{"id":"https://openalex.org/W2806054350","doi":"https://doi.org/10.3390/sym10060188","title":"Stability of the Non-Hyperbolic Zero Equilibrium of Two Close-to-Symmetric Systems of Difference Equations with Exponential Terms","display_name":"Stability of the Non-Hyperbolic Zero Equilibrium of Two Close-to-Symmetric Systems of Difference Equations with Exponential Terms","publication_year":2018,"publication_date":"2018-05-31","ids":{"openalex":"https://openalex.org/W2806054350","doi":"https://doi.org/10.3390/sym10060188","mag":"2806054350"},"language":"en","primary_location":{"id":"doi:10.3390/sym10060188","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym10060188","pdf_url":"https://www.mdpi.com/2073-8994/10/6/188/pdf?version=1527759214","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/2073-8994/10/6/188/pdf?version=1527759214","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5003279948","display_name":"Chrysoula Mylona","orcid":null},"institutions":[{"id":"https://openalex.org/I147962203","display_name":"Democritus University of Thrace","ror":"https://ror.org/03bfqnx40","country_code":"GR","type":"education","lineage":["https://openalex.org/I147962203"]}],"countries":["GR"],"is_corresponding":false,"raw_author_name":"Chrysoula Mylona","raw_affiliation_strings":["School of Engineering, Democritus University of Thrace, 67100 Xanthi, Greece"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Engineering, Democritus University of Thrace, 67100 Xanthi, Greece","institution_ids":["https://openalex.org/I147962203"]}]},{"author_position":"middle","author":{"id":null,"display_name":"Nikolaos Psarros","orcid":null},"institutions":[{"id":"https://openalex.org/I147962203","display_name":"Democritus University of Thrace","ror":"https://ror.org/03bfqnx40","country_code":"GR","type":"education","lineage":["https://openalex.org/I147962203"]}],"countries":["GR"],"is_corresponding":false,"raw_author_name":"Nikolaos Psarros","raw_affiliation_strings":["School of Engineering, Democritus University of Thrace, 67100 Xanthi, Greece"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Engineering, Democritus University of Thrace, 67100 Xanthi, Greece","institution_ids":["https://openalex.org/I147962203"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5030721192","display_name":"G. Papaschinopoulos","orcid":"https://orcid.org/0000-0001-9299-7933"},"institutions":[{"id":"https://openalex.org/I147962203","display_name":"Democritus University of Thrace","ror":"https://ror.org/03bfqnx40","country_code":"GR","type":"education","lineage":["https://openalex.org/I147962203"]}],"countries":["GR"],"is_corresponding":true,"raw_author_name":"Garyfalos Papaschinopoulos","raw_affiliation_strings":["School of Engineering, Democritus University of Thrace, 67100 Xanthi, Greece"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Engineering, Democritus University of Thrace, 67100 Xanthi, Greece","institution_ids":["https://openalex.org/I147962203"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5048939743","display_name":"C.J. Schinas","orcid":"https://orcid.org/0000-0002-1971-3137"},"institutions":[{"id":"https://openalex.org/I147962203","display_name":"Democritus University of Thrace","ror":"https://ror.org/03bfqnx40","country_code":"GR","type":"education","lineage":["https://openalex.org/I147962203"]}],"countries":["GR"],"is_corresponding":false,"raw_author_name":"Christos Schinas","raw_affiliation_strings":["School of Engineering, Democritus University of Thrace, 67100 Xanthi, Greece"],"raw_orcid":"https://orcid.org/0000-0002-1971-3137","affiliations":[{"raw_affiliation_string":"School of Engineering, Democritus University of Thrace, 67100 Xanthi, Greece","institution_ids":["https://openalex.org/I147962203"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5030721192"],"corresponding_institution_ids":["https://openalex.org/I147962203"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":1.8538,"has_fulltext":false,"cited_by_count":11,"citation_normalized_percentile":{"value":0.85707896,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":97},"biblio":{"volume":"10","issue":"6","first_page":"188","last_page":"188"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10482","display_name":"Mathematical and Theoretical Epidemiology and Ecology Models","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/2739","display_name":"Public Health, Environmental and Occupational Health"},"field":{"id":"https://openalex.org/fields/27","display_name":"Medicine"},"domain":{"id":"https://openalex.org/domains/4","display_name":"Health Sciences"}},"topics":[{"id":"https://openalex.org/T10482","display_name":"Mathematical and Theoretical Epidemiology and Ecology Models","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/2739","display_name":"Public Health, Environmental and Occupational Health"},"field":{"id":"https://openalex.org/fields/27","display_name":"Medicine"},"domain":{"id":"https://openalex.org/domains/4","display_name":"Health Sciences"}},{"id":"https://openalex.org/T10541","display_name":"Nonlinear Differential Equations Analysis","score":0.9847999811172485,"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/T12396","display_name":"Advanced Differential Equations and Dynamical Systems","score":0.9593999981880188,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/zero","display_name":"Zero (linguistics)","score":0.7941721081733704},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.759049654006958},{"id":"https://openalex.org/keywords/center-manifold","display_name":"Center manifold","score":0.7130038142204285},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6991346478462219},{"id":"https://openalex.org/keywords/exponential-function","display_name":"Exponential