{"id":"https://openalex.org/W2082786393","doi":"https://doi.org/10.1137/070709128","title":"Chaotic Braided Solutions via Rigorous Numerics: Chaos in the Swift\u2013Hohenberg Equation","display_name":"Chaotic Braided Solutions via Rigorous Numerics: Chaos in the Swift\u2013Hohenberg Equation","publication_year":2008,"publication_date":"2008-01-01","ids":{"openalex":"https://openalex.org/W2082786393","doi":"https://doi.org/10.1137/070709128","mag":"2082786393"},"language":"en","primary_location":{"id":"doi:10.1137/070709128","is_oa":false,"landing_page_url":"https://doi.org/10.1137/070709128","pdf_url":null,"source":{"id":"https://openalex.org/S161160597","display_name":"SIAM Journal on Applied Dynamical Systems","issn_l":"1536-0040","issn":["1536-0040"],"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 Dynamical Systems","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/A5068716450","display_name":"Jan Bouwe van den Berg","orcid":"https://orcid.org/0000-0003-3503-7535"},"institutions":[{"id":"https://openalex.org/I865915315","display_name":"Vrije Universiteit Amsterdam","ror":"https://ror.org/008xxew50","country_code":"NL","type":"education","lineage":["https://openalex.org/I865915315"]}],"countries":["NL"],"is_corresponding":true,"raw_author_name":"Jan Bouwe van den Berg","raw_affiliation_strings":["Vrije Universiteit Amsterdam ,"],"affiliations":[{"raw_affiliation_string":"Vrije Universiteit Amsterdam ,","institution_ids":["https://openalex.org/I865915315"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5010664127","display_name":"Jean\u2010Philippe Lessard","orcid":"https://orcid.org/0000-0001-7672-6623"},"institutions":[{"id":"https://openalex.org/I102322142","display_name":"Rutgers, The State University of New Jersey","ror":"https://ror.org/05vt9qd57","country_code":"US","type":"education","lineage":["https://openalex.org/I102322142"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Jean-Philippe Lessard","raw_affiliation_strings":["Rutgers, University"],"affiliations":[{"raw_affiliation_string":"Rutgers, University","institution_ids":["https://openalex.org/I102322142"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5068716450"],"corresponding_institution_ids":["https://openalex.org/I865915315"],"apc_list":null,"apc_paid":null,"fwci":4.2325,"has_fulltext":false,"cited_by_count":49,"citation_normalized_percentile":{"value":0.94147863,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"7","issue":"3","first_page":"988","last_page":"1031"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.9990000128746033,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.9990000128746033,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/T11261","display_name":"Quantum chaos and dynamical systems","score":0.9976999759674072,"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":0.9973999857902527,"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/computer-assisted-proof","display_name":"Computer-assisted proof","score":0.6920059323310852},{"id":"https://openalex.org/keywords/braid","display_name":"Braid","score":0.6756475567817688},{"id":"https://openalex.org/keywords/chaotic","display_name":"Chaotic","score":0.6468498110771179},{"id":"https://openalex.org/keywords/continuation","display_name":"Continuation","score":0.5552939176559448},{"id":"https://openalex.org/keywords/periodic-orbits","display_name":"Periodic orbits","score":0.5449038147926331},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5323778986930847},{"id":"https://openalex.org/keywords/forcing","display_name":"Forcing (mathematics)","score":0.4878402054309845},{"id":"https://openalex.org/keywords/range","display_name":"Range (aeronautics)","score":0.4818952679634094},{"id":"https://openalex.org/keywords/orbit","display_name":"Orbit (dynamics)","score":0.4486621618270874},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4131041467189789},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.22257107496261597},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.11675229668617249}],"concepts":[{"id":"https://openalex.org/C173613240","wikidata":"https://www.wikidata.org/wiki/Q1150675","display_name":"Computer-assisted proof","level":3,"score":0.6920059323310852},{"id":"https://openalex.org/C124194741","wikidata":"https://www.wikidata.org/wiki/Q320809","display_name":"Braid","level":2,"score":0.6756475567817688},{"id":"https://openalex.org/C2777052490","wikidata":"https://www.wikidata.org/wiki/Q5072826","display_name":"Chaotic","level":2,"score":0.6468498110771179},{"id":"https://openalex.org/C88626702","wikidata":"https://www.wikidata.org/wiki/Q1128903","display_name":"Continuation","level":2,"score":0.5552939176559448},{"id":"https://openalex.org/C2984157986","wikidata":"https://www.wikidata.org/wiki/Q3884964","display_name":"Periodic orbits","level":2,"score":0.5449038147926331},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5323778986930847},{"id":"https://openalex.org/C197115733","wikidata":"https://www.wikidata.org/wiki/Q1003136","display_name":"Forcing (mathematics)","level":2,"score":0.4878402054309845},{"id":"https://openalex.org/C204323151","wikidata":"https://www.wikidata.org/wiki/Q905424","display_name":"Range (aeronautics)","level":2,"score":0.4818952679634094},{"id":"https://openalex.org/C196644772","wikidata":"https://www.wikidata.org/wiki/Q3884964","display_name":"Orbit (dynamics)","level":2,"score":0.4486621618270874},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4131041467189789},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.22257107496261597},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.11675229668617249},{"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/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0},{"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/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C146978453","wikidata":"https://www.wikidata.org/wiki/Q3798668","display_name":"Aerospace