{"id":"https://openalex.org/W7225449692","doi":"https://doi.org/10.5281/zenodo.24044646","title":"Exact Global Bound HL(5) = 2 for the Classical Li\u00e9nard Equation of Degree 5 via Hilbert-Fourier Variety Analysis","display_name":"Exact Global Bound HL(5) = 2 for the Classical Li\u00e9nard Equation of Degree 5 via Hilbert-Fourier Variety Analysis","publication_year":2026,"publication_date":"2026-11-07","ids":{"openalex":"https://openalex.org/W7225449692","doi":"https://doi.org/10.5281/zenodo.24044646"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.24044646","doi":"https://doi.org/10.5281/zenodo.24044646","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24044646","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.5281/zenodo.24044646","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5047320676","display_name":"Manuel Hern\u00e1ndez Rosales","orcid":"https://orcid.org/0000-0002-2905-7444","observed_orcids":["https://orcid.org/0000-0002-2905-7444"]},"institutions":[{"id":"https://openalex.org/I8961855","display_name":"Universidad Nacional Aut\u00f3noma de M\u00e9xico","ror":"https://ror.org/01tmp8f25","country_code":"MX","type":"education","lineage":["https://openalex.org/I8961855"]}],"countries":["MX"],"is_corresponding":true,"raw_author_name":"Hernandez Rosales, Manuel","raw_affiliation_strings":["Universidad Nacional Aut\u00f3noma de M\u00e9xico"],"raw_orcid":"https://orcid.org/0000-0002-2905-7444","affiliations":[{"raw_affiliation_string":"Universidad Nacional Aut\u00f3noma de M\u00e9xico","institution_ids":["https://openalex.org/I8961855"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5047320676"],"corresponding_institution_ids":["https://openalex.org/I8961855"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"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":0.9994999766349792,"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":0.9994999766349792,"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":9.999999747378752e-05,"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/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.0,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/lienard-equation","display_name":"Li\u00e9nard equation","score":0.8579999804496765},{"id":"https://openalex.org/keywords/limit-cycles","display_name":"limit cycles","score":0.9160000085830688},{"id":"https://openalex.org/keywords/hilbert-s-16th-problem","display_name":"Hilbert's 16th problem","score":0.8600000143051147},{"id":"https://openalex.org/keywords/planar-differential-systems","display_name":"planar differential systems","score":0.7279999852180481},{"id":"https://openalex.org/keywords/amplitude-correction","display_name":"amplitude correction","score":0.5120000243186951},{"id":"https://openalex.org/keywords/frequency-shift","display_name":"frequency shift","score":0.6840000152587891},{"id":"https://openalex.org/keywords/saddle-node-bifurcation","display_name":"saddle-node bifurcation","score":0.9039999842643738},{"id":"https://openalex.org/keywords/weak-focus","display_name":"weak focus","score":0.9089999794960022}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7767000198364258},{"id":"https://openalex.org/C99692599","wikidata":"https://www.wikidata.org/wiki/Q596577","display_name":"Dissipative system","level":2,"score":0.5805000066757202},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.5601999759674072},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5501000285148621},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.4765999913215637},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.4740999937057495},{"id":"https://openalex.org/C93357160","wikidata":"https://www.wikidata.org/wiki/Q1545916","display_name":"Limit cycle","level":3,"score":0.43619999289512634},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.4300999939441681},{"id":"https://openalex.org/C135402231","wikidata":"https://www.wikidata.org/wiki/Q898440","display_name":"Dissipation","level":2,"score":0.39570000767707825},{"id":"https://openalex.org/C41608201","wikidata":"https://www.wikidata.org/wiki/Q980509","display_name":"Embedding","level":2,"score":0.39259999990463257},{"id":"https://openalex.org/C134786449","wikidata":"https://www.wikidata.org/wiki/Q3391255","display_name":"Planar","level":2,"score":0.3375999927520752},{"id":"https://openalex.org/C192209626","wikidata":"https://www.wikidata.org/wiki/Q190909","display_name":"Focus (optics)","level":2,"score":0.336899995803833},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.33219999074935913},{"id":"https://openalex.org/C165150564","wikidata":"https://www.wikidata.org/wiki/Q1154357","display_name":"Infinite-period bifurcation","level":5,"score":0.3149999976158142},{"id":"https://openalex.org/C2781349735","wikidata":"https://www.wikidata.org/wiki/Q4904824","display_name":"Bifurcation","level":3,"score":0.31459999084472656},{"id":"https://openalex.org/C96403706","wikidata":"https://www.wikidata.org/wiki/Q543956","display_name":"Monodromy","level":2,"score":0.3109000027179718},{"id":"https://openalex.org/C180205008","wikidata":"https://www.wikidata.org/wiki/Q159190","display_name":"Amplitude","level":2,"score":0.31029999256134033},{"id":"https://openalex.org/C200331156","wikidata":"https://www.wikidata.org/wiki/Q506041","display_name":"Jacobian