{"id":"https://openalex.org/W2112177740","doi":"https://doi.org/10.1142/s0218127412501544","title":"GLOBAL DYNAMICS IN THE POINCAR\u00c9 BALL OF THE CHEN SYSTEM HAVING INVARIANT ALGEBRAIC SURFACES","display_name":"GLOBAL DYNAMICS IN THE POINCAR\u00c9 BALL OF THE CHEN SYSTEM HAVING INVARIANT ALGEBRAIC SURFACES","publication_year":2012,"publication_date":"2012-04-24","ids":{"openalex":"https://openalex.org/W2112177740","doi":"https://doi.org/10.1142/s0218127412501544","mag":"2112177740"},"language":"en","primary_location":{"id":"doi:10.1142/s0218127412501544","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127412501544","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":true,"oa_status":"green","oa_url":"https://ddd.uab.cat/record/150498","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, 08193 Bellaterra, Barcelona, Catalonia, Spain"],"affiliations":[{"raw_affiliation_string":"Departament de Matem\u00e0tiques, Universitat Aut\u00f2noma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain","institution_ids":["https://openalex.org/I123044942"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5059877036","display_name":"Marcelo Messias","orcid":"https://orcid.org/0000-0003-2269-7091"},"institutions":[{"id":"https://openalex.org/I879563668","display_name":"Universidade Estadual Paulista (Unesp)","ror":"https://ror.org/00987cb86","country_code":"BR","type":"education","lineage":["https://openalex.org/I879563668"]}],"countries":["BR"],"is_corresponding":false,"raw_author_name":"MARCELO MESSIAS","raw_affiliation_strings":["Departamento de Matem\u00e1tica, Estat\u00edstica e Computa\u00e7\u00e3o, Faculdade de Ci\u00eancias e Tecnologia, UNESP \u2013 Univ Estadual Paulista, Cx. Postal 266, 19060-900, Presidente Prudente, S\u00e3o Paulo, Brazil"],"affiliations":[{"raw_affiliation_string":"Departamento de Matem\u00e1tica, Estat\u00edstica e Computa\u00e7\u00e3o, Faculdade de Ci\u00eancias e Tecnologia, UNESP \u2013 Univ Estadual Paulista, Cx. Postal 266, 19060-900, Presidente Prudente, S\u00e3o Paulo, Brazil","institution_ids":["https://openalex.org/I879563668"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5001277261","display_name":"Paulo Ricardo da Silva","orcid":"https://orcid.org/0000-0002-1430-5986"},"institutions":[{"id":"https://openalex.org/I879563668","display_name":"Universidade Estadual Paulista (Unesp)","ror":"https://ror.org/00987cb86","country_code":"BR","type":"education","lineage":["https://openalex.org/I879563668"]}],"countries":["BR"],"is_corresponding":false,"raw_author_name":"PAULO RICARDO DA SILVA","raw_affiliation_strings":["Departamento de Matem\u00e1tica, Instituto de Bioci\u00eancias, Letras e Ci\u00eancias Exatas, UNESP \u2013 Univ Estadual Paulista, Rua C. Colombo, 2265, 15054-000, S. J. Rio Preto, S\u00e3o Paulo, Brazil"],"affiliations":[{"raw_affiliation_string":"Departamento de Matem\u00e1tica, Instituto de Bioci\u00eancias, Letras e Ci\u00eancias Exatas, UNESP \u2013 Univ Estadual Paulista, Rua C. Colombo, 2265, 15054-000, S. J. Rio Preto, S\u00e3o Paulo, Brazil","institution_ids":["https://openalex.org/I879563668"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5046015548"],"corresponding_institution_ids":["https://openalex.org/I123044942"],"apc_list":null,"apc_paid":null,"fwci":2.9465,"has_fulltext":true,"cited_by_count":31,"citation_normalized_percentile":{"value":0.90370104,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"22","issue":"06","first_page":"1250154","last_page":"1250154"},"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/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.9987999796867371,"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"}},{"id":"https://openalex.org/T11261","display_name":"Quantum chaos and dynamical systems","score":0.9983000159263611,"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/mathematics","display_name":"Mathematics","score":0.6252732276916504},{"id":"https://openalex.org/keywords/phase-portrait","display_name":"Phase portrait","score":0.6171109080314636},{"id":"https://openalex.org/keywords/chen","display_name":"Chen","score":0.5835941433906555},{"id":"https://openalex.org/keywords/gravitational-singularity","display_name":"Gravitational singularity","score":0.5754291415214539},{"id":"https://openalex.org/keywords/ball","display_name":"Ball (mathematics)","score":0.5676838159561157},{"id":"https://openalex.org/keywords/invariant","display_name":"Invariant (physics)","score":0.556348979473114},{"id":"https://openalex.org/keywords/algebraic-number","display_name":"Algebraic number","score":0.5383594632148743},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4834235906600952},{"id":"https://openalex.org/keywords/compactification","display_name":"Compactification (mathematics)","score":0.461603581905365},{"id":"https://openalex.org/keywords/poincar\u00e9-conjecture","display_name":"Poincar\u00e9 conjecture","score":0.4584142863750458},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.42035096883773804},{"id":"https://openalex.org/keywords/mathematical-physics","display_name":"Mathematical