{"id":"https://openalex.org/W2112157264","doi":"https://doi.org/10.1142/s0218127406016628","title":"NULLCLINES AND NULLCLINE INTERSECTIONS","display_name":"NULLCLINES AND NULLCLINE INTERSECTIONS","publication_year":2006,"publication_date":"2006-10-01","ids":{"openalex":"https://openalex.org/W2112157264","doi":"https://doi.org/10.1142/s0218127406016628","mag":"2112157264"},"language":"en","primary_location":{"id":"doi:10.1142/s0218127406016628","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127406016628","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":true,"raw_author_name":"REN\u00c9 THOMAS","raw_affiliation_strings":["Center for Nonlinear Phenomena and Complex Systems, Universit\u00e9 de Bruxelles, Bd. du Triomphe, B-1050 Bruxelles, Belgium"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Center for Nonlinear Phenomena and Complex Systems, Universit\u00e9 de Bruxelles, Bd. du Triomphe, B-1050 Bruxelles, Belgium","institution_ids":["https://openalex.org/I132053463"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5030463072"],"corresponding_institution_ids":["https://openalex.org/I132053463"],"apc_list":null,"apc_paid":null,"fwci":0.2462,"has_fulltext":false,"cited_by_count":4,"citation_normalized_percentile":{"value":0.58516599,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":95},"biblio":{"volume":"16","issue":"10","first_page":"3023","last_page":"3033"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11261","display_name":"Quantum chaos and dynamical systems","score":0.9979000091552734,"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"}},"topics":[{"id":"https://openalex.org/T11261","display_name":"Quantum chaos and dynamical systems","score":0.9979000091552734,"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/T10244","display_name":"Chaos control and synchronization","score":0.9937000274658203,"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/T11187","display_name":"Nonlinear Dynamics and Pattern Formation","score":0.988099992275238,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/jacobian-matrix-and-determinant","display_name":"Jacobian matrix and determinant","score":0.8339518904685974},{"id":"https://openalex.org/keywords/partial-derivative","display_name":"Partial derivative","score":0.583805501461029},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.563564658164978},{"id":"https://openalex.org/keywords/sign","display_name":"Sign (mathematics)","score":0.5625768899917603},{"id":"https://openalex.org/keywords/electronic-circuit","display_name":"Electronic circuit","score":0.5563911199569702},{"id":"https://openalex.org/keywords/phase-space","display_name":"Phase space","score":0.5319643020629883},{"id":"https://openalex.org/keywords/dynamical-systems-theory","display_name":"Dynamical systems theory","score":0.46082791686058044},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.45491719245910645},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.44412916898727417},{"id":"https://openalex.org/keywords/chaotic","display_name":"Chaotic","score":0.4213627576828003},{"id":"https://openalex.org/keywords/space","display_name":"Space (punctuation)","score":0.41525518894195557},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.35661619901657104},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3505983352661133},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3265121579170227},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.21110090613365173},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.12699702382087708}],"concepts":[{"id":"https://openalex.org/C200331156","wikidata":"https://www.wikidata.org/wiki/Q506041","display_name":"Jacobian matrix and determinant","level":2,"score":0.8339518904685974},{"id":"https://openalex.org/C53846429","wikidata":"https://www.wikidata.org/wiki/Q186475","display_name":"Partial derivative","level":2,"score":0.583805501461029},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.563564658164978},{"id":"https://openalex.org/C139676723","wikidata":"https://www.wikidata.org/wiki/Q1193832","display_name":"Sign (mathematics)","level":2,"score":0.5625768899917603},{"id":"https://openalex.org/C134146338","wikidata":"https://www.wikidata.org/wiki/Q1815901","display_name":"Electronic circuit","level":2,"score":0.5563911199569702},{"id":"https://openalex.org/C151342819","wikidata":"https://www.wikidata.org/wiki/Q62542","display_name":"Phase space","level":2,"score":0.5319643020629883},{"id":"https://openalex.org/C79379906","wikidata":"https://www.wikidata.org/wiki/Q3174497","display_name":"Dynamical systems