{"id":"https://openalex.org/W7125498630","doi":"https://doi.org/10.3390/e28010131","title":"Some New Maximally Chaotic Discrete Maps","display_name":"Some New Maximally Chaotic Discrete Maps","publication_year":2026,"publication_date":"2026-01-22","ids":{"openalex":"https://openalex.org/W7125498630","doi":"https://doi.org/10.3390/e28010131","pmid":"https://pubmed.ncbi.nlm.nih.gov/41594038"},"language":"en","primary_location":{"id":"doi:10.3390/e28010131","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e28010131","pdf_url":"https://www.mdpi.com/1099-4300/28/1/131/pdf?version=1769091974","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj","pubmed"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/1099-4300/28/1/131/pdf?version=1769091974","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5030189822","display_name":"Hyojeong Choi","orcid":"https://orcid.org/0000-0003-2305-5111"},"institutions":[{"id":"https://openalex.org/I193775966","display_name":"Yonsei University","ror":"https://ror.org/01wjejq96","country_code":"KR","type":"education","lineage":["https://openalex.org/I193775966"]}],"countries":["KR"],"is_corresponding":false,"raw_author_name":"Hyojeong Choi","raw_affiliation_strings":["Department of Electrical and Electronic Engineering, Yonsei University, Seoul 03722, Republic of Korea"],"raw_orcid":"https://orcid.org/0000-0003-2305-5111","affiliations":[{"raw_affiliation_string":"Department of Electrical and Electronic Engineering, Yonsei University, Seoul 03722, Republic of Korea","institution_ids":["https://openalex.org/I193775966"]}]},{"author_position":"middle","author":{"id":null,"display_name":"Gangsan Kim","orcid":"https://orcid.org/0000-0002-3864-5379"},"institutions":[{"id":"https://openalex.org/I193775966","display_name":"Yonsei University","ror":"https://ror.org/01wjejq96","country_code":"KR","type":"education","lineage":["https://openalex.org/I193775966"]}],"countries":["KR"],"is_corresponding":false,"raw_author_name":"Gangsan Kim","raw_affiliation_strings":["Department of Electrical and Electronic Engineering, Yonsei University, Seoul 03722, Republic of Korea"],"raw_orcid":"https://orcid.org/0000-0002-3864-5379","affiliations":[{"raw_affiliation_string":"Department of Electrical and Electronic Engineering, Yonsei University, Seoul 03722, Republic of Korea","institution_ids":["https://openalex.org/I193775966"]}]},{"author_position":"middle","author":{"id":null,"display_name":"Hong-Yeop Song","orcid":"https://orcid.org/0000-0001-8764-9424"},"institutions":[{"id":"https://openalex.org/I193775966","display_name":"Yonsei University","ror":"https://ror.org/01wjejq96","country_code":"KR","type":"education","lineage":["https://openalex.org/I193775966"]}],"countries":["KR"],"is_corresponding":true,"raw_author_name":"Hong-Yeop Song","raw_affiliation_strings":["Department of Electrical and Electronic Engineering, Yonsei University, Seoul 03722, Republic of Korea"],"raw_orcid":"https://orcid.org/0000-0001-8764-9424","affiliations":[{"raw_affiliation_string":"Department of Electrical and Electronic Engineering, Yonsei University, Seoul 03722, Republic of Korea","institution_ids":["https://openalex.org/I193775966"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5113102354","display_name":"Sangung Shin","orcid":null},"institutions":[{"id":"https://openalex.org/I206093572","display_name":"Nexen (Canada)","ror":"https://ror.org/04zwxgg32","country_code":"CA","type":"company","lineage":["https://openalex.org/I206093572"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Sangung Shin","raw_affiliation_strings":["Department of C4I R&D Center, LIG Nex1, 333 Pangyo-ro, Bundang-gu, Seongnam-si 13488, Republic of Korea"],"raw_orcid":"https://orcid.org/0009-0008-1662-5327","affiliations":[{"raw_affiliation_string":"Department of C4I R&D