physics","score":0.2698401212692261},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.21564981341362},{"id":"https://openalex.org/keywords/bifurcation","display_name":"Bifurcation","score":0.16819331049919128}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6252732276916504},{"id":"https://openalex.org/C122283620","wikidata":"https://www.wikidata.org/wiki/Q2532943","display_name":"Phase portrait","level":4,"score":0.6171109080314636},{"id":"https://openalex.org/C2776085556","wikidata":"https://www.wikidata.org/wiki/Q183361","display_name":"Chen","level":2,"score":0.5835941433906555},{"id":"https://openalex.org/C12843","wikidata":"https://www.wikidata.org/wiki/Q201721","display_name":"Gravitational singularity","level":2,"score":0.5754291415214539},{"id":"https://openalex.org/C122041747","wikidata":"https://www.wikidata.org/wiki/Q838611","display_name":"Ball (mathematics)","level":2,"score":0.5676838159561157},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.556348979473114},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.5383594632148743},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4834235906600952},{"id":"https://openalex.org/C2779165322","wikidata":"https://www.wikidata.org/wiki/Q546145","display_name":"Compactification (mathematics)","level":2,"score":0.461603581905365},{"id":"https://openalex.org/C88221313","wikidata":"https://www.wikidata.org/wiki/Q203586","display_name":"Poincar\u00e9 conjecture","level":2,"score":0.4584142863750458},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.42035096883773804},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.2698401212692261},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.21564981341362},{"id":"https://openalex.org/C2781349735","wikidata":"https://www.wikidata.org/wiki/Q4904824","display_name":"Bifurcation","level":3,"score":0.16819331049919128},{"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/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","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/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.1142/s0218127412501544","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127412501544","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"},{"id":"pmh:oai:ddd.uab.cat:150498","is_oa":true,"landing_page_url":"https://ddd.uab.cat/record/150498","pdf_url":"https://ddd.uab.cat/record/150498","source":{"id":"https://openalex.org/S4306400381","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"},{"id":"pmh:oai:acervodigital.unesp.br:11449/22166","is_oa":false,"landing_page_url":"http://acervodigital.unesp.br/handle/11449/22166","pdf_url":null,"source":{"id":"https://openalex.org/S4306400256","display_name":"Acervo Digital da Universidade Estadual Paulista (Universidade Estadual Paulista)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210115194","host_organization_name":"Funda\u00e7\u00e3o para o Desenvolvimento da UNESP","host_organization_lineage":["https://openalex.org/I4210115194"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"outro"},{"id":"pmh:oai:dnet:uabarcelona_::5a818b31eacbea3e856c24b2a240f6ba","is_oa":false,"landing_page_url":"http://ddd.uab.cat/record/150498","pdf_url":null,"source":{"id":"https://openalex.org/S4306402641","display_name":"LA Referencia (Red Federada de Repositorios Institucionales de Publicaciones Cient\u00edficas)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4383465926","host_organization_name":"LA Referencia","host_organization_lineage":["https://openalex.org/I4383465926"],"host_organization_lineage_names":[],"type":"repository"},"license":"public-domain","license_id":"https://openalex.org/licenses/public-domain","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"info:eu-repo/semantics/article"},{"id":"pmh:oai:repositorio.unesp.br:11449/22166","is_oa":false,"landing_page_url":"http://hdl.handle.net/11449/22166","pdf_url":null,"source":{"id":"https://openalex.org/S4377196277","display_name":"UNESP Institutional Repository (S\u00e3o Paulo State University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I879563668","host_organization_name":"Universidade Estadual Paulista (Unesp)","host_organization_lineage":["https://openalex.org/I879563668"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":{"id":"pmh:oai:ddd.uab.cat:150498","is_oa":true,"landing_page_url":"https://ddd.uab.cat/record/150498","pdf_url":"https://ddd.uab.cat/record/150498","source":{"id":"https://openalex.org/S4306400381","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/G1224945033","display_name":null,"funder_award_id":"FEDER","funder_id":"https://openalex.org/F4320322930","funder_display_name":"Ministerio de Ciencia e Innovaci\u00f3n"},{"id":"https://openalex.org/G1809062499","display_name":null,"funder_award_id":"MTM2008-03437","funder_id":"https://openalex.org/F4320322930","funder_display_name":"Ministerio de Ciencia e Innovaci\u00f3n"},{"id":"https://openalex.org/G196208344","display_name":null,"funder_award_id":"2009SGR","funder_id":"https://openalex.org/F4320321505","funder_display_name":"Generalitat