{"id":"https://openalex.org/W2000849354","doi":"https://doi.org/10.1142/s0218127409024037","title":"THE SEPARATRIX VALUES OF A PLANAR HOMOCLINIC LOOP","display_name":"THE SEPARATRIX VALUES OF A PLANAR HOMOCLINIC LOOP","publication_year":2009,"publication_date":"2009-07-01","ids":{"openalex":"https://openalex.org/W2000849354","doi":"https://doi.org/10.1142/s0218127409024037","mag":"2000849354"},"language":"en","primary_location":{"id":"doi:10.1142/s0218127409024037","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127409024037","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/A5101684735","display_name":"Liqin Zhao","orcid":"https://orcid.org/0009-0007-5008-2621"},"institutions":[{"id":"https://openalex.org/I25254941","display_name":"Beijing Normal University","ror":"https://ror.org/022k4wk35","country_code":"CN","type":"education","lineage":["https://openalex.org/I25254941"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"LIQIN ZHAO","raw_affiliation_strings":["School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, P. R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, P. R. China","institution_ids":["https://openalex.org/I25254941"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5081848844","display_name":"Xuexing Wang","orcid":null},"institutions":[{"id":"https://openalex.org/I25254941","display_name":"Beijing Normal University","ror":"https://ror.org/022k4wk35","country_code":"CN","type":"education","lineage":["https://openalex.org/I25254941"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"XUEXING WANG","raw_affiliation_strings":["School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, P. R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, P. R. China","institution_ids":["https://openalex.org/I25254941"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I25254941"],"apc_list":null,"apc_paid":null,"fwci":0.3788,"has_fulltext":false,"cited_by_count":2,"citation_normalized_percentile":{"value":0.55670861,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"19","issue":"07","first_page":"2233","last_page":"2247"},"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/homoclinic-orbit","display_name":"Homoclinic orbit","score":0.9667288064956665},{"id":"https://openalex.org/keywords/separatrix","display_name":"Separatrix","score":0.8591041564941406},{"id":"https://openalex.org/keywords/saddle","display_name":"Saddle","score":0.7900855541229248},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6308252811431885},{"id":"https://openalex.org/keywords/homoclinic-bifurcation","display_name":"Homoclinic bifurcation","score":0.6306424736976624},{"id":"https://openalex.org/keywords/loop","display_name":"Loop (graph theory)","score":0.6273523569107056},{"id":"https://openalex.org/keywords/planar","display_name":"Planar","score":0.5689542889595032},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.5087020993232727},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.4151771664619446},{"id":"https://openalex.org/keywords/bifurcation","display_name":"Bifurcation","score":0.3862702548503876},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.35143715143203735},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.2342088520526886},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.07839566469192505},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.06848195195198059},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.05461719632148743},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.0516258180141449}],"concepts":[{"id":"https://openalex.org/C200581526","wikidata":"https://www.wikidata.org/wiki/Q690494","display_name":"Homoclinic orbit","level":4,"score":0.9667288064956665},{"id":"https://openalex.org/C76379757","wikidata":"https://www.wikidata.org/wiki/Q2902561","display_name":"Separatrix","level":3,"score":0.8591041564941406},{"id":"https://openalex.org/C2777127463","wikidata":"https://www.wikidata.org/wiki/Q10862618","display_name":"Saddle","level":2,"score":0.7900855541229248},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6308252811431885},{"id":"https://openalex.org/C155405617","wikidata":"https://www.wikidata.org/wiki/Q1154357","display_name":"Homoclinic bifurcation","level":5,"score":0.6306424736976624},{"id":"https://openalex.org/C184670325","wikidata":"https://www.wikidata.org/wiki/Q512604","display_name":"Loop (graph