},"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"}},{"id":"https://openalex.org/T13080","display_name":"Advanced Differential Geometry Research","score":0.9541000127792358,"subfield":{"id":"https://openalex.org/subfields/3103","display_name":"Astronomy and Astrophysics"},"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/T11261","display_name":"Quantum chaos and dynamical systems","score":0.9114000201225281,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/homoclinic-orbit","display_name":"Homoclinic orbit","score":0.9495692849159241},{"id":"https://openalex.org/keywords/loop","display_name":"Loop (graph theory)","score":0.8067164421081543},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6774144172668457},{"id":"https://openalex.org/keywords/sequence","display_name":"Sequence (biology)","score":0.6333345174789429},{"id":"https://openalex.org/keywords/planar","display_name":"Planar","score":0.6206456422805786},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.5489601492881775},{"id":"https://openalex.org/keywords/vector-field","display_name":"Vector field","score":0.469893217086792},{"id":"https://openalex.org/keywords/homoclinic-bifurcation","display_name":"Homoclinic bifurcation","score":0.4633304178714752},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4000563323497772},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3335399031639099},{"id":"https://openalex.org/keywords/control-theory","display_name":"Control theory (sociology)","score":0.3320878744125366},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.29716092348098755},{"id":"https://openalex.org/keywords/bifurcation","display_name":"Bifurcation","score":0.25185662508010864},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.19112014770507812},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.141657292842865},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.12177953124046326},{"id":"https://openalex.org/keywords/control","display_name":"Control (management)","score":0.06938457489013672}],"concepts":[{"id":"https://openalex.org/C200581526","wikidata":"https://www.wikidata.org/wiki/Q690494","display_name":"Homoclinic orbit","level":4,"score":0.9495692849159241},{"id":"https://openalex.org/C184670325","wikidata":"https://www.wikidata.org/wiki/Q512604","display_name":"Loop (graph theory)","level":2,"score":0.8067164421081543},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6774144172668457},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.6333345174789429},{"id":"https://openalex.org/C134786449","wikidata":"https://www.wikidata.org/wiki/Q3391255","display_name":"Planar","level":2,"score":0.6206456422805786},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.5489601492881775},{"id":"https://openalex.org/C91188154","wikidata":"https://www.wikidata.org/wiki/Q186247","display_name":"Vector field","level":2,"score":0.469893217086792},{"id":"https://openalex.org/C155405617","wikidata":"https://www.wikidata.org/wiki/Q1154357","display_name":"Homoclinic bifurcation","level":5,"score":0.4633304178714752},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4000563323497772},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3335399031639099},{"id":"https://openalex.org/C47446073","wikidata":"https://www.wikidata.org/wiki/Q5165890","display_name":"Control theory (sociology)","level":3,"score":0.3320878744125366},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.29716092348098755},{"id":"https://openalex.org/C2781349735","wikidata":"https://www.wikidata.org/wiki/Q4904824","display_name":"Bifurcation","level":3,"score":0.25185662508010864},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.19112014770507812},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.141657292842865},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.12177953124046326},{"id":"https://openalex.org/C2775924081","wikidata":"https://www.wikidata.org/wiki/Q55608371","display_name":"Control (management)","level":2,"score":0.06938457489013672},{"id":"https://openalex.org/C54355233","wikidata":"https://www.wikidata.org/wiki/Q7162","display_name":"Genetics","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/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"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/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"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/s0218127418501018","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127418501018","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":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":23,"referenced_works":["https://openalex.org/W1603340857","https://openalex.org/W1978541442","https://openalex.org/W1991574094","https://openalex.org/W1999617885","https://openalex.org/W2000849354","https://openalex.org/W2001428827","https://openalex.org/W2003626137","https://openalex.org/W2029829738","https://openalex.org/W2031038544","https://openalex.org/W2036381498","https://openalex.org/W2042498362","https://openalex.org/W2049815318","https://openalex.org/W2061809639","https://openalex.org/W2069228691","https://openalex.org/W2079151093","https://openalex.org/W2089532288","https://openalex.org/W2142664960","https://openalex.org/W2207632376","https://openalex.org/W2316492659","https://openalex.org/W2584645706","https://openalex.org/W2774033034","https://openalex.org/W3123094903","https://openalex.org/W4292283611"],"related_works":["https://openalex.org/W2890492950","https://openalex.org/W3115184012","https://openalex.org/W2010606039","https://openalex.org/W352136236","https://openalex.org/W1999123830","https://openalex.org/W2042800579","https://openalex.org/W944345475","https://openalex.org/W2021681126","https://openalex.org/W2142664960","https://openalex.org/W3150320995"],"abstract_inverted_index":{"This":[0],"paper":[1],"is":[2],"devoted":[3],"to":[4],"the":[5,33,38,43,48,52,60,71,76],"study":[6],"of":[7,11,25,37,47],"stability":[8,34],"and":[9,35,78],"bifurcations":[10,36],"homoclinic":[12,21,80],"loops":[13],"for":[14,51,70,74],"planar":[15],"vector":[16],"fields.":[17],"For":[18],"a":[19,23],"given":[20],"loop,":[22],"sequence":[24],"loop":[26,39,55,72],"numbers":[27,56,73],"can":[28],"be":[29],"defined":[30],"such":[31],"that":[32],"are":[40],"determined":[41],"by":[42],"first":[44,53],"nonzero":[45],"term":[46],"sequence.":[49],"Formulas":[50],"several":[54],"were":[57],"established":[58],"in":[59],"past.":[61],"In":[62],"this":[63],"paper,":[64],"we":[65],"will":[66],"introduce":[67],"general":[68],"formulas":[69],"both":[75],"single":[77],"double":[79],"loops.":[81]},"counts_by_year":[],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}