{"id":"https://openalex.org/W7204948533","doi":"https://doi.org/10.48550/arxiv.2608.29150","title":"Exact Rayleigh Reduction for Direct Detection of Turning Bifurcations","display_name":"Exact Rayleigh Reduction for Direct Detection of Turning Bifurcations","publication_year":2026,"publication_date":"2026-08-29","ids":{"openalex":"https://openalex.org/W7204948533","doi":"https://doi.org/10.48550/arxiv.2608.29150"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2608.29150","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.29150","pdf_url":"https://arxiv.org/pdf/2608.29150","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2608.29150","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5064689944","display_name":"Yavdat Shavkatovich Il'yasov","orcid":"https://orcid.org/0000-0002-6310-9164","observed_orcids":["https://orcid.org/0000-0002-6310-9164"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Il'yasov, Yavdat","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5064689944"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"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/T10194","display_name":"Nonlinear Partial Differential Equations","score":0.8827999830245972,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/T10194","display_name":"Nonlinear Partial Differential Equations","score":0.8827999830245972,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/T12396","display_name":"Advanced Differential Equations and Dynamical Systems","score":0.05139999836683273,"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/T10541","display_name":"Nonlinear Differential Equations Analysis","score":0.040300000458955765,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/kirchhoff-equations","display_name":"Kirchhoff equations","score":0.7300000190734863},{"id":"https://openalex.org/keywords/nonlinear-elliptic-equations","display_name":"nonlinear elliptic equations","score":0.6349999904632568},{"id":"https://openalex.org/keywords/saddle-node-bifurcation","display_name":"saddle-node bifurcation","score":0.621999979019165},{"id":"https://openalex.org/keywords/turning-points","display_name":"turning points","score":0.492000013589859},{"id":"https://openalex.org/keywords/parameter-dependent-equations","display_name":"parameter-dependent equations","score":0.45500001311302185},{"id":"https://openalex.org/keywords/amplitude-scaling","display_name":"amplitude scaling","score":0.49000000953674316}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7563999891281128},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.6292999982833862},{"id":"https://openalex.org/C2781349735","wikidata":"https://www.wikidata.org/wiki/Q4904824","display_name":"Bifurcation","level":3,"score":0.5841000080108643},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.5741999745368958},{"id":"https://openalex.org/C138187205","wikidata":"https://www.wikidata.org/wiki/Q131251","display_name":"Tangent","level":2,"score":0.5268999934196472},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.4244999885559082},{"id":"https://openalex.org/C46244369","wikidata":"https://www.wikidata.org/wiki/Q193709","display_name":"Rayleigh scattering","level":2,"score":0.4147999882698059},{"id":"https://openalex.org/C99844830","wikidata":"https://www.wikidata.org/wiki/Q102441924","display_name":"Scaling","level":2,"score":0.34459999203681946},{"id":"https://openalex.org/C85075877","wikidata":"https://www.wikidata.org/wiki/Q1154357","display_name":"Bifurcation theory","level":4,"score":0.3327000141143799},{"id":"https://openalex.org/C101225446","wikidata":"https://www.wikidata.org/wiki/Q740190","display_name":"Saddle-node bifurcation","level":4,"score":0.32580000162124634},{"id":"https://openalex.org/C57691317","wikidata":"https://www.wikidata.org/wiki/Q1289248","display_name":"Scalar (mathematics)","level":2,"score":0.3086000084877014},{"id":"https://openalex.org/C194271172","wikidata":"https://www.wikidata.org/wiki/Q2181373","display_name":"Period-doubling bifurcation","level":4,"score":0.2842999994754791},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.2825999855995178},{"id":"https://openalex.org/C180205008","wikidata":"https://www.wikidata.org/wiki/Q159190","display_name":"Amplitude","level":2,"score":0.28189998865127563},{"id":"https://openalex.org/C111335779","wikidata":"https://www.wikidata.org/wiki/Q3454686","display_name":"Reduction (mathematics)","level":2,"score":0.2808000147342682},{"id":"https://openalex.org/C155405617","wikidata":"https://www.wikidata.org/wiki/Q1154357","display_name":"Homoclinic