{"id":"https://openalex.org/W7138891025","doi":"https://doi.org/10.3390/e28030350","title":"Exact Response Theory for Delay Equations","display_name":"Exact Response Theory for Delay Equations","publication_year":2026,"publication_date":"2026-03-20","ids":{"openalex":"https://openalex.org/W7138891025","doi":"https://doi.org/10.3390/e28030350","pmid":"https://pubmed.ncbi.nlm.nih.gov/41900001"},"language":"en","primary_location":{"id":"doi:10.3390/e28030350","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e28030350","pdf_url":"https://www.mdpi.com/1099-4300/28/3/350/pdf?version=1773994853","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/1099-4300/28/3/350/pdf?version=1773994853","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5130210911","display_name":"Federico Gollinucci","orcid":null},"institutions":[{"id":"https://openalex.org/I177477856","display_name":"Politecnico di Torino","ror":"https://ror.org/00bgk9508","country_code":"IT","type":"education","lineage":["https://openalex.org/I177477856"]},{"id":"https://openalex.org/I4210099850","display_name":"Istituto Nazionale di Fisica Nucleare, Sezione di Torino","ror":"https://ror.org/01vj6ck58","country_code":"IT","type":"facility","lineage":["https://openalex.org/I160013858","https://openalex.org/I4210099850"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Federico Gollinucci","raw_affiliation_strings":["Department of Mathematical Sciences \u201cGiuseppe Luigi Lagrange\u201d, Politecnico di Torino, 10129 Torino, Italy","Istituto Nazionale di Fisica Nucleare, Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy"],"raw_orcid":"https://orcid.org/0009-0003-5664-2248","affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences \u201cGiuseppe Luigi Lagrange\u201d, Politecnico di Torino, 10129 Torino, Italy","institution_ids":["https://openalex.org/I177477856"]},{"raw_affiliation_string":"Istituto Nazionale di Fisica Nucleare, Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy","institution_ids":["https://openalex.org/I4210099850"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5130017519","display_name":"Enrico Ortu","orcid":null},"institutions":[{"id":"https://openalex.org/I177477856","display_name":"Politecnico di Torino","ror":"https://ror.org/00bgk9508","country_code":"IT","type":"education","lineage":["https://openalex.org/I177477856"]},{"id":"https://openalex.org/I30771326","display_name":"Italian Institute of Technology","ror":"https://ror.org/042t93s57","country_code":"IT","type":"facility","lineage":["https://openalex.org/I30771326"]},{"id":"https://openalex.org/I4210099850","display_name":"Istituto Nazionale di Fisica Nucleare, Sezione di Torino","ror":"https://ror.org/01vj6ck58","country_code":"IT","type":"facility","lineage":["https://openalex.org/I160013858","https://openalex.org/I4210099850"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Enrico Ortu","raw_affiliation_strings":["CONCEPT Lab, Fondazione Istituto Italiano di Tecnologia, Via E. Melen 83, 16152 Genova, Italy","Department of Mathematical Sciences \u201cGiuseppe Luigi Lagrange\u201d, Politecnico di Torino, 10129 Torino, Italy","Istituto Nazionale di Fisica Nucleare, Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy"],"raw_orcid":"https://orcid.org/0009-0002-9298-5603","affiliations":[{"raw_affiliation_string":"CONCEPT Lab, Fondazione Istituto Italiano di Tecnologia, Via E. Melen 83, 16152 Genova, Italy","institution_ids":["https://openalex.org/I30771326"]},{"raw_affiliation_string":"Department of Mathematical Sciences \u201cGiuseppe Luigi Lagrange\u201d, Politecnico di Torino, 10129 Torino, Italy","institution_ids":["https://openalex.org/I177477856"]},{"raw_affiliation_string":"Istituto Nazionale di Fisica Nucleare, Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy","institution_ids":["https://openalex.org/I4210099850"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5037162554","display_name":"Lamberto Rondoni","orcid":"https://orcid.org/0000-0002-4223-6279"},"institutions":[{"id":"https://openalex.org/I177477856","display_name":"Politecnico di Torino","ror":"https://ror.org/00bgk9508","country_code":"IT","type":"education","lineage":["https://openalex.org/I177477856"]},{"id":"https://openalex.org/I4210099850","display_name":"Istituto Nazionale di Fisica Nucleare, Sezione di Torino","ror":"https://ror.org/01vj6ck58","country_code":"IT","type":"facility","lineage":["https://openalex.org/I160013858","https://openalex.org/I4210099850"]}],"countries":["IT"],"is_corresponding":true,"raw_author_name":"Lamberto Rondoni","raw_affiliation_strings":["Department of Mathematical Sciences \u201cGiuseppe Luigi Lagrange\u201d, Politecnico di Torino, 10129 Torino, Italy","Istituto Nazionale di Fisica Nucleare, Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy"],"raw_orcid":"https://orcid.org/0000-0002-4223-6279","affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences \u201cGiuseppe Luigi Lagrange\u201d, Politecnico di Torino, 10129 Torino, Italy","institution_ids":["https://openalex.org/I177477856"]},{"raw_affiliation_string":"Istituto Nazionale di Fisica