{"id":"https://openalex.org/W1978133537","doi":"https://doi.org/10.1142/s0218127414300213","title":"Chaotic Oscillations of Solutions of the Klein\u2013Gordon Equation Due to Imbalance of Distributed and Boundary Energy Flows","display_name":"Chaotic Oscillations of Solutions of the Klein\u2013Gordon Equation Due to Imbalance of Distributed and Boundary Energy Flows","publication_year":2014,"publication_date":"2014-07-01","ids":{"openalex":"https://openalex.org/W1978133537","doi":"https://doi.org/10.1142/s0218127414300213","mag":"1978133537"},"language":"en","primary_location":{"id":"doi:10.1142/s0218127414300213","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127414300213","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/A5113524040","display_name":"Goong Chen","orcid":null},"institutions":[{"id":"https://openalex.org/I58152225","display_name":"Texas A&M University at Qatar","ror":"https://ror.org/03vb4dm14","country_code":"QA","type":"education","lineage":["https://openalex.org/I58152225","https://openalex.org/I91045830"]},{"id":"https://openalex.org/I91045830","display_name":"Texas A&M University","ror":"https://ror.org/01f5ytq51","country_code":"US","type":"education","lineage":["https://openalex.org/I91045830"]}],"countries":["QA","US"],"is_corresponding":true,"raw_author_name":"Goong Chen","raw_affiliation_strings":["Institute for Quantum Science and Engineering (IQSE), Department of Mathematics, Texas A&M University, College Station, TX 77843, USA","Science Program, Texas A&M University at Qatar, Education City, Doha, Qatar"],"affiliations":[{"raw_affiliation_string":"Institute for Quantum Science and Engineering (IQSE), Department of Mathematics, Texas A&M University, College Station, TX 77843, USA","institution_ids":["https://openalex.org/I91045830"]},{"raw_affiliation_string":"Science Program, Texas A&M University at Qatar, Education City, Doha, Qatar","institution_ids":["https://openalex.org/I58152225"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5101965121","display_name":"Bo Sun","orcid":"https://orcid.org/0000-0002-3287-0858"},"institutions":[{"id":"https://openalex.org/I91045830","display_name":"Texas A&M University","ror":"https://ror.org/01f5ytq51","country_code":"US","type":"education","lineage":["https://openalex.org/I91045830"]},{"id":"https://openalex.org/I56934997","display_name":"Changsha University of Science and Technology","ror":"https://ror.org/03yph8055","country_code":"CN","type":"education","lineage":["https://openalex.org/I56934997"]}],"countries":["CN","US"],"is_corresponding":false,"raw_author_name":"Bo Sun","raw_affiliation_strings":["Department of Mathematics, Changsha University of Science and Technology, Changsha, Hunan, P. R. China","Department of Mathematics, Texas A&M University, College Station, TX 77843, USA"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Changsha University of Science and Technology, Changsha, Hunan, P. R. China","institution_ids":["https://openalex.org/I56934997"]},{"raw_affiliation_string":"Department of Mathematics, Texas A&M University, College Station, TX 77843, USA","institution_ids":["https://openalex.org/I91045830"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5074290686","display_name":"Tingwen Huang","orcid":"https://orcid.org/0000-0001-9610-846X"},"institutions":[{"id":"https://openalex.org/I58152225","display_name":"Texas A&M University at Qatar","ror":"https://ror.org/03vb4dm14","country_code":"QA","type":"education","lineage":["https://openalex.org/I58152225","https://openalex.org/I91045830"]}],"countries":["QA"],"is_corresponding":false,"raw_author_name":"Tingwen Huang","raw_affiliation_strings":["Science Program, Texas A&M University at Qatar, Education City, Doha, Qatar"],"affiliations":[{"raw_affiliation_string":"Science Program, Texas A&M University at Qatar, Education City, Doha, Qatar","institution_ids":["https://openalex.org/I58152225"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5113524040"],"corresponding_institution_ids":["https://openalex.org/I58152225","https://openalex.org/I91045830"],"apc_list":null,"apc_paid":null,"fwci":1.0344,"has_fulltext":false,"cited_by_count":16,"citation_normalized_percentile":{"value":0.78488572,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"24","issue":"07","first_page":"1430021","last_page":"1430021"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11187","display_name":"Nonlinear Dynamics and Pattern Formation","score":0.9958000183105469,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"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/T11187","display_name":"Nonlinear Dynamics and Pattern Formation","score":0.9958000183105469,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"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/T11261","display_name":"Quantum chaos and dynamical systems","score":0.9957000017166138,"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/T11110","display_name":"Stability and Controllability of Differential Equations","score":0.9926000237464905,"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/nonlinear-system","display_name":"Nonlinear system","score":0.5702437162399292},{"id":"https://openalex.org/keywords/chaotic","display_name":"Chaotic","score":0.5614935159683228},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.5481038093566895},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5446748733520508},{"id":"https://openalex.org/keywords/boundary","display_name":"Boundary (topology)","score":0.5027611255645752},{"id":"https://openalex.org/keywords/boundary-value-problem","display_name":"Boundary value problem","score":0.46176877617836},{"id":"https://openalex.org/keywords/klein\u2013gordon-equation","display_name":"Klein\u2013Gordon equation","score":0.4548199772834778},{"id":"https://openalex.org/keywords/wave-equation","display_name":"Wave equation","score":0.44532740116119385},{"id":"https://openalex.org/keywords/term","display_name":"Term (time)","score":0.42689618468284607},{"id":"https://openalex.org/keywords/mathematical-proof","display_name":"Mathematical proof","score":0.41817033290863037},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.41743892431259155},{"id":"https://openalex.org/keywords/chaos","display_name":"CHAOS (operating system)","score":0.4108163118362427},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.3002638816833496},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.1422552764415741},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.11902058124542236}],"concepts":[{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.5702437162399292},{"id":"https://openalex.org/C2777052490","wikidata":"https://www.wikidata.org/wiki/Q5072826","display_name":"Chaotic","level":2,"score":0.5614935159683228},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5481038093566895},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5446748733520508},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.5027611255645752},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.46176877617836},{"id":"https://openalex.org/C101682238","wikidata":"https://www.wikidata.org/wiki/Q868967","display_name":"Klein\u2013Gordon equation","level":3,"score":0.4548199772834778},{"id":"https://openalex.org/C59696629","wikidata":"https://www.wikidata.org/wiki/Q193846","display_name":"Wave equation","level":2,"score":0.44532740116119385},{"id":"https://openalex.org/C61797465","wikidata":"https://www.wikidata.org/wiki/Q1188986","display_name":"Term (time)","level":2,"score":0.42689618468284607},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.41817033290863037},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.41743892431259155},{"id":"https://openalex.org/C2779374083","wikidata":"https://www.wikidata.org/wiki/Q5011038","display_name":"CHAOS (operating system)","level":2,"score":0.4108163118362427},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.3002638816833496},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.1422552764415741},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.11902058124542236},{"id":"https://openalex.org/C38652104","wikidata":"https://www.wikidata.org/wiki/Q3510521","display_name":"Computer security","level":1,"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}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1142/s0218127414300213","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127414300213","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":[{"score":0.5,"display_name":"Affordable and clean energy","id":"https://metadata.un.org/sdg/7"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":7,"referenced_works":["https://openalex.org/W1641808155","https://openalex.org/W1961956412","https://openalex.org/W1990608211","https://openalex.org/W2058791052","https://openalex.org/W2080858310","https://openalex.org/W2127497910","https://openalex.org/W2315759974"],"related_works":["https://openalex.org/W3146590310","https://openalex.org/W2006544532","https://openalex.org/W2043702156","https://openalex.org/W2129259516","https://openalex.org/W4313196245","https://openalex.org/W2904592692","https://openalex.org/W3040632610","https://openalex.org/W2028085703","https://openalex.org/W4293689345","https://openalex.org/W2439622949"],"abstract_inverted_index":{"Chaos":[0],"in":[1,39,51],"physical":[2],"systems":[3],"governed":[4],"by":[5,128],"nonlinear":[6],"PDEs":[7],"have":[8],"been":[9],"amply":[10],"observed":[11],"and":[12,45,110,121],"reported.":[13],"However,":[14],"rigorous":[15],"proofs":[16],"for":[17,24,58,99,105],"their":[18],"occurrences":[19],"are":[20,126],"challenging.":[21],"In":[22,79],"particular,":[23],"a":[25,33,46,52,106,117],"second":[26],"order":[27],"PDE":[28],"of":[29,41,61,108,124],"hyperbolic":[30],"type":[31],"with":[32,88,91],"van":[34],"der":[35],"Pol":[36],"cubic":[37],"nonlinearity":[38],"one":[40],"the":[42,59,73,89,92,100],"boundary":[43,111],"conditions":[44,112],"spatially":[47],"distributed":[48],"antidamping":[49,74],"term":[50,75],"linear":[53],"governing":[54],"equation,":[55],"no":[56],"proof":[57],"onset":[60],"chaos":[62,67,103],"was":[63,68],"available":[64],"even":[65],"though":[66],"expected":[69],"to":[70,96],"occur":[71],"when":[72,113],"becomes":[76],"sufficiently":[77],"strong.":[78],"this":[80],"paper,":[81],"we":[82],"use":[83],"an":[84],"operator-factoring":[85],"technique":[86],"together":[87],"analogy":[90],"one-dimensional":[93],"wave":[94],"equation":[95,102],"prove":[97],"that":[98],"Klein\u2013Gordon":[101],"occurs":[104],"class":[107],"equations":[109],"system":[114],"parameters":[115],"enter":[116],"certain":[118],"regime.":[119],"Chaotic":[120],"nonchaotic":[122],"profiles":[123],"solutions":[125],"illustrated":[127],"computer":[129],"graphics.":[130]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":4},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":2},{"year":2015,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}