{"id":"https://openalex.org/W2105191447","doi":"https://doi.org/10.1137/s0036139902418419","title":"Divergence Criterion for Generic Planar Systems","display_name":"Divergence Criterion for Generic Planar Systems","publication_year":2003,"publication_date":"2003-01-01","ids":{"openalex":"https://openalex.org/W2105191447","doi":"https://doi.org/10.1137/s0036139902418419","mag":"2105191447"},"language":"en","primary_location":{"id":"doi:10.1137/s0036139902418419","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0036139902418419","pdf_url":null,"source":{"id":"https://openalex.org/S77200987","display_name":"SIAM Journal on Applied Mathematics","issn_l":"0036-1399","issn":["0036-1399","1095-712X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Applied Mathematics","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/A5043740268","display_name":"Paul Waltman","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Paul Waltman","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5073662392","display_name":"Sergei S. Pilyugin","orcid":"https://orcid.org/0000-0001-5014-2232"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Sergei S. Pilyugin","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":1.3989,"has_fulltext":false,"cited_by_count":5,"citation_normalized_percentile":{"value":0.84672742,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"64","issue":"1","first_page":"81","last_page":"93"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10928","display_name":"Probabilistic and Robust Engineering Design","score":0.9624000191688538,"subfield":{"id":"https://openalex.org/subfields/1804","display_name":"Statistics, Probability and Uncertainty"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},"topics":[{"id":"https://openalex.org/T10928","display_name":"Probabilistic and Robust Engineering Design","score":0.9624000191688538,"subfield":{"id":"https://openalex.org/subfields/1804","display_name":"Statistics, Probability and Uncertainty"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T13539","display_name":"thermodynamics and calorimetric analyses","score":0.9444000124931335,"subfield":{"id":"https://openalex.org/subfields/1606","display_name":"Physical and Theoretical Chemistry"},"field":{"id":"https://openalex.org/fields/16","display_name":"Chemistry"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11188","display_name":"Coagulation and Flocculation Studies","score":0.9369000196456909,"subfield":{"id":"https://openalex.org/subfields/2312","display_name":"Water Science and Technology"},"field":{"id":"https://openalex.org/fields/23","display_name":"Environmental Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/hopf-bifurcation","display_name":"Hopf bifurcation","score":0.7651907205581665},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7555843591690063},{"id":"https://openalex.org/keywords/divergence","display_name":"Divergence (linguistics)","score":0.7284912467002869},{"id":"https://openalex.org/keywords/sign","display_name":"Sign (mathematics)","score":0.5350115895271301},{"id":"https://openalex.org/keywords/bifurcation","display_name":"Bifurcation","score":0.5241760611534119},{"id":"https://openalex.org/keywords/planar","display_name":"Planar","score":0.48108431696891785},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4765530228614807},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.47274670004844666},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.4308282136917114},{"id":"https://openalex.org/keywords/simple","display_name":"Simple (philosophy)","score":0.4173962473869324},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.13943645358085632},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.1065312922000885},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.08859050273895264}],"concepts":[{"id":"https://openalex.org/C172435161","wikidata":"https://www.wikidata.org/wiki/Q531126","display_name":"Hopf bifurcation","level":4,"score":0.7651907205581665},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7555843591690063},{"id":"https://openalex.org/C207390915","wikidata":"https://www.wikidata.org/wiki/Q1230525","display_name":"Divergence (linguistics)","level":2,"score":0.7284912467002869},{"id":"https://openalex.org/C139676723","wikidata":"https://www.wikidata.org/wiki/Q1193832","display_name":"Sign (mathematics)","level":2,"score":0.5350115895271301},{"id":"https://openalex.org/C2781349735","wikidata":"https://www.wikidata.org/wiki/Q4904824","display_name":"Bifurcation","level":3,"score":0.5241760611534119},{"id":"https://openalex.org/C134786449","wikidata":"https://www.wikidata.org/wiki/Q3391255","display_name":"Planar","level":2,"score":0.48108431696891785},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4765530228614807},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.47274670004844666},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4308282136917114},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.4173962473869324},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.13943645358085632},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.1065312922000885},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.08859050273895264},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","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/C121684516","wikidata":"https://www.wikidata.org/wiki/Q7600677","display_name":"Computer graphics (images)","level":1,"score":0.0},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/s0036139902418419","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0036139902418419","pdf_url":null,"source":{"id":"https://openalex.org/S77200987","display_name":"SIAM Journal on Applied Mathematics","issn_l":"0036-1399","issn":["0036-1399","1095-712X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Applied Mathematics","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":20,"referenced_works":["https://openalex.org/W2264312","https://openalex.org/W642788517","https://openalex.org/W1531715726","https://openalex.org/W1596321532","https://openalex.org/W1603340857","https://openalex.org/W1966200641","https://openalex.org/W1970961543","https://openalex.org/W1981550616","https://openalex.org/W1992450291","https://openalex.org/W1995336298","https://openalex.org/W2002616979","https://openalex.org/W2016913956","https://openalex.org/W2023893235","https://openalex.org/W2035695789","https://openalex.org/W2037581075","https://openalex.org/W2058925994","https://openalex.org/W2140135910","https://openalex.org/W2146322847","https://openalex.org/W3010782316","https://openalex.org/W4214513274"],"related_works":["https://openalex.org/W3208095355","https://openalex.org/W4206327962","https://openalex.org/W2052507016","https://openalex.org/W2180461068","https://openalex.org/W2110143569","https://openalex.org/W2010009304","https://openalex.org/W3122938442","https://openalex.org/W2177370417","https://openalex.org/W2368054172","https://openalex.org/W2045638785"],"abstract_inverted_index":{"The":[0,58],"divergence":[1,25,42],"criterion":[2,43],"has":[3],"been":[4],"shown":[5,119],"to":[6,22,36,50,86],"be":[7],"helpful":[8],"in":[9,73,106],"distinguishing":[10],"between":[11],"sub-":[12,138],"and":[13,124,131,139],"supercritical":[14,103,140],"Hopf":[15,95,141],"bifurcations,":[16],"but":[17],"its":[18],"applicability":[19,39],"is":[20,26,44,60,83,99,117],"limited":[21],"systems":[23,91],"whose":[24],"sign":[27,55],"definite.":[28],"A":[29],"step-by-step":[30],"computational":[31],"procedure":[32,59,82],"which":[33],"allows":[34],"one":[35,53],"extend":[37],"the":[38,41,48,64,77,94,121,125,151,154],"of":[40,76,89,153],"derived":[45],"by":[46,68],"altering":[47],"system":[49,108],"an":[51],"equivalent":[52],"with":[54,113,128],"definite":[56],"divergence.":[57],"based":[61],"on":[62],"multiplying":[63],"original":[65],"vector":[66],"field":[67],"a":[69,74,107],"positive":[70],"quadratic":[71],"function":[72],"neighborhood":[75],"bifurcating":[78],"rest":[79],"point.":[80],"This":[81],"then":[84],"applied":[85],"several":[87],"examples":[88],"planar":[90],"that":[92,101,120],"exhibit":[93,136],"bifurcation.":[96],"Specifically,":[97],"it":[98],"demonstrated":[100],"only":[102],"bifurcations":[104],"occur":[105],"modeling":[109],"specific":[110],"immune":[111],"responses":[112],"handling":[114],"time.":[115],"It":[116],"also":[118],"FitzHugh-Nagumo":[122],"equations":[123,127],"chemostat":[126],"substrate":[129],"inhibition":[130],"linear":[132],"yield":[133],"coefficient":[134],"may":[135],"both":[137,144],"bifurcations.":[142],"In":[143],"cases,":[145],"simple":[146],"analytic":[147],"criteria":[148],"for":[149],"determining":[150],"criticality":[152],"bifurcation":[155],"are":[156],"presented.":[157]},"counts_by_year":[{"year":2013,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}