{"id":"https://openalex.org/W2062771516","doi":"https://doi.org/10.1142/s0218127405014039","title":"FRONTIER DIAGRAMS: PARTITION OF PHASE SPACE ACCORDING TO THE SIGNS OF EIGENVALUES OR SIGN PATTERNS OF THE CIRCUITS","display_name":"FRONTIER DIAGRAMS: PARTITION OF PHASE SPACE ACCORDING TO THE SIGNS OF EIGENVALUES OR SIGN PATTERNS OF THE CIRCUITS","publication_year":2005,"publication_date":"2005-10-01","ids":{"openalex":"https://openalex.org/W2062771516","doi":"https://doi.org/10.1142/s0218127405014039","mag":"2062771516"},"language":"en","primary_location":{"id":"doi:10.1142/s0218127405014039","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127405014039","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/A5030463072","display_name":"Ren\u00e9 Thomas","orcid":null},"institutions":[{"id":"https://openalex.org/I132053463","display_name":"Universit\u00e9 Libre de Bruxelles","ror":"https://ror.org/01r9htc13","country_code":"BE","type":"education","lineage":["https://openalex.org/I132053463"]}],"countries":["BE"],"is_corresponding":false,"raw_author_name":"REN\u00c9 THOMAS","raw_affiliation_strings":["Universit\u00e9 Libre de Bruxelles, Centre for Nonlinear Phenomena and Complex Systems CP 231, Boulevard du Triomphe, 1050 Brussels, Belgium"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Universit\u00e9 Libre de Bruxelles, Centre for Nonlinear Phenomena and Complex Systems CP 231, Boulevard du Triomphe, 1050 Brussels, Belgium","institution_ids":["https://openalex.org/I132053463"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5108699282","display_name":"Marcelle Kaufman","orcid":null},"institutions":[{"id":"https://openalex.org/I132053463","display_name":"Universit\u00e9 Libre de Bruxelles","ror":"https://ror.org/01r9htc13","country_code":"BE","type":"education","lineage":["https://openalex.org/I132053463"]}],"countries":["BE"],"is_corresponding":false,"raw_author_name":"MARCELLE KAUFMAN","raw_affiliation_strings":["Universit\u00e9 Libre de Bruxelles, Centre for Nonlinear Phenomena and Complex Systems CP 231, Boulevard du Triomphe, 1050 Brussels, Belgium"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Universit\u00e9 Libre de Bruxelles, Centre for Nonlinear Phenomena and Complex Systems CP 231, Boulevard du Triomphe, 1050 Brussels, Belgium","institution_ids":["https://openalex.org/I132053463"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I132053463"],"apc_list":null,"apc_paid":null,"fwci":1.0813,"has_fulltext":false,"cited_by_count":11,"citation_normalized_percentile":{"value":0.81665475,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"15","issue":"10","first_page":"3051","last_page":"3074"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11187","display_name":"Nonlinear Dynamics and Pattern Formation","score":0.9927999973297119,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11187","display_name":"Nonlinear Dynamics and Pattern Formation","score":0.9927999973297119,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10244","display_name":"Chaos control and synchronization","score":0.9894000291824341,"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"}},{"id":"https://openalex.org/T12813","display_name":"Plant and Biological Electrophysiology Studies","score":0.9746000170707703,"subfield":{"id":"https://openalex.org/subfields/1110","display_name":"Plant Science"},"field":{"id":"https://openalex.org/fields/11","display_name":"Agricultural and Biological Sciences"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/jacobian-matrix-and-determinant","display_name":"Jacobian matrix and determinant","score":0.7996501922607422},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.6979986429214478},{"id":"https://openalex.org/keywords/phase-space","display_name":"Phase space","score":0.5741783976554871},{"id":"https://openalex.org/keywords/sign","display_name":"Sign (mathematics)","score":0.5318287014961243},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.5174513459205627},{"id":"https://openalex.org/keywords/electronic-circuit","display_name":"Electronic circuit","score":0.5072178840637207},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5061618685722351},{"id":"https://openalex.org/keywords/curse-of-dimensionality","display_name":"Curse of dimensionality","score":0.4687768518924713},{"id":"https://openalex.org/keywords/partition","display_name":"Partition (number theory)","score":0.4399188756942749},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.33435553312301636},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.24282649159431458},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.19699299335479736},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.14723554253578186},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.1117885410785675}],"concepts":[{"id":"https://openalex.org/C200331156","wikidata":"https://www.wikidata.org/wiki/Q506041","display_name":"Jacobian matrix and determinant","level":2,"score":0.7996501922607422},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.6979986429214478},{"id":"https://openalex.org/C151342819","wikidata":"https://www.wikidata.org/wiki/Q62542","display_name":"Phase space","level":2,"score":0.5741783976554871},{"id":"https://openalex.org/C139676723","wikidata":"https://www.wikidata.org/wiki/Q1193832","display_name":"Sign (mathematics)","level":2,"score":0.5318287014961243},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.5174513459205627},{"id":"https://openalex.org/C134146338","wikidata":"https://www.wikidata.org/wiki/Q1815901","display_name":"Electronic circuit","level":2,"score":0.5072178840637207},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5061618685722351},{"id":"https://openalex.org/C111030470","wikidata":"https://www.wikidata.org/wiki/Q1430460","display_name":"Curse of dimensionality","level":2,"score":0.4687768518924713},{"id":"https://openalex.org/C42812","wikidata":"https://www.wikidata.org/wiki/Q1082910","display_name":"Partition (number theory)","level":2,"score":0.4399188756942749},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.33435553312301636},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.24282649159431458},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.19699299335479736},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.14723554253578186},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.1117885410785675},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1142/s0218127405014039","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127405014039","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":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":17,"referenced_works":["https://openalex.org/W344332445","https://openalex.org/W1965360919","https://openalex.org/W1974011730","https://openalex.org/W1977358249","https://openalex.org/W1979230873","https://openalex.org/W1994859578","https://openalex.org/W2027510758","https://openalex.org/W2033925647","https://openalex.org/W2043346919","https://openalex.org/W2044419029","https://openalex.org/W2047575710","https://openalex.org/W2069103848","https://openalex.org/W2080931300","https://openalex.org/W2084734494","https://openalex.org/W2094114128","https://openalex.org/W2164348551","https://openalex.org/W2168364602"],"related_works":["https://openalex.org/W2007405763","https://openalex.org/W1486373823","https://openalex.org/W3136087161","https://openalex.org/W2053762185","https://openalex.org/W2145980196","https://openalex.org/W2018238589","https://openalex.org/W1984283682","https://openalex.org/W2235865024","https://openalex.org/W2125326499","https://openalex.org/W2951193830"],"abstract_inverted_index":{"Nonlinear":[0],"dynamic":[1],"systems":[2,130,406,437],"can":[3,22,90,109,281,516],"be":[4,23,110,276,282,401,517],"considered":[5],"in":[6,58,63,72,79,115,119,124,155,184,206,218,278,291,295,357,438,497,519,530,542],"terms":[7,175,543],"of":[8,31,42,50,82,105,131,149,153,162,174,176,209,212,243,247,261,270,288,303,307,323,335,337,343,366,369,378,381,388,414,436,453,484,487,508,510,544],"feedback":[9],"circuits":[10,39],"(for":[11],"short,":[12],"circuits),":[13],"which":[14,439],"are":[15,37,76,226,253,459,495,499,528,540],"circular":[16],"interactions":[17],"between":[18],"variables.":[19],"Each":[20],"circuit":[21],"identified":[24,283],"without":[25],"ambiguity":[26],"from":[27],"the":[28,32,48,51,59,65,80,102,106,150,159,168,172,177,182,213,241,244,259,262,267,285,304,321,332,338,344,367,379,382,392,417,454,478,485,488,502,506,535,545],"Jacobian":[29,151,178,339,393],"matrix":[30,152,179],"system.":[33,52],"Of":[34],"special":[35],"relevance":[36],"those":[38,492],"(or":[40,246],"unions":[41],"disjoint":[43],"circuits)":[44],"that":[45,235,274,292,348,355