{"id":"https://openalex.org/W3199351936","doi":"https://doi.org/10.3390/sym13091736","title":"Integrability and Limit Cycles via First Integrals","display_name":"Integrability and Limit Cycles via First Integrals","publication_year":2021,"publication_date":"2021-09-18","ids":{"openalex":"https://openalex.org/W3199351936","doi":"https://doi.org/10.3390/sym13091736","mag":"3199351936"},"language":"en","primary_location":{"id":"doi:10.3390/sym13091736","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym13091736","pdf_url":"https://www.mdpi.com/2073-8994/13/9/1736/pdf?version=1632374762","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/13/9/1736/pdf?version=1632374762","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5046015548","display_name":"Jaume Llibre","orcid":"https://orcid.org/0000-0002-9511-5999"},"institutions":[{"id":"https://openalex.org/I123044942","display_name":"Universitat Aut\u00f2noma de Barcelona","ror":"https://ror.org/052g8jq94","country_code":"ES","type":"education","lineage":["https://openalex.org/I123044942"]}],"countries":["ES"],"is_corresponding":true,"raw_author_name":"Jaume Llibre","raw_affiliation_strings":["Departament de Matem\u00e0tiques, Universitat Aut\u00f2noma de Barcelona, Bellaterra, 08193 Barcelona, Spain"],"raw_orcid":"https://orcid.org/0000-0002-9511-5999","affiliations":[{"raw_affiliation_string":"Departament de Matem\u00e0tiques, Universitat Aut\u00f2noma de Barcelona, Bellaterra, 08193 Barcelona, Spain","institution_ids":["https://openalex.org/I123044942"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5046015548"],"corresponding_institution_ids":["https://openalex.org/I123044942"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":4.2203,"has_fulltext":false,"cited_by_count":10,"citation_normalized_percentile":{"value":0.94894309,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":98},"biblio":{"volume":"13","issue":"9","first_page":"1736","last_page":"1736"},"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/T11844","display_name":"Lipid metabolism and biosynthesis","score":0.9556000232696533,"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":0.6732906699180603},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.601068377494812},{"id":"https://openalex.org/keywords/phase-portrait","display_name":"Phase portrait","score":0.5810709595680237},{"id":"https://openalex.org/keywords/integrating-factor","display_name":"Integrating factor","score":0.5039018988609314},{"id":"https://openalex.org/keywords/limit","display_name":"Limit (mathematics)","score":0.49837255477905273},{"id":"https://openalex.org/keywords/piecewise","display_name":"Piecewise","score":0.4902413487434387},{"id":"https://openalex.org/keywords/phase-space","display_name":"Phase space","score":0.47546786069869995},{"id":"https://openalex.org/keywords/differential","display_name":"Differential (mechanical device)","score":0.47383078932762146},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.45428845286369324},{"id":"https://openalex.org/keywords/ordinary-differential-equation","display_name":"Ordinary differential equation","score":0.4478135406970978},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.43682169914245605},{"id":"https://openalex.org/keywords/manifold","display_name":"Manifold (fluid mechanics)","score":0.41572558879852295},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3879760801792145},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.36377909779548645},{"id":"https://openalex.org/keywords/differential-algebraic-equation","display_name":"Differential algebraic equation","score":0.1607668399810791},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.1310696303844452},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.07933351397514343}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6732906699180603},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.601068377494812},{"id":"https://openalex.org/C122283620","wikidata":"https://www.wikidata.org/wiki/Q2532943","display_name":"Phase portrait","level":4,"score":0.5810709595680237},{"id":"https://openalex.org/C163105828","wikidata":"https://www.wikidata.org/wiki/Q2734005","display_name":"Integrating factor","level":5,"score":0.5039018988609314},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.49837255477905273},{"id":"https://openalex.org/C164660894","wikidata":"https://www.wikidata.org/wiki/Q2037833","display_name":"Piecewise","level":2,"score":0.4902413487434387},{"id":"https://openalex.org/C151342819","wikidata":"https://www.wikidata.org/wiki/Q62542","display_name":"Phase space","level":2,"score":0.47546786069869995},{"id":"https://openalex.org/C93226319","wikidata":"https://www.wikidata.org/wiki/Q193137","display_name":"Differential (mechanical device)","level":2,"score":0.47383078932762146},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.45428845286369324},{"id":"https://openalex.org/C51544822","wikidata":"https://www.wikidata.org/wiki/Q465274","display_name":"Ordinary differential equation","level":3,"score":0.4478135406970978},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.43682169914245605},{"id":"https://openalex.org/C529865628","wikidata":"https://www.wikidata.org/wiki/Q1790740","display_name":"Manifold (fluid mechanics)","level":2,"score":0.41572558879852295},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3879760801792145},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.36377909779548645},{"id":"https://openalex.org/C186219872","wikidata":"https://www.wikidata.org/wiki/Q955889","display_name":"Differential algebraic