{"id":"https://openalex.org/W2047389619","doi":"https://doi.org/10.1137/0153037","title":"The Numerical Computation of Heteroclinic Connections in Systems of Gradient Partial Differential Equations","display_name":"The Numerical Computation of Heteroclinic Connections in Systems of Gradient Partial Differential Equations","publication_year":1993,"publication_date":"1993-06-01","ids":{"openalex":"https://openalex.org/W2047389619","doi":"https://doi.org/10.1137/0153037","mag":"2047389619"},"language":"en","primary_location":{"id":"doi:10.1137/0153037","is_oa":false,"landing_page_url":"https://doi.org/10.1137/0153037","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/A5102215816","display_name":"Fengshan Bai","orcid":null},"institutions":[{"id":"https://openalex.org/I51601045","display_name":"University of Bath","ror":"https://ror.org/002h8g185","country_code":"GB","type":"education","lineage":["https://openalex.org/I51601045"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"Fengshan Bai","raw_affiliation_strings":["School of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom","United Kingdom Science Engineering Research Council","Mathematics Department, Tsinghua University, Beijing 100084, People's Republic of China","Division of Applied Mechanics, Program in Scientific Computing and Compu-tational Mathematics, Durand-252, Stanford University, Stanford, CMifornia 94305"],"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom","institution_ids":["https://openalex.org/I51601045"]},{"raw_affiliation_string":"United Kingdom Science Engineering Research Council","institution_ids":[]},{"raw_affiliation_string":"Mathematics Department, Tsinghua University, Beijing 100084, People's Republic of China","institution_ids":[]},{"raw_affiliation_string":"Division of Applied Mechanics, Program in Scientific Computing and Compu-tational Mathematics, Durand-252, Stanford University, Stanford, CMifornia 94305","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5109439078","display_name":"A. Spence","orcid":null},"institutions":[{"id":"https://openalex.org/I51601045","display_name":"University of Bath","ror":"https://ror.org/002h8g185","country_code":"GB","type":"education","lineage":["https://openalex.org/I51601045"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Alastair Spence","raw_affiliation_strings":["Division of Applied Mechanics, Program in Scientific Computing and Compu-tational Mathematics, Durand-252, Stanford University, Stanford, CMifornia 94305","School of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom","Mathematics Department, Tsinghua University, Beijing 100084, People's Republic of China","United Kingdom Science Engineering Research Council"],"affiliations":[{"raw_affiliation_string":"Division of Applied Mechanics, Program in Scientific Computing and Compu-tational Mathematics, Durand-252, Stanford University, Stanford, CMifornia 94305","institution_ids":[]},{"raw_affiliation_string":"School of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom","institution_ids":["https://openalex.org/I51601045"]},{"raw_affiliation_string":"Mathematics Department, Tsinghua University, Beijing 100084, People's Republic of China","institution_ids":[]},{"raw_affiliation_string":"United Kingdom Science Engineering Research Council","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5032114852","display_name":"Andrew M. Stuart","orcid":"https://orcid.org/0000-0001-9091-7266"},"institutions":[{"id":"https://openalex.org/I51601045","display_name":"University of Bath","ror":"https://ror.org/002h8g185","country_code":"GB","type":"education","lineage":["https://openalex.org/I51601045"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Andrew M. Stuart","raw_affiliation_strings":["School of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom","Mathematics Department, Tsinghua University, Beijing 100084, People's Republic of China","Division of Applied Mechanics, Program in Scientific Computing and Compu-tational Mathematics, Durand-252, Stanford University, Stanford, CMifornia 94305","United Kingdom Science Engineering Research Council"],"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom","institution_ids":["https://openalex.org/I51601045"]},{"raw_affiliation_string":"Mathematics Department, Tsinghua University, Beijing 100084, People's Republic of China","institution_ids":[]},{"raw_affiliation_string":"Division of Applied Mechanics, Program in Scientific Computing and Compu-tational Mathematics, Durand-252, Stanford University, Stanford, CMifornia 94305","institution_ids":[]},{"raw_affiliation_string":"United Kingdom Science Engineering Research