{"id":"https://openalex.org/W2153641542","doi":"https://doi.org/10.1142/s0218127414500837","title":"Seven Limit Cycles Around a Focus Point in a Simple Three-Dimensional Quadratic Vector Field","display_name":"Seven Limit Cycles Around a Focus Point in a Simple Three-Dimensional Quadratic Vector Field","publication_year":2014,"publication_date":"2014-06-01","ids":{"openalex":"https://openalex.org/W2153641542","doi":"https://doi.org/10.1142/s0218127414500837","mag":"2153641542"},"language":"en","primary_location":{"id":"doi:10.1142/s0218127414500837","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127414500837","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/A5017250935","display_name":"Yun Tian","orcid":"https://orcid.org/0000-0002-3557-4384"},"institutions":[{"id":"https://openalex.org/I125749732","display_name":"Western University","ror":"https://ror.org/02grkyz14","country_code":"CA","type":"education","lineage":["https://openalex.org/I125749732"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Yun Tian","raw_affiliation_strings":["Department of Applied Mathematics, Western University, London, Ontario N6A 5B7, Canada"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics, Western University, London, Ontario N6A 5B7, Canada","institution_ids":["https://openalex.org/I125749732"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5100652002","display_name":"Pei Yu","orcid":"https://orcid.org/0000-0002-8848-0584"},"institutions":[{"id":"https://openalex.org/I125749732","display_name":"Western University","ror":"https://ror.org/02grkyz14","country_code":"CA","type":"education","lineage":["https://openalex.org/I125749732"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Pei Yu","raw_affiliation_strings":["Department of Applied Mathematics, Western University, London, Ontario N6A 5B7, Canada"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics, Western University, London, Ontario N6A 5B7, Canada","institution_ids":["https://openalex.org/I125749732"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I125749732"],"apc_list":null,"apc_paid":null,"fwci":1.3381,"has_fulltext":false,"cited_by_count":12,"citation_normalized_percentile":{"value":0.80440493,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"24","issue":"06","first_page":"1450083","last_page":"1450083"},"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/T11261","display_name":"Quantum chaos and dynamical systems","score":0.9695000052452087,"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"}},{"id":"https://openalex.org/T11844","display_name":"Lipid metabolism and biosynthesis","score":0.965499997138977,"subfield":{"id":"https://openalex.org/subfields/1303","display_name":"Biochemistry"},"field":{"id":"https://openalex.org/fields/13","display_name":"Biochemistry, Genetics and Molecular Biology"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6831966042518616},{"id":"https://openalex.org/keywords/center-manifold","display_name":"Center manifold","score":0.6634472608566284},{"id":"https://openalex.org/keywords/vector-field","display_name":"Vector field","score":0.6543524265289307},{"id":"https://openalex.org/keywords/focus","display_name":"Focus (optics)","score":0.6540654897689819},{"id":"https://openalex.org/keywords/quadratic-equation","display_name":"Quadratic equation","score":0.6279044151306152},{"id":"https://openalex.org/keywords/limit","display_name":"Limit (mathematics)","score":0.5782833099365234},{"id":"https://openalex.org/keywords/manifold","display_name":"Manifold (fluid mechanics)","score":0.45923882722854614},{"id":"https://openalex.org/keywords/computation","display_name":"Computation","score":0.445825457572937},{"id":"https://openalex.org/keywords/field","display_name":"Field (mathematics)","score":0.43965667486190796},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.4173164665699005},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.3620189428329468},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3453007936477661},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.2815718650817871},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.18179979920387268},{"id":"https://openalex.org/keywords/hopf-bifurcation","display_name":"Hopf bifurcation","score":0.13885292410850525},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.12445369362831116},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.11431211233139038},{"id":"https://openalex.org/keywords/bifurcation","display_name":"Bifurcation","score":0.09847551584243774}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6831966042518616},{"id":"https://openalex.org/C74774103","wikidata":"https://www.wikidata.org/wiki/Q4504202","display_name":"Center manifold","level":5,"score":0.6634472608566284},{"id":"https://openalex.org/C91188154","wikidata":"https://www.wikidata.org/wiki/Q186247","display_name":"Vector