theory","level":2,"score":0.46082791686058044},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.45491719245910645},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.44412916898727417},{"id":"https://openalex.org/C2777052490","wikidata":"https://www.wikidata.org/wiki/Q5072826","display_name":"Chaotic","level":2,"score":0.4213627576828003},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.41525518894195557},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.35661619901657104},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3505983352661133},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3265121579170227},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.21110090613365173},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.12699702382087708},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"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/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1142/s0218127406016628","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127406016628","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:dipot.ulb.ac.be:2013/128693","is_oa":false,"landing_page_url":"http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/128693","pdf_url":null,"source":{"id":"https://openalex.org/S4306401063","display_name":"D\u00e9p\u00f4t institutionnel de l'Universit\u00e9 libre de Bruxelles (Universit\u00e9 Libre de Bruxelles)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I132053463","host_organization_name":"Universit\u00e9 Libre de Bruxelles","host_organization_lineage":["https://openalex.org/I132053463"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"International journal of bifurcation and chaos in applied sciences and engineering, 16","raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":null,"sustainable_development_goals":[{"score":0.47999998927116394,"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":13,"referenced_works":["https://openalex.org/W344332445","https://openalex.org/W577093083","https://openalex.org/W617581742","https://openalex.org/W1516866308","https://openalex.org/W1979230873","https://openalex.org/W1992731445","https://openalex.org/W2027510758","https://openalex.org/W2036009506","https://openalex.org/W2043346919","https://openalex.org/W2047575710","https://openalex.org/W2062771516","https://openalex.org/W2164348551","https://openalex.org/W2168364602"],"related_works":["https://openalex.org/W2007405763","https://openalex.org/W3136087161","https://openalex.org/W2053762185","https://openalex.org/W2145980196","https://openalex.org/W2018238589","https://openalex.org/W1486373823","https://openalex.org/W1984283682","https://openalex.org/W2027776969","https://openalex.org/W2374427492","https://openalex.org/W2081531990"],"abstract_inverted_index":{"One":[0],"purpose":[1],"of":[2,47,58,70,72,80,84,102,108,119,127,137,140,196,203],"this":[3],"paper":[4],"is":[5,113,179,198],"to":[6,78,155,165],"document":[7],"the":[8,19,40,45,48,56,81,109,116,120,124,138,145,187,204,217],"fact":[9],"that,":[10],"in":[11,43,65,88,100,189,194],"dynamical":[12],"systems":[13],"described":[14],"by":[15,51,75,199],"ordinary":[16],"differential":[17],"equations,":[18],"trajectories":[20],"can":[21,130],"be":[22],"organized":[23],"not":[24],"only":[25,158],"around":[26,33],"fixed":[27],"points":[28],"(steady":[29],"states),":[30],"but":[31],"also":[32],"lines.":[34],"In":[35,207],"2D,":[36],"these":[37],"lines":[38],"are":[39,98],"nullclines":[41,49,94],"themselves,":[42],"3D,":[44],"intersections":[46,97],"two":[50,218],"two,":[52],"etc.":[53],"We":[54],"precise":[55],"concepts":[57],"\"partial":[59,85],"steady":[60,63],"states\"":[61],"(i.e.":[62,181],"states":[64],"a":[66,90,152,162,172,210],"subsystem":[67],"that":[68,147],"consists":[69],"sections":[71],"phase":[73,190],"space":[74],"planes":[76],"normal":[77],"one":[79],"axes)":[82],"and":[83,123,161],"multistationarity\"":[86],"(multistationarity":[87],"such":[89],"subsystem).":[91],"Steady":[92],"states,":[93],"or":[95,183],"their":[96],"revisited":[99],"terms":[101,195],"circuits,":[103],"defined":[104],"from":[105],"nonzero":[106],"elements":[107],"Jacobian":[110,121],"matrix.":[111],"It":[112],"shown":[114,170],"how":[115],"mere":[117],"examination":[118],"matrix":[122],"sign":[125],"patterns":[126],"its":[128],"circuits":[129,197],"help":[131],"interpreting":[132],"(and":[133],"often":[134],"predicting)":[135],"aspects":[136],"dynamics":[139,149],"systems.":[141],"The":[142,192],"results":[143],"reinforce":[144],"idea":[146],"chaotic":[148],"requires":[150],"both":[151],"positive":[153,182],"circuit,":[154,164],"provide":[156,166],"(if":[157],"partial)":[159],"multistationarity,":[160],"negative":[163,184],"sustained":[167],"oscillations.":[168],"As":[169],"elsewhere,":[171],"single":[173],"circuit":[174],"may":[175],"suffice":[176],"if":[177],"it":[178],"ambiguous":[180],"depending":[185],"on":[186],"location":[188],"space).":[191],"description":[193],"no":[200],"means":[201],"exclusive":[202],"classical":[205],"description.":[206],"many":[208],"cases,":[209],"fruitful":[211],"approach":[212],"involves":[213],"repeated":[214],"feedback":[215],"between":[216],"viewpoints.":[219]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}