{"id":"https://openalex.org/W4382402519","doi":"https://doi.org/10.3390/sym15071311","title":"A Family of 1D Chaotic Maps without Equilibria","display_name":"A Family of 1D Chaotic Maps without Equilibria","publication_year":2023,"publication_date":"2023-06-27","ids":{"openalex":"https://openalex.org/W4382402519","doi":"https://doi.org/10.3390/sym15071311"},"language":"en","primary_location":{"id":"doi:10.3390/sym15071311","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15071311","pdf_url":"https://www.mdpi.com/2073-8994/15/7/1311/pdf?version=1687854664","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"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":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/15/7/1311/pdf?version=1687854664","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5034132103","display_name":"Marcin Lawnik","orcid":"https://orcid.org/0000-0002-0235-0878"},"institutions":[{"id":"https://openalex.org/I119004910","display_name":"Silesian University of Technology","ror":"https://ror.org/02dyjk442","country_code":"PL","type":"education","lineage":["https://openalex.org/I119004910"]}],"countries":["PL"],"is_corresponding":true,"raw_author_name":"Marcin Lawnik","raw_affiliation_strings":["Department of Mathematics Applications and Methods for Artificial Intelligence, Faculty of Applied Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland"],"raw_orcid":"https://orcid.org/0000-0002-0235-0878","affiliations":[{"raw_affiliation_string":"Department of Mathematics Applications and Methods for Artificial Intelligence, Faculty of Applied Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland","institution_ids":["https://openalex.org/I119004910"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5055872900","display_name":"Lazaros Moysis","orcid":"https://orcid.org/0000-0002-5652-2532"},"institutions":[{"id":"https://openalex.org/I21370196","display_name":"Aristotle University of Thessaloniki","ror":"https://ror.org/02j61yw88","country_code":"GR","type":"education","lineage":["https://openalex.org/I21370196"]},{"id":"https://openalex.org/I89506807","display_name":"University of Western Macedonia","ror":"https://ror.org/00a5pe906","country_code":"GR","type":"education","lineage":["https://openalex.org/I89506807"]}],"countries":["GR"],"is_corresponding":true,"raw_author_name":"Lazaros Moysis","raw_affiliation_strings":["Department of Mechanical Engineering, University of Western Macedonia, 50100 Kozani, Greece","Laboratory of Nonlinear Systems-Circuits & Complexity, Physics Department, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece"],"raw_orcid":"https://orcid.org/0000-0002-5652-2532","affiliations":[{"raw_affiliation_string":"Department of Mechanical Engineering, University of Western Macedonia, 50100 Kozani, Greece","institution_ids":["https://openalex.org/I89506807"]},{"raw_affiliation_string":"Laboratory of Nonlinear Systems-Circuits & Complexity, Physics Department, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece","institution_ids":["https://openalex.org/I21370196"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5061639181","display_name":"Christos Volos","orcid":"https://orcid.org/0000-0001-8763-7255"},"institutions":[{"id":"https://openalex.org/I21370196","display_name":"Aristotle University of Thessaloniki","ror":"https://ror.org/02j61yw88","country_code":"GR","type":"education","lineage":["https://openalex.org/I21370196"]}],"countries":["GR"],"is_corresponding":false,"raw_author_name":"Christos Volos","raw_affiliation_strings":["Laboratory of Nonlinear Systems-Circuits & Complexity, Physics Department, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece"],"raw_orcid":"https://orcid.org/0000-0001-8763-7255","affiliations":[{"raw_affiliation_string":"Laboratory of Nonlinear Systems-Circuits & Complexity, Physics Department, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece","institution_ids":["https://openalex.org/I21370196"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5034132103","https://openalex.org/A5055872900"],"corresponding_institution_ids":["https://openalex.org/I119004910","https://openalex.org/I21370196","https://openalex.org/I89506807"],"apc_list":{"value":2400,"currency":"CHF","value_usd":2896},"apc_paid":{"value":2400,"currency":"CHF","value_usd":2896},"fwci":1.4247,"has_fulltext":true,"cited_by_count":15,"citation_normalized_percentile":{"value":0.81579489,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":96,"max":99},"biblio":{"volume":"15","issue":"7","first_page":"1311","last_page":"1311"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11017","display_name":"Chaos-based Image/Signal