matrix and determinant","level":2,"score":0.31029999256134033},{"id":"https://openalex.org/C186370098","wikidata":"https://www.wikidata.org/wiki/Q442787","display_name":"Energy (signal processing)","level":2,"score":0.3082999885082245},{"id":"https://openalex.org/C25878781","wikidata":"https://www.wikidata.org/wiki/Q1473607","display_name":"Degree of a polynomial","level":3,"score":0.30379998683929443},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.29420000314712524},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.29010000824928284},{"id":"https://openalex.org/C62799726","wikidata":"https://www.wikidata.org/wiki/Q190056","display_name":"Hilbert space","level":2,"score":0.2896000146865845},{"id":"https://openalex.org/C17103678","wikidata":"https://www.wikidata.org/wiki/Q1474074","display_name":"Homogeneous polynomial","level":4,"score":0.27160000801086426},{"id":"https://openalex.org/C92047909","wikidata":"https://www.wikidata.org/wiki/Q204034","display_name":"Hyperbolic function","level":2,"score":0.2685999870300293},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.26840001344680786},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.265500009059906},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.25859999656677246},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.25540000200271606},{"id":"https://openalex.org/C172435161","wikidata":"https://www.wikidata.org/wiki/Q531126","display_name":"Hopf bifurcation","level":4,"score":0.25360000133514404},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.2515999972820282}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.5281/zenodo.24044646","doi":"https://doi.org/10.5281/zenodo.24044646","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24044646","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},{"id":"doi:10.5281/zenodo.24044647","doi":"https://doi.org/10.5281/zenodo.24044647","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24044647","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.5281/zenodo.24044646","doi":"https://doi.org/10.5281/zenodo.24044646","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24044646","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7203911190","https://openalex.org/W4404696004","https://openalex.org/W7225010707","https://openalex.org/W3095403251","https://openalex.org/W2000286527","https://openalex.org/W2093230829","https://openalex.org/W4301481016","https://openalex.org/W2056447802","https://openalex.org/W2318145412","https://openalex.org/W7171547593"],"abstract_inverted_index":{"We":[0,74],"establish":[1],"an":[2,42],"exact":[3,89],"and":[4,58,68],"rigorous":[5],"global":[6],"proof":[7],"that":[8,76,119,152],"the":[9,15,39,46,66,94,120,123,146,153,166,183,194,197],"maximum":[10],"number":[11],"of":[12,21,122,135,196],"limit":[13,136],"cyclesfor":[14],"classical":[16],"planar":[17],"Li\u00e9nard":[18],"differential":[19],"system":[20],"degree":[22,201],"5,":[23],"x_":[24],"=":[25,30,35,107,127,156,170,188],"y":[26],"\udbc0\udc00":[27,186],"F(x),":[28],"y_":[29],"\udbc0\udc00x,":[31],"isstrictly":[32],"two:":[33],"HL(5)":[34],"2.":[36],"By":[37],"embedding":[38],"dynamics":[40],"into":[41],"infinite-dimensional":[43],"algebraicvariety":[44],"in":[45,103,172],"Hilbert":[47],"space":[48],"`2":[49],"under":[50],"a":[51,100,131,161,178],"natural":[52],"parity-scaling":[53],"hierarchy\u2014assigning":[54],"to":[55,60,88,99,177],"evendissipative":[56],"terms":[57],"2":[59],"odd":[61],"dissipative":[62],"terms\u2014we":[63],"completely":[64],"decouple":[65],"amplitudecorrections":[67],"frequency":[69],"shifts":[70],"order":[71,92],"by":[72,160,182],"order.":[73],"prove":[75],"all":[77],"odd-order":[78],"corrections":[79],"tothe":[80],"fundamental":[81,95],"amplitude":[82],"vanish":[83],"identically":[84],"(X1;2m+1":[85],"0)":[86],"due":[87],"temporal":[90],"parity.At":[91],"2,":[93],"energy":[96,105],"balance":[97],"collapses":[98],"quadratic":[101],"polynomial":[102,125],"thebasal":[104],"u":[106],"X21;0,":[108],"which":[109,173],"admits":[110],"at":[111],"most":[112],"two":[113,139],"positive":[114],"real":[115],"roots.":[116],"Furthermore,":[117],"wedemonstrate":[118],"vanishing":[121],"Jacobian":[124],"Q(X1;0)":[126],"0":[128,157],"corresponds":[129],"strictlyto":[130],"saddle-node":[132],"(fold)":[133],"bifurcation":[134],"cycles,":[137],"where":[138],"hyperbolic":[140,162],"orbits":[141],"coalesce":[142],"andannihilate":[143],"without":[144],"increasing":[145],"cycle":[147],"capacity.":[148],"Finally,":[149],"we":[150],"show":[151],"complementarybranch":[154],"X1;0":[155],"is":[158],"blocked":[159],"focus":[163,180],"barrier":[164],"unless":[165],"linear":[167],"dissipation":[168],"vanishes(a1":[169],"0),":[171],"case":[174],"it":[175],"reduces":[176],"weak":[179],"governed":[181],"Blows\u2013Lloyd":[184],"theorem(b(5":[185],"1)=2c":[187],"2).":[189],"These":[190],"results":[191],"definitively":[192],"confirm":[193],"validity":[195],"Lins\u2013de":[198],"Melo\u2013Pughconjecture":[199],"for":[200],"5.":[202]},"counts_by_year":[],"updated_date":"2026-11-09T13:22:03.289127","created_date":"2026-11-08T00:00:00"}