field","level":2,"score":0.6543524265289307},{"id":"https://openalex.org/C192209626","wikidata":"https://www.wikidata.org/wiki/Q190909","display_name":"Focus (optics)","level":2,"score":0.6540654897689819},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.6279044151306152},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.5782833099365234},{"id":"https://openalex.org/C529865628","wikidata":"https://www.wikidata.org/wiki/Q1790740","display_name":"Manifold (fluid mechanics)","level":2,"score":0.45923882722854614},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.445825457572937},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.43965667486190796},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.4173164665699005},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3620189428329468},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3453007936477661},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.2815718650817871},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.18179979920387268},{"id":"https://openalex.org/C172435161","wikidata":"https://www.wikidata.org/wiki/Q531126","display_name":"Hopf bifurcation","level":4,"score":0.13885292410850525},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.12445369362831116},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.11431211233139038},{"id":"https://openalex.org/C2781349735","wikidata":"https://www.wikidata.org/wiki/Q4904824","display_name":"Bifurcation","level":3,"score":0.09847551584243774},{"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/C120665830","wikidata":"https://www.wikidata.org/wiki/Q14620","display_name":"Optics","level":1,"score":0.0},{"id":"https://openalex.org/C78519656","wikidata":"https://www.wikidata.org/wiki/Q101333","display_name":"Mechanical engineering","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/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1142/s0218127414500837","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127414500837","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":[{"id":"https://openalex.org/F4320334593","display_name":"Natural Sciences and Engineering Research Council of Canada","ror":"https://ror.org/01h531d29"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":17,"referenced_works":["https://openalex.org/W102382135","https://openalex.org/W625401248","https://openalex.org/W1978933280","https://openalex.org/W1998107461","https://openalex.org/W2004043483","https://openalex.org/W2009798055","https://openalex.org/W2029954120","https://openalex.org/W2031173069","https://openalex.org/W2049544062","https://openalex.org/W2053496265","https://openalex.org/W2081699720","https://openalex.org/W2129875234","https://openalex.org/W2135076964","https://openalex.org/W2140959905","https://openalex.org/W2372306478","https://openalex.org/W2963427501","https://openalex.org/W4248348571"],"related_works":["https://openalex.org/W2030527508","https://openalex.org/W2390921864","https://openalex.org/W2393436281","https://openalex.org/W840566636","https://openalex.org/W2366945349","https://openalex.org/W2065622633","https://openalex.org/W2097333826","https://openalex.org/W2051065969","https://openalex.org/W2077011938","https://openalex.org/W2380167091"],"abstract_inverted_index":{"In":[0,76],"this":[1,61],"paper,":[2],"we":[3],"show":[4],"that":[5],"a":[6,22],"simple":[7],"three-dimensional":[8,128],"quadratic":[9,36],"vector":[10,38,129],"field":[11],"can":[12,41],"have":[13,43],"at":[14],"least":[15],"seven":[16],"small-amplitude":[17,45],"limit":[18,46],"cycles,":[19],"bifurcating":[20,48],"from":[21,49],"Hopf":[23],"critical":[24],"point.":[25],"This":[26],"result":[27,34],"is":[28,119],"surprisingly":[29],"higher":[30],"than":[31],"the":[32,109,123,127],"Bautin's":[33],"for":[35,83,107],"planar":[37],"fields":[39],"which":[40],"only":[42],"three":[44],"cycles":[47],"an":[50,54],"elementary":[51,55],"focus":[52,65,81,124],"or":[53],"center.":[56],"The":[57],"methods":[58],"used":[59],"in":[60],"paper":[62],"include":[63],"computing":[64,108],"values,":[66],"and":[67,104,112],"solving":[68],"multivariate":[69],"polynomial":[70],"systems":[71,118],"using":[72],"modular":[73],"regular":[74],"chains.":[75],"order":[77],"to":[78,121],"obtain":[79],"higher-order":[80],"values":[82,125],"nonplanar":[84],"dynamical":[85],"systems,":[86],"computationally":[87],"efficient":[88],"approaches":[89],"combined":[90],"with":[91],"center":[92,113],"manifold":[93,114],"computation":[94],"must":[95],"be":[96],"adopted.":[97],"A":[98],"recently":[99],"developed":[100],"explicit,":[101],"recursive":[102],"formula":[103],"Maple":[105],"program":[106],"normal":[110],"form":[111],"of":[115,126],"general":[116],"n-dimensional":[117],"applied":[120],"compute":[122],"field.":[130]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":1},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":1},{"year":2014,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}