{"id":"https://openalex.org/W2786115491","doi":"https://doi.org/10.1142/s0218127417502248","title":"Unique Normal Form and the Associated Coefficients for a Class of Three-Dimensional Nilpotent Vector Fields","display_name":"Unique Normal Form and the Associated Coefficients for a Class of Three-Dimensional Nilpotent Vector Fields","publication_year":2017,"publication_date":"2017-12-30","ids":{"openalex":"https://openalex.org/W2786115491","doi":"https://doi.org/10.1142/s0218127417502248","mag":"2786115491"},"language":"en","primary_location":{"id":"doi:10.1142/s0218127417502248","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127417502248","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/A5100615345","display_name":"Jing Li","orcid":"https://orcid.org/0000-0002-7278-2764"},"institutions":[{"id":"https://openalex.org/I37796252","display_name":"Beijing University of Technology","ror":"https://ror.org/037b1pp87","country_code":"CN","type":"education","lineage":["https://openalex.org/I37796252"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Jing Li","raw_affiliation_strings":["College of Applied Sciences, Beijing University of Technology, Beijing 100124, P. R. China"],"affiliations":[{"raw_affiliation_string":"College of Applied Sciences, Beijing University of Technology, Beijing 100124, P. R. China","institution_ids":["https://openalex.org/I37796252"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5019460895","display_name":"Liying Kou","orcid":null},"institutions":[{"id":"https://openalex.org/I37796252","display_name":"Beijing University of Technology","ror":"https://ror.org/037b1pp87","country_code":"CN","type":"education","lineage":["https://openalex.org/I37796252"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Liying Kou","raw_affiliation_strings":["College of Applied Sciences, Beijing University of Technology, Beijing 100124, P. R. China"],"affiliations":[{"raw_affiliation_string":"College of Applied Sciences, Beijing University of Technology, Beijing 100124, P. R. China","institution_ids":["https://openalex.org/I37796252"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5100334967","display_name":"Duo Wang","orcid":"https://orcid.org/0000-0002-5151-1009"},"institutions":[{"id":"https://openalex.org/I20231570","display_name":"Peking University","ror":"https://ror.org/02v51f717","country_code":"CN","type":"education","lineage":["https://openalex.org/I20231570"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Duo Wang","raw_affiliation_strings":["School of Mathematical Sciences, Peking University, Beijing 100871, P. R. China"],"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences, Peking University, Beijing 100871, P. R. China","institution_ids":["https://openalex.org/I20231570"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5100666886","display_name":"Zhang Wei","orcid":"https://orcid.org/0000-0001-9660-1190"},"institutions":[{"id":"https://openalex.org/I37796252","display_name":"Beijing University of Technology","ror":"https://ror.org/037b1pp87","country_code":"CN","type":"education","lineage":["https://openalex.org/I37796252"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Wei Zhang","raw_affiliation_strings":["Beijing Key Laboratory on Nonlinear Vibrations and Strength of Mechanical Structures, College of Mechanical Engineering, Beijing University of Technology, Beijing 100124, P. R. China"],"affiliations":[{"raw_affiliation_string":"Beijing Key Laboratory on Nonlinear Vibrations and Strength of Mechanical Structures, College of Mechanical Engineering, Beijing University of Technology, Beijing 100124, P. R. China","institution_ids":["https://openalex.org/I37796252"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5019460895","https://openalex.org/A5100615345"],"corresponding_institution_ids":["https://openalex.org/I37796252"],"apc_list":null,"apc_paid":null,"fwci":0.8457,"has_fulltext":false,"cited_by_count":4,"citation_normalized_percentile":{"value":0.7328863,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"27","issue":"14","first_page":"1750224","last_page":"1750224"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12396","display_name":"Advanced Differential Equations and Dynamical Systems","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/T12396","display_name":"Advanced Differential Equations and Dynamical Systems","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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":0.9861000180244446,"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/T11416","display_name":"Numerical methods for differential equations","score":0.9434000253677368,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/vector-field","display_name":"Vector field","score":0.7657711505889893},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7551990747451782},{"id":"https://openalex.org/keywords/class","display_name":"Class (philosophy)","score":0.6520259380340576},{"id":"https://openalex.org/keywords/focus","display_name":"Focus (optics)","score":0.6157471537590027},{"id":"https://openalex.org/keywords/transformation","display_name":"Transformation (genetics)","score":0.5936692953109741},{"id":"https://openalex.org/keywords/nilpotent","display_name":"Nilpotent","score":0.5764362812042236},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.5278534889221191},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4539756774902344},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.45370668172836304},{"id":"https://openalex.org/keywords/field","display_name":"Field (mathematics)","score":0.43342775106430054},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4174477458000183},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.3437590003013611},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.16947486996650696},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.127064049243927},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.07352477312088013}],"concepts":[{"id":"https://openalex.org/C91188154","wikidata":"https://www.wikidata.org/wiki/Q186247","display_name":"Vector field","level":2,"score":0.7657711505889893},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7551990747451782},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.6520259380340576},{"id":"https://openalex.org/C192209626","wikidata":"https://www.wikidata.org/wiki/Q190909","display_name":"Focus (optics)","level":2,"score":0.6157471537590027},{"id":"https://openalex.org/C204241405","wikidata":"https://www.wikidata.org/wiki/Q461499","display_name":"Transformation (genetics)","level":3,"score":0.5936692953109741},{"id":"https://openalex.org/C50555996","wikidata":"https://www.wikidata.org/wiki/Q840023","display_name":"Nilpotent","level":2,"score":0.5764362812042236},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.5278534889221191},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4539756774902344},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.45370668172836304},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.43342775106430054},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4174477458000183},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3437590003013611},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.16947486996650696},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.127064049243927},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.07352477312088013},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C120665830","wikidata":"https://www.wikidata.org/wiki/Q14620","display_name":"Optics","level":1,"score":0.0},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","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/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1142/s0218127417502248","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127417502248","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":[{"id":"https://openalex.org/G3612510400","display_name":null,"funder_award_id":"1122001","funder_id":"https://openalex.org/F4320334977","funder_display_name":"Beijing Municipal Natural Science Foundation"},{"id":"https://openalex.org/G6247504843","display_name":null,"funder_award_id":"11772007","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G6937179550","display_name":null,"funder_award_id":"11372014","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G7079607317","display_name":null,"funder_award_id":"11072008","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G7587146415","display_name":null,"funder_award_id":"11290152","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G7627735582","display_name":null,"funder_award_id":"1172002","funder_id":"https://openalex.org/F4320334977","funder_display_name":"Beijing Municipal Natural Science Foundation"},{"id":"https://openalex.org/G806960711","display_name":null,"funder_award_id":"11070007","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320334977","display_name":"Beijing Municipal Natural Science Foundation","ror":null}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":13,"referenced_works":["https://openalex.org/W1515043949","https://openalex.org/W1975818603","https://openalex.org/W1995383096","https://openalex.org/W1998107461","https://openalex.org/W2030362902","https://openalex.org/W2033345573","https://openalex.org/W2033477867","https://openalex.org/W2063943049","https://openalex.org/W2084750861","https://openalex.org/W2088424770","https://openalex.org/W2093246922","https://openalex.org/W2152203759","https://openalex.org/W2743785931"],"related_works":["https://openalex.org/W4381552820","https://openalex.org/W2012531322","https://openalex.org/W2949088068","https://openalex.org/W2785900585","https://openalex.org/W2353730437","https://openalex.org/W4254387663","https://openalex.org/W2490303674","https://openalex.org/W2609066826","https://openalex.org/W2810752900","https://openalex.org/W2365677836"],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"we":[3],"mainly":[4],"focus":[5],"on":[6],"the":[7,19,34,40,53,56],"unique":[8],"normal":[9,37],"form":[10,38],"for":[11],"a":[12,58],"class":[13],"of":[14,21,55,60],"three-dimensional":[15],"vector":[16],"fields":[17],"via":[18],"method":[20],"transformation":[22],"with":[23,63],"parameters.":[24],"A":[25],"general":[26],"explicit":[27],"recursive":[28],"formula":[29],"is":[30,65],"derived":[31],"to":[32],"compute":[33],"higher":[35],"order":[36],"and":[39],"associated":[41],"coefficients,":[42],"which":[43],"can":[44],"be":[45],"achieved":[46],"easily":[47],"by":[48],"symbolic":[49],"calculations.":[50],"To":[51],"illustrate":[52],"efficiency":[54],"approach,":[57],"comparison":[59],"our":[61],"result":[62],"others":[64],"also":[66],"presented.":[67]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":1}],"updated_date":"2026-04-09T08:11:56.329763","created_date":"2025-10-10T00:00:00"}