de Catalunya"},{"id":"https://openalex.org/G3601516360","display_name":null,"funder_award_id":"Brazil","funder_id":"https://openalex.org/F4320322025","funder_display_name":"Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico"},{"id":"https://openalex.org/G3913404609","display_name":null,"funder_award_id":"CNPq-Brazil","funder_id":"https://openalex.org/F4320322025","funder_display_name":"Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico"},{"id":"https://openalex.org/G4089352303","display_name":null,"funder_award_id":"MTM2008-03437","funder_id":"https://openalex.org/F4320321505","funder_display_name":"Generalitat de Catalunya"},{"id":"https://openalex.org/G4952876371","display_name":null,"funder_award_id":"General","funder_id":"https://openalex.org/F4320322025","funder_display_name":"Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico"},{"id":"https://openalex.org/G5079005330","display_name":null,"funder_award_id":"support","funder_id":"https://openalex.org/F4320322025","funder_display_name":"Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico"},{"id":"https://openalex.org/G5301220158","display_name":null,"funder_award_id":"MTM2008-03437","funder_id":"https://openalex.org/F4320322121","funder_display_name":"Instituci\u00f3 Catalana de Recerca i Estudis Avan\u00e7ats"}],"funders":[{"id":"https://openalex.org/F4320321505","display_name":"Generalitat de Catalunya","ror":"https://ror.org/01bg62x04"},{"id":"https://openalex.org/F4320322025","display_name":"Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico","ror":"https://ror.org/03swz6y49"},{"id":"https://openalex.org/F4320322121","display_name":"Instituci\u00f3 Catalana de Recerca i Estudis Avan\u00e7ats","ror":"https://ror.org/0371hy230"},{"id":"https://openalex.org/F4320322930","display_name":"Ministerio de Ciencia e Innovaci\u00f3n","ror":"https://ror.org/034900433"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2112177740.pdf","grobid_xml":"https://content.openalex.org/works/W2112177740.grobid-xml"},"referenced_works_count":14,"referenced_works":["https://openalex.org/W1965257932","https://openalex.org/W1971650444","https://openalex.org/W2014769138","https://openalex.org/W2025846951","https://openalex.org/W2035706753","https://openalex.org/W2043266210","https://openalex.org/W2050556108","https://openalex.org/W2052311828","https://openalex.org/W2069436097","https://openalex.org/W2094255955","https://openalex.org/W2104604364","https://openalex.org/W2110602566","https://openalex.org/W2141394518","https://openalex.org/W2142587090"],"related_works":["https://openalex.org/W2339458745","https://openalex.org/W2370310922","https://openalex.org/W2386975637","https://openalex.org/W3021650107","https://openalex.org/W2009968757","https://openalex.org/W4287803004","https://openalex.org/W2891436901","https://openalex.org/W2958194412","https://openalex.org/W2905738647","https://openalex.org/W1998894406"],"abstract_inverted_index":{"In":[0,59],"this":[1],"paper,":[2],"we":[3,62,121],"perform":[4],"a":[5,74,126,142,147],"global":[6,65],"analysis":[7],"of":[8,11,38,49,68,77,83,100,108,125,144,156,165],"the":[9,12,35,42,50,53,64,69,78,87,98,104,112,123,138,152,157,163],"dynamics":[10,40],"Chen":[13,70,139,158],"system":[14,54,71,140,159],"[Formula:":[15],"see":[16],"text]":[17],"where":[18],"(x,":[19],"y,":[20],"z)":[21],"\u2208":[22,29],"\u211d":[23,30,101],"3":[24,31,102],"and":[25,72,80,146],"(a,":[26],"b,":[27],"c)":[28],".":[32],"We":[33],"give":[34,73],"complete":[36,75],"description":[37,76],"its":[39,84,91],"on":[41,134],"sphere":[43,105],"at":[44],"infinity.":[45,109],"For":[46],"six":[47],"sets":[48,82],"parameter":[51],"values,":[52],"has":[55,141],"invariant":[56,135],"algebraic":[57],"surfaces.":[58],"these":[60],"cases,":[61],"provide":[63],"phase":[66],"portrait":[67],"\u03b1-":[79],"\u03c9-limit":[81],"orbits":[85,132],"in":[86,97,162],"Poincar\u00e9":[88],"ball,":[89],"including":[90],"boundary":[92],"\ud835\udd4a":[93,106],"2":[94,107],",":[95],"i.e.":[96],"compactification":[99],"with":[103,116,128],"Moreover,":[110],"combining":[111],"analytical":[113],"results":[114],"obtained":[115],"an":[117],"accurate":[118],"numerical":[119],"analysis,":[120],"prove":[122],"existence":[124],"family":[127],"infinitely":[129],"many":[130],"heteroclinic":[131],"contained":[133],"cylinders":[136],"when":[137],"line":[143],"singularities":[145],"first":[148],"integral,":[149],"which":[150],"indicates":[151],"complicated":[153],"dynamical":[154],"behavior":[155],"solutions":[160],"even":[161],"absence":[164],"chaotic":[166],"dynamics.":[167]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":2},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":4},{"year":2019,"cited_by_count":3},{"year":2018,"cited_by_count":5},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":3},{"year":2015,"cited_by_count":3},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":3}],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}
