{"id":"https://openalex.org/W1986274266","doi":"https://doi.org/10.1142/s0218127403007618","title":"Center-Focus Problem for Discontinuous Planar Differential Equations","display_name":"Center-Focus Problem for Discontinuous Planar Differential Equations","publication_year":2003,"publication_date":"2003-07-01","ids":{"openalex":"https://openalex.org/W1986274266","doi":"https://doi.org/10.1142/s0218127403007618","mag":"1986274266"},"language":"en","primary_location":{"id":"doi:10.1142/s0218127403007618","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127403007618","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/A5078018430","display_name":"Armengol Gasull","orcid":"https://orcid.org/0000-0002-1719-8231"},"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":false,"raw_author_name":"Armengol Gasull","raw_affiliation_strings":["Departament de Matem\u00e0tiques, Universitat Aut\u00f2noma de Barcelona, Edifici C, 08193 Bellaterra, Barcelona, Spain"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Departament de Matem\u00e0tiques, Universitat Aut\u00f2noma de Barcelona, Edifici C, 08193 Bellaterra, Barcelona, Spain","institution_ids":["https://openalex.org/I123044942"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5085767356","display_name":"Joan Torregrosa","orcid":"https://orcid.org/0000-0002-2753-1827"},"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":false,"raw_author_name":"Joan Torregrosa","raw_affiliation_strings":["Departament de Matem\u00e0tiques, Universitat Aut\u00f2noma de Barcelona, Edifici C, 08193 Bellaterra, Barcelona, Spain"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Departament de Matem\u00e0tiques, Universitat Aut\u00f2noma de Barcelona, Edifici C, 08193 Bellaterra, Barcelona, Spain","institution_ids":["https://openalex.org/I123044942"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I123044942"],"apc_list":null,"apc_paid":null,"fwci":0.431,"has_fulltext":false,"cited_by_count":93,"citation_normalized_percentile":{"value":0.57065933,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":94,"max":99},"biblio":{"volume":"13","issue":"07","first_page":"1755","last_page":"1765"},"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":0.9510999917984009,"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/focus","display_name":"Focus (optics)","score":0.7503461241722107},{"id":"https://openalex.org/keywords/ordinary-differential-equation","display_name":"Ordinary differential equation","score":0.6398736238479614},{"id":"https://openalex.org/keywords/ball","display_name":"Ball (mathematics)","score":0.6385405659675598},{"id":"https://openalex.org/keywords/planar","display_name":"Planar","score":0.6143625974655151},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6087482571601868},{"id":"https://openalex.org/keywords/computation","display_name":"Computation","score":0.5544122457504272},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.49596652388572693},{"id":"https://openalex.org/keywords/center","display_name":"Center (category theory)","score":0.49007976055145264},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.45957374572753906},{"id":"https://openalex.org/keywords/polar-coordinate-system","display_name":"Polar coordinate system","score":0.4356911778450012},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.43187642097473145},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.28999456763267517},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.27357324957847595},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.12275075912475586},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.11068761348724365}],"concepts":[{"id":"https://openalex.org/C192209626","wikidata":"https://www.wikidata.org/wiki/Q190909","display_name":"Focus (optics)","level":2,"score":0.7503461241722107},{"id":"https://openalex.org/C51544822","wikidata":"https://www.wikidata.org/wiki/Q465274","display_name":"Ordinary differential equation","level":3,"score":0.6398736238479614},{"id":"https://openalex.org/C122041747","wikidata":"https://www.wikidata.org/wiki/Q838611","display_name":"Ball (mathematics)","level":2,"score":0.6385405659675598},{"id":"https://openalex.org/C134786449","wikidata":"https://www.wikidata.org/wiki/Q3391255","display_name":"Planar","level":2,"score":0.6143625974655151},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6087482571601868},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.5544122457504272},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.49596652388572693},{"id":"https://openalex.org/C2779463800","wikidata":"https://www.wikidata.org/wiki/Q5062222","display_name":"Center (category theory)","level":2,"score":0.49007976055145264},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.45957374572753906},{"id":"https://openalex.org/C40069579","wikidata":"https://www.wikidata.org/wiki/Q62494","display_name":"Polar coordinate system","level":2,"score":0.4356911778450012},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.43187642097473145},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.28999456763267517},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.27357324957847595},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.12275075912475586},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.11068761348724365},{"id":"https://openalex.org/C8010536","wikidata":"https://www.wikidata.org/wiki/Q160398","display_name":"Crystallography","level":1,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"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/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/s0218127403007618","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127403007618","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":[{"score":0.44999998807907104,"id":"https://metadata.un.org/sdg/11","display_name":"Sustainable cities and communities"}],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":16,"referenced_works":["https://openalex.org/W625401248","https://openalex.org/W1539776296","https://openalex.org/W1562727027","https://openalex.org/W1593811304","https://openalex.org/W1983611771","https://openalex.org/W1985051324","https://openalex.org/W2014529799","https://openalex.org/W2042245054","https://openalex.org/W2058897776","https://openalex.org/W2063880270","https://openalex.org/W2072652000","https://openalex.org/W2143052212","https://openalex.org/W2160492379","https://openalex.org/W4232368046","https://openalex.org/W4245729428","https://openalex.org/W4293078920"],"related_works":["https://openalex.org/W4206327962","https://openalex.org/W2010009304","https://openalex.org/W1996851061","https://openalex.org/W3021694725","https://openalex.org/W2025195393","https://openalex.org/W2940695648","https://openalex.org/W1973684381","https://openalex.org/W4251403274","https://openalex.org/W2096304864","https://openalex.org/W2043671984"],"abstract_inverted_index":{"We":[0],"study":[1,78],"the":[2,8,31,35,54,57,66,70,86],"center-focus":[3],"problem":[4],"as":[5,7],"well":[6],"number":[9],"of":[10,22,30,49,56,82,88],"limit":[11],"cycles":[12],"which":[13],"bifurcate":[14],"from":[15],"a":[16,41,46,79,83,89],"weak":[17],"focus":[18],"for":[19],"several":[20],"families":[21],"planar":[23],"discontinuous":[24],"ordinary":[25],"differential":[26],"equations.":[27],"Our":[28],"computations":[29],"return":[32],"map":[33],"near":[34],"critical":[36],"point":[37],"are":[38],"performed":[39],"with":[40,53],"new":[42],"method":[43],"based":[44],"on":[45],"suitable":[47],"decomposition":[48,63],"certain":[50],"one-forms":[51],"associated":[52],"expression":[55],"system":[58],"in":[59,69],"polar":[60],"coordinates.":[61],"This":[62],"simplifies":[64],"all":[65],"expressions":[67],"involved":[68],"procedure.":[71],"Finally,":[72],"we":[73],"apply":[74],"our":[75],"results":[76],"to":[77],"mathematical":[80],"model":[81],"mechanical":[84],"problem,":[85],"movement":[87],"ball":[90],"between":[91],"two":[92],"elastic":[93],"walls.":[94]},"counts_by_year":[{"year":2026,"cited_by_count":3},{"year":2025,"cited_by_count":6},{"year":2024,"cited_by_count":8},{"year":2023,"cited_by_count":6},{"year":2022,"cited_by_count":11},{"year":2021,"cited_by_count":4},{"year":2020,"cited_by_count":4},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":4},{"year":2017,"cited_by_count":6},{"year":2016,"cited_by_count":7},{"year":2015,"cited_by_count":6},{"year":2014,"cited_by_count":3},{"year":2013,"cited_by_count":4},{"year":2012,"cited_by_count":6}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}