{"id":"https://openalex.org/W3164775515","doi":"https://doi.org/10.3390/sym13060934","title":"New Results on Qualitative Behavior of Second Order Nonlinear Neutral Impulsive Differential Systems with Canonical and Non-Canonical Conditions","display_name":"New Results on Qualitative Behavior of Second Order Nonlinear Neutral Impulsive Differential Systems with Canonical and Non-Canonical Conditions","publication_year":2021,"publication_date":"2021-05-25","ids":{"openalex":"https://openalex.org/W3164775515","doi":"https://doi.org/10.3390/sym13060934","mag":"3164775515"},"language":"en","primary_location":{"id":"doi:10.3390/sym13060934","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym13060934","pdf_url":"https://www.mdpi.com/2073-8994/13/6/934/pdf?version=1621938853","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/2073-8994/13/6/934/pdf?version=1621938853","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5060231747","display_name":"Shyam Sundar Santra","orcid":"https://orcid.org/0000-0001-9740-3081"},"institutions":[{"id":"https://openalex.org/I77501641","display_name":"University of Kalyani","ror":"https://ror.org/03v783k16","country_code":"IN","type":"education","lineage":["https://openalex.org/I77501641"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Shyam Sundar Santra","raw_affiliation_strings":["Department of Mathematics, JIS College of Engineering, Kalyani, West Bengal 741235, India"],"raw_orcid":"https://orcid.org/0000-0001-9740-3081","affiliations":[{"raw_affiliation_string":"Department of Mathematics, JIS College of Engineering, Kalyani, West Bengal 741235, India","institution_ids":["https://openalex.org/I77501641"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5073170321","display_name":"Khaled Mohamed Khedher","orcid":"https://orcid.org/0000-0002-4167-1690"},"institutions":[{"id":"https://openalex.org/I82952536","display_name":"King Khalid University","ror":"https://ror.org/052kwzs30","country_code":"SA","type":"education","lineage":["https://openalex.org/I82952536"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"Khaled Mohamed Khedher","raw_affiliation_strings":["Department of Civil Engineering, College of Engineering, King Khalid University, Abha 61421, Saudi Arabia","Department of Civil Engineering, High Institute of Technological Studies, Mrezgua University Campus, Nabeul 8000, Tunisia"],"raw_orcid":"https://orcid.org/0000-0002-4167-1690","affiliations":[{"raw_affiliation_string":"Department of Civil Engineering, College of Engineering, King Khalid University, Abha 61421, Saudi Arabia","institution_ids":["https://openalex.org/I82952536"]},{"raw_affiliation_string":"Department of Civil Engineering, High Institute of Technological Studies, Mrezgua University Campus, Nabeul 8000, Tunisia","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5089496116","display_name":"Kamsing Nonlaopon","orcid":"https://orcid.org/0000-0002-7469-5402"},"institutions":[{"id":"https://openalex.org/I179193067","display_name":"Khon Kaen University","ror":"https://ror.org/03cq4gr50","country_code":"TH","type":"education","lineage":["https://openalex.org/I179193067"]}],"countries":["TH"],"is_corresponding":true,"raw_author_name":"Kamsing Nonlaopon","raw_affiliation_strings":["Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand"],"raw_orcid":"https://orcid.org/0000-0002-7469-5402","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand","institution_ids":["https://openalex.org/I179193067"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5114376646","display_name":"Hijaz Ahmad","orcid":"https://orcid.org/0000-0002-5438-5407"},"institutions":[{"id":"https://openalex.org/I90610274","display_name":"University of Engineering and Technology Peshawar","ror":"https://ror.org/00p034093","country_code":"PK","type":"education","lineage":["https://openalex.org/I90610274"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Hijaz Ahmad","raw_affiliation_strings":["Department of Basic Sciences, University of Engineering and Technology, Peshawar 25000, Pakistan"],"raw_orcid":"https://orcid.org/0000-0002-5438-5407","affiliations":[{"raw_affiliation_string":"Department of Basic Sciences, University of Engineering and Technology, Peshawar 25000, Pakistan","institution_ids":["https://openalex.org/I90610274"]}]}],"institutions":[],"countries_distinct_count":4,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5089496116"],"corresponding_institution_ids":["https://openalex.org/I179193067"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2227},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2227},"fwci":2.8755,"has_fulltext":false,"cited_by_count":9,"citation_normalized_percentile":{"value":0.91238333,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"13","issue":"6","first_page":"934","last_page":"934"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10541","display_name":"Nonlinear Differential Equations Analysis","score":0.9997000098228455,"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/T10541","display_name":"Nonlinear Differential Equations Analysis","score":0.9997000098228455,"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/T12727","display_name":"Differential Equations and Numerical Methods","score":0.9991000294685364,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"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/T12589","display_name":"Differential Equations and Boundary Problems","score":0.9965000152587891,"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/mathematics","display_name":"Mathematics","score":0.6943524479866028},{"id":"https://openalex.org/keywords/canonical-form","display_name":"Canonical form","score":0.6423660516738892},{"id":"https://openalex.org/keywords/impulse","display_name":"Impulse (physics)","score":0.6350244283676147},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.6271235346794128},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.6251170039176941},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.5506977438926697},{"id":"https://openalex.org/keywords/oscillation","display_name":"Oscillation (cell