Council","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5102215816"],"corresponding_institution_ids":["https://openalex.org/I51601045"],"apc_list":null,"apc_paid":null,"fwci":4.3365,"has_fulltext":false,"cited_by_count":34,"citation_normalized_percentile":{"value":0.9380929,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"53","issue":"3","first_page":"743","last_page":"769"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11110","display_name":"Stability and Controllability of Differential Equations","score":0.9970999956130981,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11110","display_name":"Stability and Controllability of Differential Equations","score":0.9970999956130981,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11187","display_name":"Nonlinear Dynamics and Pattern Formation","score":0.9868000149726868,"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/T12100","display_name":"Advanced Mathematical Modeling in Engineering","score":0.9864000082015991,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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/mathematics","display_name":"Mathematics","score":0.6933999061584473},{"id":"https://openalex.org/keywords/heteroclinic-cycle","display_name":"Heteroclinic cycle","score":0.6498019695281982},{"id":"https://openalex.org/keywords/ode","display_name":"Ode","score":0.6319352984428406},{"id":"https://openalex.org/keywords/attractor","display_name":"Attractor","score":0.621446430683136},{"id":"https://openalex.org/keywords/heteroclinic-orbit","display_name":"Heteroclinic orbit","score":0.5913218259811401},{"id":"https://openalex.org/keywords/ordinary-differential-equation","display_name":"Ordinary differential equation","score":0.5864529609680176},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.5657432675361633},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.526726245880127},{"id":"https://openalex.org/keywords/discretization","display_name":"Discretization","score":0.46921291947364807},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.3402601480484009},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.3303040862083435},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.3171631097793579},{"id":"https://openalex.org/keywords/homoclinic-orbit","display_name":"Homoclinic orbit","score":0.1940775215625763},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.10017311573028564},{"id":"https://openalex.org/keywords/bifurcation","display_name":"Bifurcation","score":0.1000145673751831}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6933999061584473},{"id":"https://openalex.org/C142115181","wikidata":"https://www.wikidata.org/wiki/Q16948232","display_name":"Heteroclinic cycle","level":5,"score":0.6498019695281982},{"id":"https://openalex.org/C34862557","wikidata":"https://www.wikidata.org/wiki/Q178985","display_name":"Ode","level":2,"score":0.6319352984428406},{"id":"https://openalex.org/C164380108","wikidata":"https://www.wikidata.org/wiki/Q507187","display_name":"Attractor","level":2,"score":0.621446430683136},{"id":"https://openalex.org/C184481792","wikidata":"https://www.wikidata.org/wiki/Q1992087","display_name":"Heteroclinic orbit","level":5,"score":0.5913218259811401},{"id":"https://openalex.org/C51544822","wikidata":"https://www.wikidata.org/wiki/Q465274","display_name":"Ordinary differential equation","level":3,"score":0.5864529609680176},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.5657432675361633},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.526726245880127},{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.46921291947364807},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3402601480484009},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.3303040862083435},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.3171631097793579},{"id":"https://openalex.org/C200581526","wikidata":"https://www.wikidata.org/wiki/Q690494","display_name":"Homoclinic orbit","level":4,"score":0.1940775215625763},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.10017311573028564},{"id":"https://openalex.org/C2781349735","wikidata":"https://www.wikidata.org/wiki/Q4904824","display_name":"Bifurcation","level":3,"score":0.1000145673751831},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/0153037","is_oa":false,"landing_page_url":"https://doi.org/10.1137/0153037","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"},{"id":"pmh:oai:authors.library.caltech.edu:78145","is_oa":false,"landing_page_url":"https://resolver.caltech.edu/CaltechAUTHORS:20170613-082542564","pdf_url":null,"source":{"id":"https://openalex.org/S4306402161","display_name":"CaltechAUTHORS (California Institute of Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I122411786","host_organization_name":"California Institute of