,477,494],"involve":[46],"all":[47],"variables":[49,493,509,527],"We":[53,346,431],"call":[54],"them":[55],"\"nuclei\"":[56],"because,":[57],"same":[60],"way":[61],"as,":[62],"Biology,":[64],"cell":[66],"nucleus":[67,87],"contains":[68],"essential":[69,474],"genetic":[70],"information,":[71],"nonlinear":[73,166],"dynamics,":[74],"nuclei":[75,114],"crucial":[77],"elements":[78,336],"genesis":[81],"steady":[83,95,210,272,308,429,441],"states.":[84,309],"Indeed,":[85,360],"each":[86,423],"taken":[88],"alone":[89],"generate":[91],"one":[92,297,428],"or":[93,446],"more":[94,316,341,352,524],"states,":[96],"whose":[97],"nature":[98,269],"is":[99,143,328,351,473],"determined":[100],"by":[101,384,451,463],"sign":[103,333,389],"patterns":[104,334,390],"nucleus.":[107,140],"There":[108],"up":[111],"to":[112,157,240,258,331,375,400,475],"two":[113,440,444,447,520,526],"2D":[116],"systems,":[117,121,154,167,415],"six":[118],"3D":[120],"\u2026":[122],"n!":[123],"nD":[125],"systems.":[126],"However,":[127],"many":[128],"interesting":[129],"high":[132],"dimensionality":[133,486],"have":[134],"only":[135,205,491],"two,":[136],"sometimes":[137],"even":[138],"one,":[139],"This":[141,395,467],"paper":[142],"based":[144,319],"on":[145,181,284,320],"an":[146,301],"extensive":[147],"exploitation":[148],"order":[156],"figure":[158],"global":[160,364],"structure":[161,368],"phase":[163,185,219,237,326,370],"space.":[164,186,220],"In":[165,187,229,310],"value,":[169],"and":[170,202,255,372,458],"often":[171],"sign,":[173],"depend":[180],"location":[183,290],"contrast":[188],"with":[189,407],"a":[190,279,315,363,385,411,434,482,511],"current":[191],"usage,":[192],"we":[193,232,313,456],"consider":[194],"this":[195,349],"matrix,":[196,340],"as":[197,199],"well":[198],"its":[200,289,513],"eigenvalues":[201,245],"eigenvectors,":[203],"not":[204,461],"close":[207],"vicinity":[208],"states":[211,442],"system,":[214,512],"but":[215,223],"also":[216,397],"everywhere":[217],"Two":[221],"distinct,":[222],"complementary":[224],"approaches":[225],"used":[227],"here.":[228],"Sec.":[230,311,358],"2,":[231],"define":[233],"frontiers":[234],"partition":[236,418,465,503,546],"space":[238,327],"according":[239,330],"signs":[242],"their":[248],"real":[249],"part":[250],"if":[251],"they":[252],"complex),":[254],"where":[256],"required,":[257],"slopes":[260],"eigenvectors.":[263],"From":[264],"then":[265],"on,":[266],"exact":[268],"any":[271],"state":[273],"might":[275],"present":[277],"domain":[280,293,424],"sole":[286],"basis":[287],"and,":[294],"addition,":[296],"has":[298,420],"at":[299,426],"least":[300],"idea":[302],"possible":[305],"number":[306,507],"3,":[312],"use":[314],"qualitative":[317],"approach":[318,350,396],"theory":[322],"circuits.":[324],"Here,":[325],"partitioned":[329],"specifically,":[342],"nuclei.":[345],"feel":[347],"generic":[353],"than":[354,525],"described":[356],"2.":[359],"it":[361],"provides":[362],"view":[365],"space,":[371],"thus":[373,460],"permits":[374],"infer":[376],"much":[377],"dynamics":[380],"system":[383],"simple":[386],"analysis":[387],"within":[391],"matrix.":[394],"turned":[398],"out":[399],"extremely":[402],"useful":[403],"for":[404,501,538],"synthesizing":[405],"preconceived":[408],"properties.":[409],"For":[410],"wide":[412],"variety":[413],"once":[416],"process":[419],"been":[421],"achieved,":[422],"comprises":[425],"most":[427],"state.":[430],"found,":[432],"however,":[433],"family":[435],"(typically,":[443],"stable":[445],"unstable":[448],"nodes)":[449],"differ":[450],"neither":[452],"criteria":[455],"use,":[457],"separated":[462],"our":[464],"processes.":[466,547],"counter-example":[468],"implies":[469],"non-polynomial":[470],"functions.":[471],"It":[472],"realize":[476],"frontier":[479,514],"diagrams":[480],"permit":[481],"reduction":[483],"analysis,":[489],"because":[490],"involved":[496,529],"nonlinearities":[498],"relevant":[500],"process.":[504],"Whatever":[505],"diagram":[515],"drawn":[518],"dimensions":[521],"whenever":[522],"no":[523],"nonlinearities.":[531],"Pre-existing":[532],"conjectures":[533],"concerning":[534],"necessary":[536],"conditions":[537],"multistationarity":[539],"discussed":[541]},"counts_by_year":[{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":2},{"year":2013,"cited_by_count":2},{"year":2012,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}