engineering","level":1,"score":0.0},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1137/070709128","is_oa":false,"landing_page_url":"https://doi.org/10.1137/070709128","pdf_url":null,"source":{"id":"https://openalex.org/S161160597","display_name":"SIAM Journal on Applied Dynamical Systems","issn_l":"1536-0040","issn":["1536-0040"],"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 Dynamical Systems","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.535.7785","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.535.7785","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.math.vu.nl/~janbouwe/pub/chaos.pdf","raw_type":"text"},{"id":"pmh:oai:research.vu.nl:publications/a2e60fbb-2349-499f-a509-2b85f24b14b8","is_oa":false,"landing_page_url":"https://hdl.handle.net/1871.1/a2e60fbb-2349-499f-a509-2b85f24b14b8","pdf_url":null,"source":{"id":"https://openalex.org/S4306400546","display_name":"Digital Academic REpository of VU University Amsterdam (Vrije Universiteit Amsterdam)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I865915315","host_organization_name":"Vrije Universiteit Amsterdam","host_organization_lineage":["https://openalex.org/I865915315"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"van den Berg, G J B 2008, 'Chaotic braided solutions via rigorous numerics: chaos in the Swift-Hohenberg equation', SIAM Journal on Applied Dynamical Systems, vol. 7, pp. 988-1031. https://doi.org/10.1137/070709128","raw_type":"article"},{"id":"pmh:vu:oai:research.vu.nl:publications/a2e60fbb-2349-499f-a509-2b85f24b14b8","is_oa":false,"landing_page_url":"https://research.vu.nl/en/publications/a2e60fbb-2349-499f-a509-2b85f24b14b8","pdf_url":null,"source":{"id":"https://openalex.org/S4306401843","display_name":"Data Archiving and Networked Services (DANS)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1322597698","host_organization_name":"Royal Netherlands Academy of Arts and Sciences","host_organization_lineage":["https://openalex.org/I1322597698"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"SIAM Journal on Applied Dynamical Systems, 7, 988 - 1031. Society of Industrial and Applied Mathematics","raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":null,"sustainable_development_goals":[{"display_name":"Affordable and clean energy","score":0.7200000286102295,"id":"https://metadata.un.org/sdg/7"}],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":39,"referenced_works":["https://openalex.org/W1589876408","https://openalex.org/W1965565198","https://openalex.org/W1965788748","https://openalex.org/W1968313583","https://openalex.org/W1970930935","https://openalex.org/W1979499447","https://openalex.org/W1981833819","https://openalex.org/W1985434612","https://openalex.org/W1988031493","https://openalex.org/W1994876152","https://openalex.org/W1997147969","https://openalex.org/W1999536242","https://openalex.org/W2000883601","https://openalex.org/W2001675348","https://openalex.org/W2002773683","https://openalex.org/W2008421829","https://openalex.org/W2019398782","https://openalex.org/W2023794945","https://openalex.org/W2023892882","https://openalex.org/W2042338802","https://openalex.org/W2049314201","https://openalex.org/W2059340344","https://openalex.org/W2062991929","https://openalex.org/W2068661834","https://openalex.org/W2070451295","https://openalex.org/W2080906097","https://openalex.org/W2086126894","https://openalex.org/W2088423883","https://openalex.org/W2093569281","https://openalex.org/W2093624109","https://openalex.org/W2101348781","https://openalex.org/W2112602754","https://openalex.org/W2122612155","https://openalex.org/W2136143898","https://openalex.org/W2159706263","https://openalex.org/W2163239611","https://openalex.org/W2165716511","https://openalex.org/W2172178670","https://openalex.org/W4239883234"],"related_works":["https://openalex.org/W2008304607","https://openalex.org/W4323697138","https://openalex.org/W4389166184","https://openalex.org/W2002451348","https://openalex.org/W1616469453","https://openalex.org/W2021846046","https://openalex.org/W4386939745","https://openalex.org/W1998827207","https://openalex.org/W4288334325","https://openalex.org/W3137808042"],"abstract_inverted_index":{"We":[0],"prove":[1,38],"that":[2,87],"the":[3,27,39,70,88],"stationary":[4],"Swift\u2013Hohenberg":[5],"equation":[6],"has":[7],"chaotic":[8],"dynamics":[9,89],"on":[10,30],"a":[11,16,31,42,60,81],"critical":[12],"energy":[13],"level":[14],"for":[15],"large":[17],"(continuous)":[18],"range":[19],"of":[20,26,41,72,83],"parameter":[21],"values.":[22],"The":[23,49],"first":[24],"step":[25,51],"method":[28,36],"relies":[29],"computer":[32],"assisted,":[33],"rigorous,":[34],"continuation":[35],"to":[37,80],"existence":[40,71],"periodic":[43,57,76],"orbit":[44],"with":[45],"certain":[46],"geometric":[47],"properties.":[48],"second":[50],"is":[52,90],"topological:":[53],"we":[54,64],"use":[55],"this":[56],"solution":[58],"as":[59],"skeleton,":[61],"through":[62],"which":[63],"braid":[65],"other":[66],"solutions,":[67],"thus":[68],"forcing":[69],"infinitely":[73],"many":[74],"braided":[75],"orbits.":[77],"A":[78],"semiconjugacy":[79],"subshift":[82],"finite":[84],"type":[85],"shows":[86],"chaotic.":[91]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":3},{"year":2018,"cited_by_count":6},{"year":2017,"cited_by_count":4},{"year":2016,"cited_by_count":4},{"year":2015,"cited_by_count":8},{"year":2014,"cited_by_count":3},{"year":2013,"cited_by_count":1}],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}