Technology","host_organization_lineage":["https://openalex.org/I122411786"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":25,"referenced_works":["https://openalex.org/W61514442","https://openalex.org/W85861326","https://openalex.org/W409831783","https://openalex.org/W1485916122","https://openalex.org/W1532361234","https://openalex.org/W1557040331","https://openalex.org/W1559403043","https://openalex.org/W1560060148","https://openalex.org/W1594759473","https://openalex.org/W1603340857","https://openalex.org/W1966227249","https://openalex.org/W1971808848","https://openalex.org/W1982779602","https://openalex.org/W2013921840","https://openalex.org/W2028296976","https://openalex.org/W2032257836","https://openalex.org/W2037913112","https://openalex.org/W2045670796","https://openalex.org/W2046528357","https://openalex.org/W2063458426","https://openalex.org/W2073686912","https://openalex.org/W2074920889","https://openalex.org/W2136919854","https://openalex.org/W2168568158","https://openalex.org/W4292366025"],"related_works":["https://openalex.org/W2048010006","https://openalex.org/W2006672103","https://openalex.org/W4375952595","https://openalex.org/W2090855536","https://openalex.org/W2909304163","https://openalex.org/W2045664867","https://openalex.org/W3102697363","https://openalex.org/W2009573643","https://openalex.org/W2066872858","https://openalex.org/W3013343808"],"abstract_inverted_index":{"The":[0,129,243],"numerical":[1,214],"computation":[2],"of":[3,22,46,57,76,94,97,113,126,144,167,189,197,208,220,249,257,272,285,306],"heteroclinic":[4,79,119,180],"connections":[5,185,221],"in":[6,19,186,230,275],"partial":[7],"differential":[8,146],"equations":[9,147],"(PDEs)":[10],"with":[11,202],"a":[12,28,51,55,136,142,195,265,276],"gradient":[13,59,204],"structure,":[14,60],"such":[15],"as":[16,237,255],"those":[17,152],"arising":[18],"the":[20,40,44,58,70,74,95,100,107,124,127,155,160,163,168,176,187,203,209,223,241,247,258,270,295,300,320,324,334],"modeling":[21],"phase":[23],"transitions,":[24],"is":[25,33,62,88,132,312,316],"considered.":[26],"Initially,":[27],"scalar":[29],"reaction":[30],"diffusion":[31],"equation":[32,131],"studied;":[34],"structural":[35],"assumptions":[36],"are":[37,68,104,157,182,191,211,228,235,253,292],"made":[38],"on":[39,222],"problem":[41,297,326],"to":[42,83,90,140,151,200,217],"ensure":[43],"existence":[45,164],"absorbing":[47],"sets":[48],"and,":[49],"consequently,":[50],"global":[52,71,169,321],"attractor.":[53,224],"As":[54],"result":[56],"it":[61,87,315],"known":[63,198],"that,":[64],"if":[65],"all":[66],"equilibria":[67,77,82,98,114,177],"hyperbolic,":[69],"attractor":[72,170,210,322],"comprises":[73],"set":[75,96],"and":[78,99,115,165,171,178,213,262,298,314],"orbits":[80,251],"connecting":[81,101,250,308],"one":[84],"another.":[85],"Thus":[86],"natural":[89],"consider":[91],"direct":[92],"approximation":[93],"orbits.":[102],"Results":[103],"proved":[105,158],"about":[106],"Fourier":[108,232],"spanning":[109,173,233],"basis":[110],"for":[111,117,154,159,175,240,268,294,299,323,333],"branches":[112],"also":[116],"certain":[118,179,231],"connections;":[120],"these":[121,309],"results":[122,150,291],"exploit":[123],"oddness":[125],"nonlinearity.":[128],"reaction-diffusion":[130],"then":[133,192],"approximated":[134],"by":[135,283],"Galerkin":[137],"spectral":[138],"discretization":[139],"produce":[141],"system":[143,188],"ordinary":[145],"(ODEs).":[148],"Analogous":[149],"holding":[153],"PDE":[156],"ODEs\u2014in":[161],"particular,":[162],"structure":[166],"appropriate":[172],"bases":[174],"connections,":[181,226],"studied.":[183],"Heteroclinic":[184],"ODEs":[190],"computed":[193],"using":[194],"generalization":[196],"methods":[199,244],"cope":[201],"structure.":[205],"Suitable":[206],"parameterizations":[207],"introduced":[212],"continuation":[215],"used":[216,236,245],"find":[218],"families":[219],"Special":[225],"which":[227],"stable":[229],"bases,":[234],"starting":[238],"points":[239],"computations.":[242],"allow":[246],"calculation":[248],"that":[252,278,319,332],"unstable":[254],"solutions":[256],"initial":[259,287],"value":[260,288],"problem,":[261],"thus":[263],"provide":[264],"computational":[266],"tool":[267],"understanding":[269],"dynamics":[271],"dissipative":[273],"problems":[274,311],"manner":[277],"could":[279],"not":[280],"be":[281,328],"achieved":[282],"use":[284],"standard":[286],"methods.":[289],"Numerical":[290],"given":[293],"Chafee\u2013Infante":[296,325],"Cahn\u2013Hilliard":[301,335],"equation.":[302,336],"A":[303],"one-parameter":[304],"family":[305],"PDEs":[307],"two":[310],"introduced,":[313],"demonstrated":[317],"numerically":[318],"can":[327],"continuously":[329],"deformed":[330],"into":[331]},"counts_by_year":[{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":3},{"year":2019,"cited_by_count":2},{"year":2017,"cited_by_count":3},{"year":2016,"cited_by_count":1},{"year":2014,"cited_by_count":3}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}