{"id":"https://openalex.org/W3093623229","doi":"https://doi.org/10.1137/20m1340083","title":"On the Number of Limit Cycles in Generalized Abel Equations","display_name":"On the Number of Limit Cycles in Generalized Abel Equations","publication_year":2020,"publication_date":"2020-01-01","ids":{"openalex":"https://openalex.org/W3093623229","doi":"https://doi.org/10.1137/20m1340083","mag":"3093623229"},"language":"en","primary_location":{"id":"doi:10.1137/20m1340083","is_oa":false,"landing_page_url":"https://doi.org/10.1137/20m1340083","pdf_url":null,"source":{"id":"https://openalex.org/S161160597","display_name":"SIAM Journal on Applied Dynamical Systems","issn_l":"1536-0040","issn":["1536-0040"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Applied Dynamical Systems","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://ddd.uab.cat/record/236658","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5101861593","display_name":"Jianfeng Huang","orcid":"https://orcid.org/0000-0001-7790-4698"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Jianfeng Huang","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5085767356","display_name":"Joan Torregrosa","orcid":"https://orcid.org/0000-0002-2753-1827"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Joan Torregrosa","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5060001114","display_name":"Jordi Villadelprat","orcid":"https://orcid.org/0000-0002-1168-9750"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Jordi Villadelprat","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5101861593"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":2.3343,"has_fulltext":true,"cited_by_count":11,"citation_normalized_percentile":{"value":0.88542609,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"19","issue":"4","first_page":"2343","last_page":"2370"},"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":1.0,"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":1.0,"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":0.9532999992370605,"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/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.9510999917984009,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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":0.6575227379798889},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.6330000162124634},{"id":"https://openalex.org/keywords/limit","display_name":"Limit (mathematics)","score":0.6177691221237183},{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.59317946434021},{"id":"https://openalex.org/keywords/context","display_name":"Context (archaeology)","score":0.573120653629303},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5331249833106995},{"id":"https://openalex.org/keywords/trigonometry","display_name":"Trigonometry","score":0.5228029489517212},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.5062769651412964},{"id":"https://openalex.org/keywords/li\u00e9nard-equation","display_name":"Li\u00e9nard equation","score":0.41924065351486206},{"id":"https://openalex.org/keywords/mathematical-physics","display_name":"Mathematical physics","score":0.39051467180252075},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3261435925960541},{"id":"https://openalex.org/keywords/linear-differential-equation","display_name":"Linear differential equation","score":0.2326962947845459},{"id":"https://openalex.org/keywords/exact-differential-equation","display_name":"Exact differential equation","score":0.06233978271484375}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6575227379798889},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.6330000162124634},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.6177691221237183},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.59317946434021},{"id":"https://openalex.org/C2779343474","wikidata":"https://www.wikidata.org/wiki/Q3109175","display_name":"Context (archaeology)","level":2,"score":0.573120653629303},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5331249833106995},{"id":"https://openalex.org/C29001434","wikidata":"https://www.wikidata.org/wiki/Q8084","display_name":"Trigonometry","level":2,"score":0.5228029489517212},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.5062769651412964},{"id":"https://openalex.org/C181154966","wikidata":"https://www.wikidata.org/wiki/Q2500406","display_name":"Li\u00e9nard equation","level":5,"score":0.41924065351486206},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.39051467180252075},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3261435925960541},{"id":"https://openalex.org/C13982400","wikidata":"https://www.wikidata.org/wiki/Q1129902","display_name":"Linear differential equation","level":3,"score":0.2326962947845459},{"id":"https://openalex.org/C185184677","wikidata":"https://www.wikidata.org/wiki/Q1366795","display_name":"Exact differential equation","level":4,"score":0.06233978271484375},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C151730666","wikidata":"https://www.wikidata.org/wiki/Q7205","display_name":"Paleontology","level":1,"score":0.0},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","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/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/20m1340083","is_oa":false,"landing_page_url":"https://doi.org/10.1137/20m1340083","pdf_url":null,"source":{"id":"https://openalex.org/S161160597","display_name":"SIAM Journal on Applied Dynamical Systems","issn_l":"1536-0040","issn":["1536-0040"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Applied Dynamical Systems","raw_type":"journal-article"},{"id":"pmh:oai:ddd.uab.cat:236658","is_oa":true,"landing_page_url":"https://ddd.uab.cat/record/236658","pdf_url":"https://ddd.uab.cat/record/236658","source":{"id":"https://openalex.org/S4306400380","display_name":"Dip\u00f2sit Digital de Documents de la UAB (Universitat Aut\u00f2noma de Barcelona)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I123044942","host_organization_name":"Universitat Aut\u00f2noma de Barcelona","host_organization_lineage":["https://openalex.org/I123044942"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Article"}],"best_oa_location":{"id":"pmh:oai:ddd.uab.cat:236658","is_oa":true,"landing_page_url":"https://ddd.uab.cat/record/236658","pdf_url":"https://ddd.uab.cat/record/236658","source":{"id":"https://openalex.org/S4306400380","display_name":"Dip\u00f2sit Digital de Documents de la UAB (Universitat Aut\u00f2noma de Barcelona)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I123044942","host_organization_name":"Universitat Aut\u00f2noma de Barcelona","host_organization_lineage":["https://openalex.org/I123044942"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1234577443","display_name":null,"funder_award_id":"MTM2016-77278-P","funder_id":"https://openalex.org/F4320315062","funder_display_name":"Ministerio de Ciencia, Innovaci\u00f3n y Universidades"},{"id":"https://openalex.org/G1423109669","display_name":null,"funder_award_id":"MSCA-RISE-2017-777911","funder_id":"https://openalex.org/F4320335598","funder_display_name":"Agencia Estatal de Investigaci\u00f3n"},{"id":"https://openalex.org/G1426318481","display_name":null,"funder_award_id":"grant","funder_id":"https://openalex.org/F4320322725","funder_display_name":"China Scholarship Council"},{"id":"https://openalex.org/G1545329600","display_name":null,"funder_award_id":"MTM2016-77278-P","funder_id":"https://openalex.org/F4320335598","funder_display_name":"Agencia Estatal de Investigaci\u00f3n"},{"id":"https://openalex.org/G2063457095","display_name":null,"funder_award_id":"11771101","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G2087396116","display_name":null,"funder_award_id":"China","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G2436355136","display_name":null,"funder_award_id":"MSCA-RISE-2017","funder_id":"https://openalex.org/F4320332999","funder_display_name":"Horizon 2020 Framework Programme"},{"id":"https://openalex.org/G3046504837","display_name":null,"funder_award_id":"MSCARISE-2017-777911","funder_id":"https://openalex.org/F4320332999","funder_display_name":"Horizon 2020 Framework