function","score":0.5664968490600586},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.5534703731536865},{"id":"https://openalex.org/keywords/exponential-stability","display_name":"Exponential stability","score":0.546029269695282},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.4662799835205078},{"id":"https://openalex.org/keywords/manifold","display_name":"Manifold (fluid mechanics)","score":0.4411662817001343},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.39632871747016907},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.2030695378780365}],"concepts":[{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.7941721081733704},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.759049654006958},{"id":"https://openalex.org/C74774103","wikidata":"https://www.wikidata.org/wiki/Q4504202","display_name":"Center manifold","level":5,"score":0.7130038142204285},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6991346478462219},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.5664968490600586},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5534703731536865},{"id":"https://openalex.org/C167964875","wikidata":"https://www.wikidata.org/wiki/Q17011487","display_name":"Exponential stability","level":3,"score":0.546029269695282},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.4662799835205078},{"id":"https://openalex.org/C529865628","wikidata":"https://www.wikidata.org/wiki/Q1790740","display_name":"Manifold (fluid mechanics)","level":2,"score":0.4411662817001343},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.39632871747016907},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.2030695378780365},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.0},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0},{"id":"https://openalex.org/C172435161","wikidata":"https://www.wikidata.org/wiki/Q531126","display_name":"Hopf bifurcation","level":4,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"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/C2781349735","wikidata":"https://www.wikidata.org/wiki/Q4904824","display_name":"Bifurcation","level":3,"score":0.0},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.0},{"id":"https://openalex.org/C78519656","wikidata":"https://www.wikidata.org/wiki/Q101333","display_name":"Mechanical engineering","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3390/sym10060188","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym10060188","pdf_url":"https://www.mdpi.com/2073-8994/10/6/188/pdf?version=1527759214","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:2e377ed8ba134c83aebec47c254631f3","is_oa":true,"landing_page_url":"https://doaj.org/article/2e377ed8ba134c83aebec47c254631f3","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":"Symmetry, Vol 10, Iss 6, p 188 (2018)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/sym10060188","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym10060188","pdf_url":"https://www.mdpi.com/2073-8994/10/6/188/pdf?version=1527759214","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":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2806054350.pdf","grobid_xml":"https://content.openalex.org/works/W2806054350.grobid-xml"},"referenced_works_count":39,"referenced_works":["https://openalex.org/W1608917902","https://openalex.org/W1908636068","https://openalex.org/W1923518972","https://openalex.org/W1962069087","https://openalex.org/W1964870349","https://openalex.org/W1968050124","https://openalex.org/W1972543563","https://openalex.org/W1978619309","https://openalex.org/W1982324233","https://openalex.org/W1983903974","https://openalex.org/W2009077957","https://openalex.org/W2024420873","https://openalex.org/W2028639944","https://openalex.org/W2028884599","https://openalex.org/W2040821291","https://openalex.org/W2043112976","https://openalex.org/W2047285883","https://openalex.org/W2050936481","https://openalex.org/W2060822013","https://openalex.org/W2074023791","https://openalex.org/W2074316841","https://openalex.org/W2076615094","https://openalex.org/W2082208294","https://openalex.org/W2086008959","https://openalex.org/W2087605746","https://openalex.org/W2087961804","https://openalex.org/W2094158268","https://openalex.org/W2094589055","https://openalex.org/W2169217668","https://openalex.org/W2186380646","https://openalex.org/W2236526038","https://openalex.org/W2296255950","https://openalex.org/W2341753742","https://openalex.org/W2519837783","https://openalex.org/W2768042676","https://openalex.org/W2955637500","https://openalex.org/W3006235878","https://openalex.org/W4213264637","https://openalex.org/W4234282985"],"related_works":["https://openalex.org/W2329408527","https://openalex.org/W2024209920","https://openalex.org/W1600405283","https://openalex.org/W4321392694","https://openalex.org/W4382516516","https://openalex.org/W2032133996","https://openalex.org/W2348798467","https://openalex.org/W2046248813","https://openalex.org/W1983459863","https://openalex.org/W3152370143"],"abstract_inverted_index":{"In":[0,48],"this":[1],"paper,":[2],"we":[3,52],"study":[4],"the":[5,8,22,37,49,54],"stability":[6],"of":[7,11,15,28,44,56],"zero":[9],"equilibria":[10],"two":[12],"close-to-symmetric":[13],"systems":[14],"difference":[16],"equations":[17],"with":[18],"exponential":[19],"terms":[20],"in":[21,25],"special":[23],"case":[24],"which":[26],"one":[27],"their":[29],"eigenvalues":[30],"is":[31],"equal":[32],"to":[33],"\u2212":[34],"1":[35],"and":[36],"other":[38],"eigenvalue":[39],"has":[40],"an":[41],"absolute":[42],"value":[43],"less":[45],"than":[46],"1.":[47],"present":[50],"study,":[51],"use":[53],"approach":[55],"center":[57],"manifold":[58],"theory.":[59]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":3},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":2}],"updated_date":"2026-05-06T08:25:59.206177","created_date":"2025-10-10T00:00:00"}