{"id":"https://openalex.org/W2114259953","doi":"https://doi.org/10.1142/s0218127414300134","title":"On Analytical Routes to Chaos in Nonlinear Systems","display_name":"On Analytical Routes to Chaos in Nonlinear Systems","publication_year":2014,"publication_date":"2014-04-01","ids":{"openalex":"https://openalex.org/W2114259953","doi":"https://doi.org/10.1142/s0218127414300134","mag":"2114259953"},"language":"en","primary_location":{"id":"doi:10.1142/s0218127414300134","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127414300134","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/A5076270155","display_name":"Albert C. J. Luo","orcid":"https://orcid.org/0000-0001-8208-6108"},"institutions":[{"id":"https://openalex.org/I87547150","display_name":"Southern Illinois University Edwardsville","ror":"https://ror.org/04cqs5j56","country_code":"US","type":"education","lineage":["https://openalex.org/I2801502357","https://openalex.org/I87547150"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Albert C. J. Luo","raw_affiliation_strings":["Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL 62026-1805, USA","Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL 62026-1805, USA#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL 62026-1805, USA","institution_ids":["https://openalex.org/I87547150"]},{"raw_affiliation_string":"Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL 62026-1805, USA#TAB#","institution_ids":["https://openalex.org/I87547150"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5076270155"],"corresponding_institution_ids":["https://openalex.org/I87547150"],"apc_list":null,"apc_paid":null,"fwci":1.6447,"has_fulltext":false,"cited_by_count":24,"citation_normalized_percentile":{"value":0.8447787,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":98},"biblio":{"volume":"24","issue":"04","first_page":"1430013","last_page":"1430013"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10244","display_name":"Chaos control and synchronization","score":0.9990000128746033,"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/T10244","display_name":"Chaos control and synchronization","score":0.9990000128746033,"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/T11261","display_name":"Quantum chaos and dynamical systems","score":0.9962000250816345,"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/T11602","display_name":"Magnetic Bearings and Levitation Dynamics","score":0.9904000163078308,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/harmonic-balance","display_name":"Harmonic balance","score":0.7472637891769409},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.7034085988998413},{"id":"https://openalex.org/keywords/dynamical-systems-theory","display_name":"Dynamical systems theory","score":0.5997358560562134},{"id":"https://openalex.org/keywords/method-of-averaging","display_name":"Method of averaging","score":0.5606421828269958},{"id":"https://openalex.org/keywords/perturbation","display_name":"Perturbation (astronomy)","score":0.5591729879379272},{"id":"https://openalex.org/keywords/bifurcation","display_name":"Bifurcation","score":0.53675776720047},{"id":"https://openalex.org/keywords/chaos","display_name":"CHAOS (operating system)","score":0.5261083245277405},{"id":"https://openalex.org/keywords/control-theory","display_name":"Control theory (sociology)","score":0.42418372631073},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.42289793491363525},{"id":"https://openalex.org/keywords/motion","display_name":"Motion (physics)","score":0.42285674810409546},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4177548289299011},{"id":"https://openalex.org/keywords/periodic-function","display_name":"Periodic function","score":0.4145393669605255},{"id":"https://openalex.org/keywords/nonlinear-dynamical-systems","display_name":"Nonlinear dynamical systems","score":0.41023126244544983},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3496849536895752},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.2794708013534546},{"id":"https://openalex.org/keywords/classical-mechanics","display_name":"Classical mechanics","score":0.2297898530960083},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.2031230628490448}],"concepts":[{"id":"https://openalex.org/C121758426","wikidata":"https://www.wikidata.org/wiki/Q5659248","display_name":"Harmonic balance","level":3,"score":0.7472637891769409},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.7034085988998413},{"id":"https://openalex.org/C79379906","wikidata":"https://www.wikidata.org/wiki/Q3174497","display_name":"Dynamical systems theory","level":2,"score":0.5997358560562134},{"id":"https://openalex.org/C5555538","wikidata":"https://www.wikidata.org/wiki/Q6823703","display_name":"Method of averaging","level":3,"score":0.5606421828269958},{"id":"https://openalex.org/C177918212","wikidata":"https://www.wikidata.org/wiki/Q803623","display_name":"Perturbation (astronomy)","level":2,"score":0.5591729879379272},{"id":"https://openalex.org/C2781349735","wikidata":"https://www.wikidata.org/wiki/Q4904824","display_name":"Bifurcation","level":3,"score":0.53675776720047},{"id":"https://openalex.org/C2779374083","wikidata":"https://www.wikidata.org/wiki/Q5011038","display_name":"CHAOS (operating system)","level":2,"score":0.5261083245277405},{"id":"https://openalex.org/C47446073","wikidata":"https://www.wikidata.org/wiki/Q5165890","display_name":"Control theory (sociology)","level":3,"score":0.42418372631073},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.42289793491363525},{"id":"https://openalex.org/C104114177","wikidata":"https://www.wikidata.org/wiki/Q79782","display_name":"Motion (physics)","level":2,"score":0.42285674810409546},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4177548289299011},{"id":"https://openalex.org/C98644592","wikidata":"https://www.wikidata.org/wiki/Q184743","display_name":"Periodic function","level":2,"score":0.4145393669605255},{"id":"https://openalex.org/C2983030100","wikidata":"https://www.wikidata.org/wiki/Q638328","display_name":"Nonlinear dynamical systems","level":3,"score":0.41023126244544983},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3496849536895752},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2794708013534546},{"id":"https://openalex.org/C74650414","wikidata":"https://www.wikidata.org/wiki/Q11397","display_name":"Classical mechanics","level":1,"score":0.2297898530960083},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.2031230628490448},{"id":"https://openalex.org/C2775924081","wikidata":"https://www.wikidata.org/wiki/Q55608371","display_name":"Control (management)","level":2,"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/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0},{"id":"https://openalex.org/C38652104","wikidata":"https://www.wikidata.org/wiki/Q3510521","display_name":"Computer security","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1142/s0218127414300134","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127414300134","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":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":75,"referenced_works":["https://openalex.org/W7313025","https://openalex.org/W89702005","https://openalex.org/W403112863","https://openalex.org/W562652402","https://openalex.org/W569192763","https://openalex.org/W579093851","https://openalex.org/W621740939","https://openalex.org/W1521581363","https://openalex.org/W1546721432","https://openalex.org/W1556059214","https://openalex.org/W1638842865","https://openalex.org/W1966535862","https://openalex.org/W1972931998","https://openalex.org/W1977111161","https://openalex.org/W1978001141","https://openalex.org/W1979400963","https://openalex.org/W1982280336","https://openalex.org/W1983654567","https://openalex.org/W1985148757","https://openalex.org/W1986177202","https://openalex.org/W1987882966","https://openalex.org/W2010834348","https://openalex.org/W2011406684","https://openalex.org/W2011790690","https://openalex.org/W2013972821","https://openalex.org/W2018895750","https://openalex.org/W2020336658","https://openalex.org/W2023490801","https://openalex.org/W2025178995","https://openalex.org/W2029280483","https://openalex.org/W2030465009","https://openalex.org/W2032215782","https://openalex.org/W2035415155","https://openalex.org/W2041932726","https://openalex.org/W2045704042","https://openalex.org/W2050624985","https://openalex.org/W2056529601","https://openalex.org/W2059146087","https://openalex.org/W2059266872","https://openalex.org/W2063301097","https://openalex.org/W2065325403","https://openalex.org/W2065412553","https://openalex.org/W2068003521","https://openalex.org/W2069641344","https://openalex.org/W2071623546","https://openalex.org/W2071670511","https://openalex.org/W2073589779","https://openalex.org/W2078830190","https://openalex.org/W2081368350","https://openalex.org/W2083633830","https://openalex.org/W2085713131","https://openalex.org/W2092512369","https://openalex.org/W2094458350","https://openalex.org/W2114628938","https://openalex.org/W2119368478","https://openalex.org/W2124484443","https://openalex.org/W2134713733","https://openalex.org/W2150625252","https://openalex.org/W2295169085","https://openalex.org/W2315938601","https://openalex.org/W2318999407","https://openalex.org/W2325589530","https://openalex.org/W2325936659","https://openalex.org/W2328059447","https://openalex.org/W2496740228","https://openalex.org/W2575434776","https://openalex.org/W2584984113","https://openalex.org/W2903767847","https://openalex.org/W3134182129","https://openalex.org/W3141786155","https://openalex.org/W4229552796","https://openalex.org/W4245643085","https://openalex.org/W4252911570","https://openalex.org/W4297939945","https://openalex.org/W4300681176"],"related_works":["https://openalex.org/W897029668","https://openalex.org/W2115722512","https://openalex.org/W2990483082","https://openalex.org/W2035599257","https://openalex.org/W3185784018","https://openalex.org/W1520853953","https://openalex.org/W1966535862","https://openalex.org/W562652402","https://openalex.org/W2894938230","https://openalex.org/W2114259953"],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"the":[3,23,28,50,62,66,81],"analytical":[4,25,67,82,93],"methods":[5,26,44,56],"for":[6,84],"approximate":[7,43,55],"solutions":[8,68,83],"of":[9,36,42,52,69,96],"periodic":[10,70],"motions":[11,71],"to":[12,87,99],"chaos":[13,100],"in":[14,45,72],"nonlinear":[15,73],"dynamical":[16,74],"systems":[17,75],"are":[18,22,78,90,101],"reviewed.":[19],"Briefly":[20],"discussed":[21],"traditional":[24],"including":[27],"Lagrange":[29],"stand":[30],"form,":[31],"perturbation":[32],"methods,":[33],"and":[34,49,80],"method":[35],"averaging.":[37],"A":[38],"brief":[39],"literature":[40],"survey":[41],"application":[46],"is":[47,57],"completed,":[48],"weakness":[51],"current":[53],"existing":[54],"also":[58],"discussed.":[59,91],"Based":[60],"on":[61],"generalized":[63],"harmonic":[64],"balance,":[65],"with/without":[76],"time-delay":[77],"reviewed,":[79],"period-m":[85],"motion":[86,89,98],"quasi-periodic":[88],"The":[92],"bifurcation":[94],"trees":[95],"period-1":[97],"presented":[102],"as":[103],"an":[104],"application.":[105]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":4},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":4},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":4},{"year":2014,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}