Programme"},{"id":"https://openalex.org/G3216677882","display_name":null,"funder_award_id":"201606","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G3243401500","display_name":null,"funder_award_id":"MSCA-RISE-2017-777911","funder_id":"https://openalex.org/F4320334830","funder_display_name":"Ag\u00e8ncia de Gesti\u00f3 d'Ajuts Universitaris i de Recerca"},{"id":"https://openalex.org/G3345895242","display_name":null,"funder_award_id":"H2020-MSCA","funder_id":"https://openalex.org/F4320332999","funder_display_name":"Horizon 2020 Framework Programme"},{"id":"https://openalex.org/G3366058276","display_name":null,"funder_award_id":"2017SGR1617","funder_id":"https://openalex.org/F4320335598","funder_display_name":"Agencia Estatal de Investigaci\u00f3n"},{"id":"https://openalex.org/G3477743816","display_name":null,"funder_award_id":"H2020-MSCA-RISE","funder_id":"https://openalex.org/F4320332999","funder_display_name":"Horizon 2020 Framework Programme"},{"id":"https://openalex.org/G3682693491","display_name":null,"funder_award_id":"MTM2017-86795-C3-2-P","funder_id":"https://openalex.org/F4320315062","funder_display_name":"Ministerio de Ciencia, Innovaci\u00f3n y Universidades"},{"id":"https://openalex.org/G3886670469","display_name":null,"funder_award_id":"2017SGR161","funder_id":"https://openalex.org/F4320334830","funder_display_name":"Ag\u00e8ncia de Gesti\u00f3 d'Ajuts Universitaris i de Recerca"},{"id":"https://openalex.org/G4132985236","display_name":null,"funder_award_id":"unknown","funder_id":"https://openalex.org/F4320322725","funder_display_name":"China Scholarship Council"},{"id":"https://openalex.org/G4224773232","display_name":null,"funder_award_id":"2017SGR1617","funder_id":"https://openalex.org/F4320334830","funder_display_name":"Ag\u00e8ncia de Gesti\u00f3 d'Ajuts Universitaris i de Recerca"},{"id":"https://openalex.org/G4790777438","display_name":null,"funder_award_id":"201606785007","funder_id":"https://openalex.org/F4320322725","funder_display_name":"China Scholarship Council"},{"id":"https://openalex.org/G4816244515","display_name":null,"funder_award_id":"FEDER","funder_id":"https://openalex.org/F4320335598","funder_display_name":"Agencia Estatal de Investigaci\u00f3n"},{"id":"https://openalex.org/G526235227","display_name":null,"funder_award_id":"777911","funder_id":"https://openalex.org/F4320332999","funder_display_name":"Horizon 2020 Framework Programme"},{"id":"https://openalex.org/G5554424168","display_name":null,"funder_award_id":"2017SGR","funder_id":"https://openalex.org/F4320334830","funder_display_name":"Ag\u00e8ncia de Gesti\u00f3 d'Ajuts Universitaris i de Recerca"},{"id":"https://openalex.org/G5859239756","display_name":null,"funder_award_id":"MSCA-RISE","funder_id":"https://openalex.org/F4320332999","funder_display_name":"Horizon 2020 Framework Programme"},{"id":"https://openalex.org/G6059518138","display_name":null,"funder_award_id":"MSCA-RISE-2017-777911","funder_id":"https://openalex.org/F4320332999","funder_display_name":"Horizon 2020 Framework Programme"},{"id":"https://openalex.org/G683931746","display_name":null,"funder_award_id":"H2020","funder_id":"https://openalex.org/F4320335598","funder_display_name":"Agencia Estatal de Investigaci\u00f3n"},{"id":"https://openalex.org/G7015168732","display_name":null,"funder_award_id":"H2020-MSCA-RISE-2017","funder_id":"https://openalex.org/F4320332999","funder_display_name":"Horizon 2020 Framework Programme"},{"id":"https://openalex.org/G7517184171","display_name":null,"funder_award_id":"H2020-MSCA-RISE-201","funder_id":"https://openalex.org/F4320332999","funder_display_name":"Horizon 2020 Framework Programme"},{"id":"https://openalex.org/G8246274189","display_name":null,"funder_award_id":"MTM2016-77278-P","funder_id":"https://openalex.org/F4320334830","funder_display_name":"Ag\u00e8ncia de Gesti\u00f3 d'Ajuts Universitaris i de Recerca"},{"id":"https://openalex.org/G8589651859","display_name":null,"funder_award_id":"China","funder_id":"https://openalex.org/F4320322725","funder_display_name":"China Scholarship Council"},{"id":"https://openalex.org/G8646112941","display_name":null,"funder_award_id":"H2020-MSCA-RISE-2017-777911","funder_id":"https://openalex.org/F4320332999","funder_display_name":"Horizon 2020 Framework