theory)","level":2,"score":0.6273523569107056},{"id":"https://openalex.org/C134786449","wikidata":"https://www.wikidata.org/wiki/Q3391255","display_name":"Planar","level":2,"score":0.5689542889595032},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5087020993232727},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.4151771664619446},{"id":"https://openalex.org/C2781349735","wikidata":"https://www.wikidata.org/wiki/Q4904824","display_name":"Bifurcation","level":3,"score":0.3862702548503876},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.35143715143203735},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.2342088520526886},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.07839566469192505},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.06848195195198059},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.05461719632148743},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.0516258180141449},{"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/C82706917","wikidata":"https://www.wikidata.org/wiki/Q10251","display_name":"Plasma","level":2,"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}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1142/s0218127409024037","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127409024037","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":17,"referenced_works":["https://openalex.org/W1564263151","https://openalex.org/W1975136536","https://openalex.org/W2009734469","https://openalex.org/W2017348434","https://openalex.org/W2042498362","https://openalex.org/W2045256358","https://openalex.org/W2049815318","https://openalex.org/W2061809639","https://openalex.org/W2069228691","https://openalex.org/W2077189269","https://openalex.org/W2078522451","https://openalex.org/W2142664960","https://openalex.org/W2242179722","https://openalex.org/W2584645706","https://openalex.org/W3123094903","https://openalex.org/W3144514274","https://openalex.org/W6600234944"],"related_works":["https://openalex.org/W2890492950","https://openalex.org/W3115184012","https://openalex.org/W2010606039","https://openalex.org/W2397038510","https://openalex.org/W2000849354","https://openalex.org/W9644552","https://openalex.org/W1906329374","https://openalex.org/W1964759045","https://openalex.org/W2313034084","https://openalex.org/W2009600486"],"abstract_inverted_index":{"It":[0],"is":[1,15,95,224],"well":[2],"known":[3],"that":[4,213],"the":[5,19,35,41,65,68,74,78,87,90,99,120,188,191,205,214],"stability":[6,88],"of":[7,21,31,83,89,125,179,208,216],"a":[8,26,32,110],"homoclinic":[9,23,28,79,91,206,217,222],"loop":[10,29,36,44,218],"for":[11,106,190],"planar":[12,27,112],"vector":[13,113],"fields":[14],"closely":[16],"related":[17,60,72],"to":[18,40,61,73,97,118],"cyclicity":[20,215],"this":[22,183],"loop.":[24,80,92],"For":[25,109],"consisting":[30],"hyperbolic":[33],"saddle,":[34,66],"values":[37,45,52,58,70,101,122],"are":[38,46,59,71],"crucial":[39],"stability.":[42],"The":[43,56,81,140,148,166],"divided":[47],"into":[48],"two":[49],"classes:":[50],"saddle":[51,57,121],"and":[53,67,105,137,156,159,162,174,193,211],"separatrix":[54,69,100,142,150,168,195],"values.":[55,196],"Dulac":[62],"map":[63,76],"near":[64,77],"regular":[75],"alternation":[82],"these":[84],"quantities":[85],"determines":[86],"So,":[93],"it":[94],"important":[96],"investigate":[98],"in":[102],"both":[103],"theory":[104],"practical":[107],"applications.":[108],"given":[111,153],"field,":[114],"we":[115,185,199],"can":[116],"try":[117],"calculate":[119],"by":[123,130,135,146,154,160,172,177],"means":[124,178],"dual":[126],"Liapunov":[127],"constants":[128],"or":[129],"finding":[131],"elementary":[132],"invariants":[133],"developed":[134],"Liu":[136],"Li":[138,175],"[1990].":[139],"first":[141],"value":[143,151,169],"was":[144,152,170],"obtained":[145,171],"Dulac.":[147],"second":[149],"Han":[155],"Zhu":[157],"[2007]":[158],"Hu":[161],"Feng":[163],"[2001]":[164],"independently.":[165],"third":[167,192],"Luo":[173],"[2005]":[176],"Tkachev's":[180],"method.":[181],"In":[182],"paper,":[184],"shall":[186],"establish":[187],"formulae":[189],"fourth":[194],"As":[197],"applications,":[198],"will":[200],"give":[201],"an":[202],"example":[203],"with":[204,220],"bifurcation":[207],"order":[209],"9":[210],"prove":[212],"together":[219],"double":[221],"loops":[223],"57.":[225]},"counts_by_year":[{"year":2018,"cited_by_count":1},{"year":2012,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}