Encryption","score":0.9991999864578247,"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"}},"topics":[{"id":"https://openalex.org/T11017","display_name":"Chaos-based Image/Signal Encryption","score":0.9991999864578247,"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/T10244","display_name":"Chaos control and synchronization","score":0.9965999722480774,"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/T12162","display_name":"Cellular Automata and Applications","score":0.9811999797821045,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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/chaotic","display_name":"Chaotic","score":0.8606315851211548},{"id":"https://openalex.org/keywords/parameterized-complexity","display_name":"Parameterized complexity","score":0.656954288482666},{"id":"https://openalex.org/keywords/attractor","display_name":"Attractor","score":0.6423726677894592},{"id":"https://openalex.org/keywords/piecewise","display_name":"Piecewise","score":0.5866005420684814},{"id":"https://openalex.org/keywords/chaotic-map","display_name":"Chaotic map","score":0.5674870014190674},{"id":"https://openalex.org/keywords/generator","display_name":"Generator (circuit theory)","score":0.552496075630188},{"id":"https://openalex.org/keywords/constraint","display_name":"Constraint (computer-aided design)","score":0.5384542346000671},{"id":"https://openalex.org/keywords/piecewise-linear-function","display_name":"Piecewise linear function","score":0.46621543169021606},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.4620721936225891},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.4573367238044739},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.45485734939575195},{"id":"https://openalex.org/keywords/fractal","display_name":"Fractal","score":0.43340739607810974},{"id":"https://openalex.org/keywords/topology","display_name":"Topology (electrical circuits)","score":0.41479790210723877},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.3274208605289459},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3202930986881256},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.26022475957870483},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.20055049657821655},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.15897870063781738},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.10387128591537476},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.09742739796638489}],"concepts":[{"id":"https://openalex.org/C2777052490","wikidata":"https://www.wikidata.org/wiki/Q5072826","display_name":"Chaotic","level":2,"score":0.8606315851211548},{"id":"https://openalex.org/C165464430","wikidata":"https://www.wikidata.org/wiki/Q1570441","display_name":"Parameterized complexity","level":2,"score":0.656954288482666},{"id":"https://openalex.org/C164380108","wikidata":"https://www.wikidata.org/wiki/Q507187","display_name":"Attractor","level":2,"score":0.6423726677894592},{"id":"https://openalex.org/C164660894","wikidata":"https://www.wikidata.org/wiki/Q2037833","display_name":"Piecewise","level":2,"score":0.5866005420684814},{"id":"https://openalex.org/C2983793180","wikidata":"https://www.wikidata.org/wiki/Q166314","display_name":"Chaotic map","level":3,"score":0.5674870014190674},{"id":"https://openalex.org/C2780992000","wikidata":"https://www.wikidata.org/wiki/Q17016113","display_name":"Generator (circuit theory)","level":3,"score":0.552496075630188},{"id":"https://openalex.org/C2776036281","wikidata":"https://www.wikidata.org/wiki/Q48769818","display_name":"Constraint (computer-aided design)","level":2,"score":0.5384542346000671},{"id":"https://openalex.org/C17095337","wikidata":"https://www.wikidata.org/wiki/Q2375229","display_name":"Piecewise linear function","level":2,"score":0.46621543169021606},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.4620721936225891},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4573367238044739},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.45485734939575195},{"id":"https://openalex.org/C40636538","wikidata":"https://www.wikidata.org/wiki/Q81392","display_name":"Fractal","level":2,"score":0.43340739607810974},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.41479790210723877},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3274208605289459},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3202930986881256},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.26022475957870483},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.20055049657821655},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.15897870063781738},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.10387128591537476},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.09742739796638489},{"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/C163258240","wikidata":"https://www.wikidata.org/wiki/Q25342","display_name":"Power (physics)","level":2,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym15071311","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15071311","pdf_url":"https://www.mdpi.com/2073-8994/15/7/1311/pdf?version=1687854664","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"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":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:87ce5ad2e91f4a2aa828dfa98d92bec1","is_oa":false,"landing_page_url":"https://doaj.org/article/87ce5ad2e91f4a2aa828dfa98d92bec1","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 15, Iss 7, p 1311 (2023)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/15/7/1311/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym15071311","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"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":"Symmetry","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/sym15071311","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15071311","pdf_url":"https://www.mdpi.com/2073-8994/15/7/1311/pdf?version=1687854664","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"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":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[{"display_name":"Reduced inequalities","score":0.4000000059604645,"id":"https://metadata.un.org/sdg/10"}],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4382402519.pdf","grobid_xml":"https://content.openalex.org/works/W4382402519.grobid-xml"},"referenced_works_count":37,"referenced_works":["https://openalex.org/W1517403092","https://openalex.org/W1551822527","https://openalex.org/W2004821187","https://openalex.org/W2029998950","https://openalex.org/W2043117900","https://openalex.org/W2516950210","https://openalex.org/W2565482338","https://openalex.org/W2585787607","https://openalex.org/W2613566302","https://openalex.org/W2802907414","https://openalex.org/W2867359686","https://openalex.org/W2901263295","https://openalex.org/W2921716797","https://openalex.org/W2946943331","https://openalex.org/W2990382114","https://openalex.org/W2991798982","https://openalex.org/W3003027562","https://openalex.org/W3009527247","https://openalex.org/W3015780933","https://openalex.org/W3017678437","https://openalex.org/W3025587973","https://openalex.org/W3175033625","https://openalex.org/W3178866630","https://openalex.org/W3209425238","https://openalex.org/W3213404197","https://openalex.org/W4214521195","https://openalex.org/W4250883246","https://openalex.org/W4285175747","https://openalex.org/W4303627750","https://openalex.org/W4310452265","https://openalex.org/W4315628355","https://openalex.org/W4316467626","https://openalex.org/W4318576820","https://openalex.org/W4323663555","https://openalex.org/W4324359158","https://openalex.org/W6760440480","https://openalex.org/W6770062310"],"related_works":["https://openalex.org/W2064165820","https://openalex.org/W2087352669","https://openalex.org/W2133947431","https://openalex.org/W2490196280","https://openalex.org/W3182827720","https://openalex.org/W1596838019","https://openalex.org/W3208509670","https://openalex.org/W2019185558","https://openalex.org/W2109304297","https://openalex.org/W2100864966"],"abstract_inverted_index":{"In":[0],"this":[1],"work,":[2],"a":[3,36,94],"family":[4,12,56],"of":[5,13,25,41,62,69,89],"piecewise":[6],"chaotic":[7,53,70,82],"maps":[8,14,46,71,92],"is":[9,15,98],"proposed.":[10],"This":[11,55],"parameterized":[16],"by":[17],"the":[18,26,39,44,60,79,87,90],"nonlinear":[19],"functions":[20],"used":[21],"for":[22],"each":[23,42],"piece":[24],"mapping,":[27],"which":[28],"can":[29,51],"be":[30],"either":[31],"symmetric":[32],"or":[33],"non-symmetric.":[34],"Applying":[35],"constraint":[37],"on":[38],"shape":[40],"piece,":[43],"generated":[45],"have":[47],"no":[48],"equilibria":[49],"and":[50,83],"showcase":[52],"behavior.":[54,85],"thus":[57],"belongs":[58],"to":[59,93],"category":[61],"systems":[63],"with":[64],"hidden":[65],"attractors.":[66],"Numerous":[67],"examples":[68],"are":[72],"provided,":[73],"showcasing":[74],"fractal-like,":[75],"symmetrical":[76],"patterns":[77],"at":[78],"interchange":[80],"between":[81],"non-chaotic":[84],"Moreover,":[86],"application":[88],"proposed":[91],"pseudorandom":[95],"bit":[96],"generator":[97],"successfully":[99],"performed.":[100]},"counts_by_year":[{"year":2026,"cited_by_count":3},{"year":2025,"cited_by_count":6},{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":3}],"updated_date":"2026-08-21T09:56:20.448147","created_date":"2025-10-10T00:00:00"}