Programme"},{"id":"https://openalex.org/G990333805","display_name":null,"funder_award_id":"Innovaci\u00f3n y Universidades","funder_id":"https://openalex.org/F4320335598","funder_display_name":"Agencia Estatal de Investigaci\u00f3n"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320315062","display_name":"Ministerio de Ciencia, Innovaci\u00f3n y Universidades","ror":null},{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320322725","display_name":"China Scholarship Council","ror":"https://ror.org/04atp4p48"},{"id":"https://openalex.org/F4320332999","display_name":"Horizon 2020 Framework Programme","ror":"https://ror.org/00k4n6c32"},{"id":"https://openalex.org/F4320334830","display_name":"Ag\u00e8ncia de Gesti\u00f3 d'Ajuts Universitaris i de Recerca","ror":"https://ror.org/01n4pqe45"},{"id":"https://openalex.org/F4320335598","display_name":"Agencia Estatal de Investigaci\u00f3n","ror":null}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3093623229.pdf","grobid_xml":"https://content.openalex.org/works/W3093623229.grobid-xml"},"referenced_works_count":24,"referenced_works":["https://openalex.org/W1984301218","https://openalex.org/W1991638364","https://openalex.org/W1991679586","https://openalex.org/W1992538858","https://openalex.org/W2010995477","https://openalex.org/W2025644638","https://openalex.org/W2034478771","https://openalex.org/W2037024263","https://openalex.org/W2040736253","https://openalex.org/W2040844160","https://openalex.org/W2044413922","https://openalex.org/W2046383562","https://openalex.org/W2047361099","https://openalex.org/W2056860023","https://openalex.org/W2069582844","https://openalex.org/W2125797455","https://openalex.org/W2317553223","https://openalex.org/W2328818716","https://openalex.org/W2543855378","https://openalex.org/W2607171337","https://openalex.org/W2733850088","https://openalex.org/W2963798734","https://openalex.org/W2976651471","https://openalex.org/W4238937393"],"related_works":["https://openalex.org/W2314814023","https://openalex.org/W332723364","https://openalex.org/W1761788542","https://openalex.org/W2953174714","https://openalex.org/W401950388","https://openalex.org/W1973069571","https://openalex.org/W2048701188","https://openalex.org/W3204216370","https://openalex.org/W4254482063","https://openalex.org/W2204769580"],"abstract_inverted_index":{"Given":[0],"$p,q\\in\\mathbb{Z}_{\\geq":[1],"2}$":[2],"with":[3],"$p\\neq":[4],"q$,":[5],"we":[6,27,68,98,132],"study":[7],"generalized":[8],"Abel":[9,125],"differential":[10,76,126],"equations":[11,127],"$\\frac{dx}{d\\theta}=A(\\theta)x^p+B(\\theta)x^q,$":[12],"where":[13],"$A$":[14],"and":[15,26,64,66,130],"$B$":[16],"are":[17,28,111],"trigonometric":[18],"polynomials":[19],"of":[20,33,57,90,102,108],"degrees":[21],"$n,":[22],"m\\ge":[23],"1,$":[24],"respectively,":[25],"interested":[29],"in":[30,46,118],"the":[31,55,74,85,106,114,119],"number":[32],"limit":[34,82],"cycles":[35],"(i.e.,":[36,128],"isolated":[37],"periodic":[38],"orbits)":[39],"that":[40,67,73,134],"they":[41],"can":[42],"have.":[43],"More":[44],"concretely,":[45],"this":[47],"context,":[48],"an":[49,58],"open":[50],"problem":[51],"is":[52],"to":[53,105],"prove":[54,133],"existence":[56],"integer,":[59],"depending":[60],"only":[61],"on":[62],"$p,q,m$,":[63],"$n$":[65],"denote":[69],"by":[70,88],"$\\mathcal{H}_{p,q}(n,m)$,":[71],"such":[72],"above":[75],"equation":[77],"has":[78],"at":[79],"most":[80],"$\\mathcal{H}_{p,q}(n,m)$":[81,103],"cycles.":[83],"In":[84,121],"present":[86],"paper,":[87],"means":[89],"a":[91],"second":[92],"order":[93],"analysis":[94],"using":[95],"Melnikov":[96],"functions,":[97],"provide":[99],"lower":[100],"bounds":[101],"that,":[104],"best":[107],"our":[109],"knowledge,":[110],"larger":[112],"than":[113],"previous":[115],"ones":[116],"appearing":[117],"literature.":[120],"particular,":[122],"for":[123],"classical":[124],"$p=3$":[129],"$q=2$),":[131],"$\\mathcal{H}_{3,2}(n,m)\\geq":[135],"2(n+m)-1.$":[136]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2024,"cited_by_count":4},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1}],"updated_date":"2026-03-18T14:38:29.013473","created_date":"2025-10-10T00:00:00"}