{"id":"https://openalex.org/W2021462989","doi":"https://doi.org/10.1142/s0218127414501089","title":"Chaos for Cosine Operator Functions Generated by Shifts","display_name":"Chaos for Cosine Operator Functions Generated by Shifts","publication_year":2014,"publication_date":"2014-09-01","ids":{"openalex":"https://openalex.org/W2021462989","doi":"https://doi.org/10.1142/s0218127414501089","mag":"2021462989"},"language":"en","primary_location":{"id":"doi:10.1142/s0218127414501089","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127414501089","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/A5055201977","display_name":"Chung-Chuan Chen","orcid":"https://orcid.org/0000-0003-4297-3305"},"institutions":[{"id":"https://openalex.org/I34574549","display_name":"National Taichung University of Education","ror":"https://ror.org/006tnfe02","country_code":"TW","type":"education","lineage":["https://openalex.org/I34574549"]}],"countries":["TW"],"is_corresponding":true,"raw_author_name":"Chung-Chuan Chen","raw_affiliation_strings":["Department of Mathematics Education, National Taichung University of Education, Taichung, 403, Taiwan"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics Education, National Taichung University of Education, Taichung, 403, Taiwan","institution_ids":["https://openalex.org/I34574549"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5055201977"],"corresponding_institution_ids":["https://openalex.org/I34574549"],"apc_list":null,"apc_paid":null,"fwci":0.9266,"has_fulltext":false,"cited_by_count":13,"citation_normalized_percentile":{"value":0.72707059,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"24","issue":"09","first_page":"1450108","last_page":"1450108"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10884","display_name":"Holomorphic and Operator Theory","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/T10884","display_name":"Holomorphic and Operator Theory","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/T11022","display_name":"Spectral Theory in Mathematical Physics","score":0.9995999932289124,"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/T12037","display_name":"Algebraic and Geometric Analysis","score":0.9993000030517578,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/chaotic","display_name":"Chaotic","score":0.8121618032455444},{"id":"https://openalex.org/keywords/trigonometric-functions","display_name":"Trigonometric functions","score":0.7402352690696716},{"id":"https://openalex.org/keywords/operator","display_name":"Operator (biology)","score":0.706974983215332},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6635816693305969},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.4551430344581604},{"id":"https://openalex.org/keywords/shift-operator","display_name":"Shift operator","score":0.45148569345474243},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.4088042378425598},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.34298190474510193},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.20442938804626465},{"id":"https://openalex.org/keywords/compact-operator","display_name":"Compact operator","score":0.18103715777397156},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.08797401189804077},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.056778788566589355}],"concepts":[{"id":"https://openalex.org/C2777052490","wikidata":"https://www.wikidata.org/wiki/Q5072826","display_name":"Chaotic","level":2,"score":0.8121618032455444},{"id":"https://openalex.org/C178009071","wikidata":"https://www.wikidata.org/wiki/Q93344","display_name":"Trigonometric functions","level":2,"score":0.7402352690696716},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.706974983215332},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6635816693305969},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4551430344581604},{"id":"https://openalex.org/C65630602","wikidata":"https://www.wikidata.org/wiki/Q2278680","display_name":"Shift operator","level":4,"score":0.45148569345474243},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4088042378425598},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.34298190474510193},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.20442938804626465},{"id":"https://openalex.org/C79013869","wikidata":"https://www.wikidata.org/wiki/Q1780743","display_name":"Compact operator","level":3,"score":0.18103715777397156},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.08797401189804077},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.056778788566589355},{"id":"https://openalex.org/C2778029271","wikidata":"https://www.wikidata.org/wiki/Q5421931","display_name":"Extension (predicate logic)","level":2,"score":0.0},{"id":"https://openalex.org/C158448853","wikidata":"https://www.wikidata.org/wiki/Q425218","display_name":"Repressor","level":4,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"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/C86339819","wikidata":"https://www.wikidata.org/wiki/Q407384","display_name":"Transcription factor","level":3,"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/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1142/s0218127414501089","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127414501089","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":[{"id":"https://openalex.org/F4320321040","display_name":"National Science Council","ror":"https://ror.org/02kv4zf79"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":16,"referenced_works":["https://openalex.org/W248771341","https://openalex.org/W626893493","https://openalex.org/W1549649164","https://openalex.org/W1650352370","https://openalex.org/W1973822038","https://openalex.org/W1984136276","https://openalex.org/W1992908827","https://openalex.org/W1994129707","https://openalex.org/W2006541944","https://openalex.org/W2009303370","https://openalex.org/W2022662265","https://openalex.org/W2058118875","https://openalex.org/W2062332991","https://openalex.org/W2086474835","https://openalex.org/W2146561617","https://openalex.org/W4240509971"],"related_works":["https://openalex.org/W268247387","https://openalex.org/W2092836648","https://openalex.org/W2001119360","https://openalex.org/W2351515196","https://openalex.org/W2063570710","https://openalex.org/W2079191368","https://openalex.org/W2050709486","https://openalex.org/W2391621545","https://openalex.org/W1017172903","https://openalex.org/W2364824809"],"abstract_inverted_index":{"Let":[0],"1":[1],"\u2264":[2],"p":[3,24],"<":[4],"\u221e.":[5],"We":[6],"give":[7],"the":[8],"sufficient":[9],"and":[10,38],"necessary":[11],"condition":[12],"for":[13],"cosine":[14,32],"operator":[15,33],"functions,":[16],"generated":[17],"by":[18],"bilateral":[19],"weighted":[20,42],"shifts":[21],"on":[22],"\u2113":[23],"(\u2124),":[25],"to":[26],"be":[27],"chaotic.":[28,45],"Moreover,":[29],"such":[30],"a":[31],"function":[34],"is":[35,44],"chaotic":[36],"if,":[37,40],"only":[39],"its":[41],"shift":[43]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}