signaling)","score":0.5033285021781921},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.49095895886421204},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.36692196130752563},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.1913890540599823},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.1730365753173828},{"id":"https://openalex.org/keywords/classical-mechanics","display_name":"Classical mechanics","score":0.07717710733413696}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6943524479866028},{"id":"https://openalex.org/C204707403","wikidata":"https://www.wikidata.org/wiki/Q1152398","display_name":"Canonical form","level":2,"score":0.6423660516738892},{"id":"https://openalex.org/C70836080","wikidata":"https://www.wikidata.org/wiki/Q837940","display_name":"Impulse (physics)","level":2,"score":0.6350244283676147},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.6271235346794128},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.6251170039176941},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.5506977438926697},{"id":"https://openalex.org/C2778439541","wikidata":"https://www.wikidata.org/wiki/Q7106412","display_name":"Oscillation (cell signaling)","level":2,"score":0.5033285021781921},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.49095895886421204},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.36692196130752563},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.1913890540599823},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.1730365753173828},{"id":"https://openalex.org/C74650414","wikidata":"https://www.wikidata.org/wiki/Q11397","display_name":"Classical mechanics","level":1,"score":0.07717710733413696},{"id":"https://openalex.org/C54355233","wikidata":"https://www.wikidata.org/wiki/Q7162","display_name":"Genetics","level":1,"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/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym13060934","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym13060934","pdf_url":"https://www.mdpi.com/2073-8994/13/6/934/pdf?version=1621938853","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:ee7ab0ffd9a147c8b53c7e4addb551a2","is_oa":true,"landing_page_url":"https://doaj.org/article/ee7ab0ffd9a147c8b53c7e4addb551a2","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 13, Iss 6, p 934 (2021)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/13/6/934/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym13060934","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/sym13060934","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym13060934","pdf_url":"https://www.mdpi.com/2073-8994/13/6/934/pdf?version=1621938853","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":false},"content_urls":{"pdf":"https://content.openalex.org/works/W3164775515.pdf"},"referenced_works_count":42,"referenced_works":["https://openalex.org/W1583067131","https://openalex.org/W1892917588","https://openalex.org/W1982613386","https://openalex.org/W1995182165","https://openalex.org/W2013589574","https://openalex.org/W2024740056","https://openalex.org/W2062541933","https://openalex.org/W2096402868","https://openalex.org/W2096573765","https://openalex.org/W2188565877","https://openalex.org/W2346427662","https://openalex.org/W2588856696","https://openalex.org/W2606557080","https://openalex.org/W2606796707","https://openalex.org/W2766862070","https://openalex.org/W2792799191","https://openalex.org/W2799541626","https://openalex.org/W2953446726","https://openalex.org/W2971355763","https://openalex.org/W3006981577","https://openalex.org/W3012487426","https://openalex.org/W3016370273","https://openalex.org/W3027298723","https://openalex.org/W3084889420","https://openalex.org/W3095877397","https://openalex.org/W3097811719","https://openalex.org/W3105320740","https://openalex.org/W3106064471","https://openalex.org/W3106391841","https://openalex.org/W3112257729","https://openalex.org/W3120844561","https://openalex.org/W3131193315","https://openalex.org/W3205449582","https://openalex.org/W3206602597","https://openalex.org/W4214655746","https://openalex.org/W4319865669","https://openalex.org/W6634978176","https://openalex.org/W6649040901","https://openalex.org/W6687327942","https://openalex.org/W6734187965","https://openalex.org/W6801833183","https://openalex.org/W6802478376"],"related_works":["https://openalex.org/W3013085049","https://openalex.org/W3158420315","https://openalex.org/W3160917256","https://openalex.org/W2356614427","https://openalex.org/W2072100489","https://openalex.org/W2076457760","https://openalex.org/W2294866254","https://openalex.org/W2950671338","https://openalex.org/W2596503610","https://openalex.org/W2058905970"],"abstract_inverted_index":{"The":[0,18],"oscillation":[1],"of":[2,30,32,49,58,89,100,116],"impulsive":[3,33,52,102],"differential":[4,34,53,76],"equations":[5,77],"plays":[6],"an":[7],"important":[8],"role":[9],"in":[10,13],"many":[11],"applications":[12],"physics,":[14],"biology":[15],"and":[16,66],"engineering.":[17],"symmetry":[19],"helps":[20],"to":[21,26,82],"deciding":[22],"the":[23,59,64,75,86,95,101,107,117],"right":[24],"way":[25],"study":[27],"oscillatory":[28,45,79,96],"behavior":[29,48,99],"solutions":[31],"equations.":[35],"In":[36],"this":[37],"work,":[38],"several":[39],"sufficient":[40],"conditions":[41],"are":[42,120],"established":[43],"for":[44,55],"or":[46,97],"asymptotic":[47,98],"second-order":[50],"neutral":[51,61],"systems":[54],"various":[56],"ranges":[57],"bounded":[60],"coefficient":[62],"under":[63],"canonical":[65],"non-canonical":[67],"conditions.":[68],"Here,":[69],"one":[70],"can":[71],"see":[72],"that":[73],"if":[74],"is":[78],"(or":[80],"converges":[81],"zero":[83],"asymptotically),":[84],"then":[85],"discrete":[87,108],"equation":[88],"similar":[90],"type":[91],"do":[92],"not":[93],"disturb":[94],"system,":[103],"when":[104],"impulse":[105],"satisfies":[106],"equation.":[109],"Further,":[110],"some":[111],"illustrative":[112],"examples":[113],"showing":[114],"applicability":[115],"new":[118],"results":[119],"included.":[121]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":4},{"year":2021,"cited_by_count":4}],"updated_date":"2026-07-22T07:51:19.307946","created_date":"2025-10-10T00:00:00"}