Center, LIG Nex1, 333 Pangyo-ro, Bundang-gu, Seongnam-si 13488, Republic of Korea","institution_ids":["https://openalex.org/I206093572"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5123700147","display_name":"Chulho Lee","orcid":null},"institutions":[{"id":"https://openalex.org/I206093572","display_name":"Nexen (Canada)","ror":"https://ror.org/04zwxgg32","country_code":"CA","type":"company","lineage":["https://openalex.org/I206093572"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Chulho Lee","raw_affiliation_strings":["Department of C4I R&D Center, LIG Nex1, 333 Pangyo-ro, Bundang-gu, Seongnam-si 13488, Republic of Korea"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of C4I R&D Center, LIG Nex1, 333 Pangyo-ro, Bundang-gu, Seongnam-si 13488, Republic of Korea","institution_ids":["https://openalex.org/I206093572"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5093767705","display_name":"Hongjun Noh","orcid":"https://orcid.org/0000-0001-6138-742X"},"institutions":[{"id":"https://openalex.org/I206093572","display_name":"Nexen (Canada)","ror":"https://ror.org/04zwxgg32","country_code":"CA","type":"company","lineage":["https://openalex.org/I206093572"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Hongjun Noh","raw_affiliation_strings":["Department of C4I R&D Center, LIG Nex1, 333 Pangyo-ro, Bundang-gu, Seongnam-si 13488, Republic of Korea"],"raw_orcid":"https://orcid.org/0000-0001-6138-742X","affiliations":[{"raw_affiliation_string":"Department of C4I R&D Center, LIG Nex1, 333 Pangyo-ro, Bundang-gu, Seongnam-si 13488, Republic of Korea","institution_ids":["https://openalex.org/I206093572"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":6,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I193775966"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":25.6462,"has_fulltext":true,"cited_by_count":2,"citation_normalized_percentile":{"value":0.98839814,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":98,"max":99},"biblio":{"volume":"28","issue":"1","first_page":"131","last_page":"131"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10244","display_name":"Chaos control and synchronization","score":0.5175999999046326,"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":0.5175999999046326,"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/T11017","display_name":"Chaos-based Image/Signal Encryption","score":0.3538999855518341,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"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/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.05719999969005585,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/tent-map","display_name":"Tent map","score":0.6772000193595886},{"id":"https://openalex.org/keywords/bijection","display_name":"Bijection","score":0.6371999979019165},{"id":"https://openalex.org/keywords/permutation","display_name":"Permutation (music)","score":0.46129998564720154},{"id":"https://openalex.org/keywords/entropy","display_name":"Entropy (arrow of time)","score":0.44859999418258667},{"id":"https://openalex.org/keywords/lyapunov-exponent","display_name":"Lyapunov exponent","score":0.4406999945640564},{"id":"https://openalex.org/keywords/skew","display_name":"Skew","score":0.4345000088214874},{"id":"https://openalex.org/keywords/chaotic","display_name":"Chaotic","score":0.39809998869895935},{"id":"https://openalex.org/keywords/chaotic-map","display_name":"Chaotic map","score":0.39410001039505005}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.791100025177002},{"id":"https://openalex.org/C15998118","wikidata":"https://www.wikidata.org/wiki/Q11319849","display_name":"Tent