bifurcation","level":5,"score":0.26820001006126404},{"id":"https://openalex.org/C72169020","wikidata":"https://www.wikidata.org/wiki/Q194404","display_name":"Monotonic function","level":2,"score":0.26010000705718994},{"id":"https://openalex.org/C63548660","wikidata":"https://www.wikidata.org/wiki/Q858223","display_name":"Limit point","level":2,"score":0.2529999911785126},{"id":"https://openalex.org/C520416788","wikidata":"https://www.wikidata.org/wiki/Q5419229","display_name":"Exact solutions in general relativity","level":2,"score":0.25200000405311584},{"id":"https://openalex.org/C110729354","wikidata":"https://www.wikidata.org/wiki/Q185478","display_name":"Countable set","level":2,"score":0.250900000333786}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2608.29150","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.29150","pdf_url":"https://arxiv.org/pdf/2608.29150","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2608.29150","doi":"https://doi.org/10.48550/arxiv.2608.29150","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2608.29150","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2608.29150","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.29150","pdf_url":"https://arxiv.org/pdf/2608.29150","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7204948533.pdf","grobid_xml":"https://content.openalex.org/works/W7204948533.grobid-xml"},"referenced_works_count":14,"referenced_works":["https://openalex.org/W1554132433","https://openalex.org/W1972594862","https://openalex.org/W1977212986","https://openalex.org/W2064296987","https://openalex.org/W2092339127","https://openalex.org/W2171998161","https://openalex.org/W2401259934","https://openalex.org/W2490028710","https://openalex.org/W2732743443","https://openalex.org/W2953975239","https://openalex.org/W2963281268","https://openalex.org/W3125048870","https://openalex.org/W4220965354","https://openalex.org/W7161745336"],"related_works":["https://openalex.org/W2002115361","https://openalex.org/W3204113203","https://openalex.org/W2100150950","https://openalex.org/W3165674835","https://openalex.org/W4220965354","https://openalex.org/W7161745336","https://openalex.org/W2068336910","https://openalex.org/W3176094723","https://openalex.org/W2060834047","https://openalex.org/W2094586029"],"abstract_inverted_index":{"We":[0,32],"introduce":[1],"an":[2,144],"exact":[3,48,117],"Rayleigh":[4,21,71],"reduction":[5],"for":[6],"the":[7,47,82,87,94,129,134,140,162],"direct":[8],"construction":[9],"and":[10,85,91,109,122],"detection":[11],"of":[12,54,68,81,125,161],"turning":[13,126,146],"bifurcations":[14],"in":[15],"nonlinear":[16],"parameter-dependent":[17],"equations.":[18],"A":[19],"generalized":[20,130],"functional":[22],"ordinarily":[23],"provides":[24],"only":[25],"a":[26,36,55,78,98,154],"scalar":[27],"necessary":[28],"constraint":[29],"on":[30,106,112],"solutions.":[31],"show":[33],"that,":[34],"along":[35],"suitable":[37],"one-parameter":[38],"transformation":[39],"orbit":[40],"$u=\\mathcal":[41],"G_s\u03d5$,":[42],"it":[43],"can":[44],"instead":[45],"become":[46],"parameter":[49,120],"law":[50,132],"$\u03bb=\\mathcal":[51],"R(\\mathcal":[52],"G_s\u03d5)$":[53],"genuine":[56],"solution":[57],"branch.":[58],"Turning":[59],"points":[60,67],"are":[61],"then":[62],"found":[63],"from":[64],"nondegenerate":[65],"critical":[66],"this":[69],"one-dimensional":[70],"profile.":[72],"The":[73],"branch":[74,135],"tangent":[75],"automatically":[76],"yields":[77],"kernel":[79],"direction":[80],"state":[83],"linearization,":[84],"under":[86],"usual":[88],"Fredholm,":[89],"kernel-simplicity,":[90],"parameter-transversality":[92],"assumptions":[93],"detected":[95],"point":[96],"is":[97,137,159],"simple":[99],"fold.":[100],"For":[101,128],"Kirchhoff":[102,131],"equations,":[103],"amplitude":[104],"scaling":[105],"bounded":[107],"domains":[108],"spatial":[110],"dilation":[111],"$\\mathbb":[113],"R^N$":[114],"yield":[115],"explicit":[116],"branches,":[118],"global":[119],"thresholds,":[121],"countable":[123],"hierarchies":[124],"values.":[127],"$M_b(A)=a+bA^\u03b8$,":[133],"geometry":[136],"governed":[138],"by":[139],"dimension--homogeneity":[141],"index":[142],"$\u03b8(N-2)-2$:":[143],"interior":[145],"bifurcation":[147,156],"occurs":[148],"exactly":[149],"when":[150],"$\u03b8(N-2)>2$.":[151],"This":[152],"gives":[153],"universal":[155],"trichotomy":[157],"that":[158],"independent":[160],"particular":[163],"Berestycki--Lions":[164],"nonlinearity.":[165]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-02T00:00:00"}