equation","level":4,"score":0.1607668399810791},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.1310696303844452},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.07933351397514343},{"id":"https://openalex.org/C2781349735","wikidata":"https://www.wikidata.org/wiki/Q4904824","display_name":"Bifurcation","level":3,"score":0.0},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"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/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.3390/sym13091736","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym13091736","pdf_url":"https://www.mdpi.com/2073-8994/13/9/1736/pdf?version=1632374762","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:ddd.uab.cat:252022","is_oa":true,"landing_page_url":"https://ddd.uab.cat/record/252022","pdf_url":null,"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Article"},{"id":"pmh:oai:doaj.org/article:512b23b3e13147f3817ff3efa0437eaf","is_oa":true,"landing_page_url":"https://doaj.org/article/512b23b3e13147f3817ff3efa0437eaf","pdf_url":null,"source":{"id":"https://openalex.org/S112646816","display_name":"SHILAP Revista de lepidopterolog\u00eda","issn_l":"0300-5267","issn":["0300-5267","2340-4078"],"is_oa":true,"is_in_doaj":true,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 13, Iss 9, p 1736 (2021)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/13/9/1736/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym13091736","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/sym13091736","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym13091736","pdf_url":"https://www.mdpi.com/2073-8994/13/9/1736/pdf?version=1632374762","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G2133003115","display_name":null,"funder_award_id":"MSCA-RISE-2017-777911","funder_id":"https://openalex.org/F4320338335","funder_display_name":"H2020 European Research 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/G618704488","display_name":null,"funder_award_id":"PID2019-104658GB-I00","funder_id":"https://openalex.org/F4320335598","funder_display_name":"Agencia Estatal de Investigaci\u00f3n"}],"funders":[{"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},{"id":"https://openalex.org/F4320338335","display_name":"H2020 European Research Council","ror":"https://ror.org/0472cxd90"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3199351936.pdf","grobid_xml":"https://content.openalex.org/works/W3199351936.grobid-xml"},"referenced_works_count":30,"referenced_works":["https://openalex.org/W191386329","https://openalex.org/W1023286937","https://openalex.org/W1552094772","https://openalex.org/W1558079603","https://openalex.org/W1602130484","https://openalex.org/W1972241390","https://openalex.org/W1974272814","https://openalex.org/W1980228211","https://openalex.org/W2002282594","https://openalex.org/W2008393500","https://openalex.org/W2019076112","https://openalex.org/W2042192623","https://openalex.org/W2050710621","https://openalex.org/W2053658423","https://openalex.org/W2054652000","https://openalex.org/W2073072471","https://openalex.org/W2079569562","https://openalex.org/W2088842637","https://openalex.org/W2092046534","https://openalex.org/W2559924465","https://openalex.org/W2731769483","https://openalex.org/W2796609112","https://openalex.org/W2922368149","https://openalex.org/W2946518633","https://openalex.org/W3020980268","https://openalex.org/W3148988080","https://openalex.org/W4200499478","https://openalex.org/W4292170110","https://openalex.org/W4293078920","https://openalex.org/W6626537741"],"related_works":["https://openalex.org/W3161245192","https://openalex.org/W2964050831","https://openalex.org/W1626306628","https://openalex.org/W2769798300","https://openalex.org/W1660454402","https://openalex.org/W2945527131","https://openalex.org/W2375115016","https://openalex.org/W2377673787","https://openalex.org/W2077226182","https://openalex.org/W1590822228"],"abstract_inverted_index":{"In":[0],"many":[1,71,107],"problems":[2],"appearing":[3],"in":[4,7,14,69,157],"applied":[5],"mathematics":[6],"the":[8,45,64,67,87,111,135,148],"nonlinear":[9],"ordinary":[10],"differential":[11,23,51,93,108,132,159],"systems,":[12],"as":[13,70,73],"physics,":[15],"chemist,":[16],"economics,":[17],"etc.,":[18],"if":[19,98],"we":[20,77,123,143],"have":[21],"a":[22,26,34,56,92,101],"system":[24],"on":[25,53],"manifold":[27],"of":[28,31,47,55,66,106,113,139,150],"dimension,":[29],"two":[30,60,119],"them":[32],"having":[33],"first":[35,48,75,82,102,114,128,151],"integral,":[36],"then":[37],"its":[38],"phase":[39],"portrait":[40],"is":[41,84],"completely":[42],"determined.":[43],"While":[44],"existence":[46,112,149],"integrals":[49,76,83,129,152],"for":[50,130,153],"systems":[52,109,133],"manifolds":[54],"dimension":[57,65],"higher":[58],"than":[59],"allows":[61],"to":[62,80,96,126,146],"reduce":[63],"space":[68],"dimensions":[72],"independent":[74],"have.":[78],"Hence,":[79],"know":[81,97],"important,":[85],"but":[86],"following":[88],"question":[89],"appears:":[90],"Given":[91],"system,":[94],"how":[95,125,145],"it":[99],"has":[100,118],"integral?":[103],"The":[104],"symmetries":[105],"force":[110],"integrals.":[115],"This":[116],"paper":[117],"main":[120],"objectives.":[121],"First,":[122],"study":[124],"compute":[127],"polynomial":[131],"using":[134],"so-called":[136],"Darboux":[137],"theory":[138],"integrability.":[140],"Furthermore,":[141],"second,":[142],"show":[144],"use":[147],"finding":[154],"limit":[155],"cycles":[156],"piecewise":[158],"systems.":[160]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":5},{"year":2022,"cited_by_count":4}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}