map","level":3,"score":0.6772000193595886},{"id":"https://openalex.org/C24424167","wikidata":"https://www.wikidata.org/wiki/Q180907","display_name":"Bijection","level":2,"score":0.6371999979019165},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.47679999470710754},{"id":"https://openalex.org/C21308566","wikidata":"https://www.wikidata.org/wiki/Q7169365","display_name":"Permutation (music)","level":2,"score":0.46129998564720154},{"id":"https://openalex.org/C106301342","wikidata":"https://www.wikidata.org/wiki/Q4117933","display_name":"Entropy (arrow of time)","level":2,"score":0.44859999418258667},{"id":"https://openalex.org/C191544260","wikidata":"https://www.wikidata.org/wiki/Q1238630","display_name":"Lyapunov exponent","level":3,"score":0.4406999945640564},{"id":"https://openalex.org/C43711488","wikidata":"https://www.wikidata.org/wiki/Q7534783","display_name":"Skew","level":2,"score":0.4345000088214874},{"id":"https://openalex.org/C2777052490","wikidata":"https://www.wikidata.org/wiki/Q5072826","display_name":"Chaotic","level":2,"score":0.39809998869895935},{"id":"https://openalex.org/C2983793180","wikidata":"https://www.wikidata.org/wiki/Q166314","display_name":"Chaotic map","level":3,"score":0.39410001039505005},{"id":"https://openalex.org/C207390915","wikidata":"https://www.wikidata.org/wiki/Q1230525","display_name":"Divergence (linguistics)","level":2,"score":0.3813000023365021},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3646000027656555},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.3605000078678131},{"id":"https://openalex.org/C48372109","wikidata":"https://www.wikidata.org/wiki/Q3913","display_name":"Binary number","level":2,"score":0.35350000858306885},{"id":"https://openalex.org/C205330730","wikidata":"https://www.wikidata.org/wiki/Q587182","display_name":"Logistic map","level":3,"score":0.35019999742507935},{"id":"https://openalex.org/C55689738","wikidata":"https://www.wikidata.org/wiki/Q15963867","display_name":"Discrete time and continuous time","level":2,"score":0.31859999895095825},{"id":"https://openalex.org/C9679016","wikidata":"https://www.wikidata.org/wiki/Q1417473","display_name":"Principle of maximum entropy","level":2,"score":0.3089999854564667},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.28209999203681946},{"id":"https://openalex.org/C60640748","wikidata":"https://www.wikidata.org/wiki/Q2337858","display_name":"Lyapunov function","level":3,"score":0.2800000011920929},{"id":"https://openalex.org/C2781402108","wikidata":"https://www.wikidata.org/wiki/Q1642961","display_name":"H\u00e9non map","level":3,"score":0.27230000495910645},{"id":"https://openalex.org/C2780388253","wikidata":"https://www.wikidata.org/wiki/Q5421508","display_name":"Exponent","level":2,"score":0.27140000462532043},{"id":"https://openalex.org/C165263651","wikidata":"https://www.wikidata.org/wiki/Q2443460","display_name":"Topological conjugacy","level":2,"score":0.26750001311302185}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.3390/e28010131","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e28010131","pdf_url":"https://www.mdpi.com/1099-4300/28/1/131/pdf?version=1769091974","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},{"id":"pmid:41594038","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/41594038","pdf_url":null,"source":{"id":"https://openalex.org/S4306525036","display_name":"PubMed","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy (Basel, Switzerland)","raw_type":null},{"id":"pmh:oai:doaj.org/article:a532ef1afcbe4c8a87385f4d148111ea","is_oa":true,"landing_page_url":"https://doaj.org/article/a532ef1afcbe4c8a87385f4d148111ea","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy, Vol 28, Iss 1, p 131 (2026)","raw_type":"article"},{"id":"pmh:oai:pubmedcentral.nih.gov:12839921","is_oa":true,"landing_page_url":"https://pmc.ncbi.nlm.nih.gov/articles/PMC12839921/","pdf_url":null,"source":{"id":"https://openalex.org/S2764455111","display_name":"PubMed Central","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy (Basel)","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/e28010131","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e28010131","pdf_url":"https://www.mdpi.com/1099-4300/28/1/131/pdf?version=1769091974","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},"sustainable_development_goals":[{"score":0.49347683787345886,"id":"https://metadata.un.org/sdg/11","display_name":"Sustainable cities and communities"}],"awards":[],"funders":[{"id":"https://openalex.org/F4320334874","display_name":"Defense Acquisition Program Administration","ror":"https://ror.org/04bjg9m96"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7125498630.pdf","grobid_xml":"https://content.openalex.org/works/W7125498630.grobid-xml"},"referenced_works_count":28,"referenced_works":["https://openalex.org/W1775527311","https://openalex.org/W2001426628","https://openalex.org/W2014683958","https://openalex.org/W2015287612","https://openalex.org/W2016367480","https://openalex.org/W2077204677","https://openalex.org/W2102688401","https://openalex.org/W2166934960","https://openalex.org/W2901046412","https://openalex.org/W2904052140","https://openalex.org/W2905420870","https://openalex.org/W2946827305","https://openalex.org/W2969998594","https://openalex.org/W2976764752","https://openalex.org/W2980608281","https://openalex.org/W3120522579","https://openalex.org/W3120566145","https://openalex.org/W3126839004","https://openalex.org/W3131964039","https://openalex.org/W3166013009","https://openalex.org/W3175033625","https://openalex.org/W4312502461","https://openalex.org/W4312502759","https://openalex.org/W4312632710","https://openalex.org/W4383113562","https://openalex.org/W4389751739","https://openalex.org/W4402569701","https://openalex.org/W4412592833"],"related_works":[],"abstract_inverted_index":{"In":[0,106],"this":[1,42],"paper,":[2],"we":[3,44,141],"first":[4],"prove":[5,55,71],"(Theorem":[6,57,73],"1)":[7,48,110],"that":[8,31,56,72,134],"any":[9],"two":[10],"inputs":[11],"producing":[12],"the":[13,27,59,75,81,91,99,111,116,122,126,135,144,149,154,160,167],"same":[14,28],"output":[15,161],"in":[16],"a":[17,49,63],"symmetric":[18],"pair":[19],"of":[20,80,159,175],"discrete":[21,51,76],"skew":[22],"tent":[23],"maps":[24,97,124],"always":[25],"have":[26],"parity,":[29],"meaning":[30],"they":[32],"are":[33],"either":[34],"both":[35,38],"even":[36],"or":[37],"odd.":[39],"Building":[40],"on":[41],"property,":[43],"then":[45],"propose":[46],"(Definition":[47],"new":[50],"chaotic":[52,119],"map":[53,61,83,113,137],"and":[54,156,172],"2)":[58],"proposed":[60,82,112,136],"is":[62,84,138],"bijection":[64],"for":[65],"all":[66,95],"control":[67],"parameters.":[68],"We":[69,152,164],"further":[70,132],"3)":[74],"Lyapunov":[77],"exponent":[78],"(dLE)":[79],"not":[85],"only":[86],"positive":[87],"but":[88],"also":[89,165],"approaches":[90],"maximum":[92],"value":[93],"among":[94,121],"permutation":[96,123,157],"over":[98,125],"integers":[100,127],"{0,1,\u2026,2m\u22121}":[101],"as":[102],"m":[103],"gets":[104],"larger.":[105],"other":[107],"words,":[108],"(Corollary":[109],"asymptotically":[114],"achieves":[115],"highest":[117],"possible":[118],"divergence":[120],"{0,1,\u2026,2m\u22121}.":[128],"To":[129],"provide":[130],"some":[131,146,176],"evidence":[133],"highly":[139],"chaotic,":[140],"present":[142],"at":[143],"end":[145],"results":[147,171],"from":[148],"numerical":[150],"experiments.":[151],"calculate":[153],"approximation":[155],"entropy":[158],"integer":[162],"sequences.":[163,179],"show":[166],"NIST":[168],"SP800-22":[169],"tests":[170],"correlation":[173],"properties":[174],"derived":[177],"binary":[178]},"counts_by_year":[{"year":2026,"cited_by_count":2}],"updated_date":"2026-06-15T08:34:33.830935","created_date":"2026-01-24T00:00:00"}