Nucleare, Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy","institution_ids":["https://openalex.org/I4210099850"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5037162554"],"corresponding_institution_ids":["https://openalex.org/I177477856","https://openalex.org/I4210099850"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.4754266,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"28","issue":"3","first_page":"350","last_page":"350"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11513","display_name":"stochastic dynamics and bifurcation","score":0.1362999975681305,"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/T11513","display_name":"stochastic dynamics and bifurcation","score":0.1362999975681305,"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/T10621","display_name":"Gene Regulatory Network Analysis","score":0.1225999966263771,"subfield":{"id":"https://openalex.org/subfields/1312","display_name":"Molecular Biology"},"field":{"id":"https://openalex.org/fields/13","display_name":"Biochemistry, Genetics and Molecular Biology"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}},{"id":"https://openalex.org/T11520","display_name":"Advanced Thermodynamics and Statistical Mechanics","score":0.1145000010728836,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/observable","display_name":"Observable","score":0.5548999905586243},{"id":"https://openalex.org/keywords/simple","display_name":"Simple (philosophy)","score":0.49720001220703125},{"id":"https://openalex.org/keywords/dissipation","display_name":"Dissipation","score":0.4375},{"id":"https://openalex.org/keywords/dynamical-systems-theory","display_name":"Dynamical systems theory","score":0.4043000042438507},{"id":"https://openalex.org/keywords/perturbation","display_name":"Perturbation (astronomy)","score":0.39719998836517334},{"id":"https://openalex.org/keywords/linear-system","display_name":"Linear system","score":0.39410001039505005},{"id":"https://openalex.org/keywords/dynamics","display_name":"Dynamics (music)","score":0.39320001006126404},{"id":"https://openalex.org/keywords/perturbation-theory","display_name":"Perturbation theory (quantum mechanics)","score":0.38089999556541443},{"id":"https://openalex.org/keywords/phase-response","display_name":"Phase response","score":0.37869998812675476}],"concepts":[{"id":"https://openalex.org/C32848918","wikidata":"https://www.wikidata.org/wiki/Q845789","display_name":"Observable","level":2,"score":0.5548999905586243},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4991999864578247},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.49720001220703125},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.47600001096725464},{"id":"https://openalex.org/C135402231","wikidata":"https://www.wikidata.org/wiki/Q898440","display_name":"Dissipation","level":2,"score":0.4375},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.4323999881744385},{"id":"https://openalex.org/C79379906","wikidata":"https://www.wikidata.org/wiki/Q3174497","display_name":"Dynamical systems theory","level":2,"score":0.4043000042438507},{"id":"https://openalex.org/C177918212","wikidata":"https://www.wikidata.org/wiki/Q803623","display_name":"Perturbation (astronomy)","level":2,"score":0.39719998836517334},{"id":"https://openalex.org/C6802819","wikidata":"https://www.wikidata.org/wiki/Q1072174","display_name":"Linear system","level":2,"score":0.39410001039505005},{"id":"https://openalex.org/C145912823","wikidata":"https://www.wikidata.org/wiki/Q113558","display_name":"Dynamics (music)","level":2,"score":0.39320001006126404},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.38339999318122864},{"id":"https://openalex.org/C174256460","wikidata":"https://www.wikidata.org/wiki/Q911364","display_name":"Perturbation theory (quantum mechanics)","level":2,"score":0.38089999556541443},{"id":"https://openalex.org/C54767332","wikidata":"https://www.wikidata.org/wiki/Q1751572","display_name":"Phase response","level":3,"score":0.37869998812675476},{"id":"https://openalex.org/C85761212","wikidata":"https://www.wikidata.org/wiki/Q1974593","display_name":"Transient response","level":2,"score":0.3723999857902527},{"id":"https://openalex.org/C74193536","wikidata":"https://www.wikidata.org/wiki/Q574844","display_name":"Kernel (algebra)","level":2,"score":0.36000001430511475},{"id":"https://openalex.org/C114275822","wikidata":"https://www.wikidata.org/wiki/Q621512","display_name":"Linear dynamical system","level":3,"score":0.3434000015258789},{"id":"https://openalex.org/C2994474493","wikidata":"https://www.wikidata.org/wiki/Q3261000","display_name":"Linear response theory","level":2,"score":0.3231000006198883},{"id":"https://openalex.org/C2780799671","wikidata":"https://www.wikidata.org/wiki/Q17087362","display_name":"Transient (computer programming)","level":2,"score":0.3206999897956848},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.3183000087738037},{"id":"https://openalex.org/C47446073","wikidata":"https://www.wikidata.org/wiki/Q5165890","display_name":"Control theory (sociology)","level":3,"score":0.30320000648498535},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.2912999987602234},{"id":"https://openalex.org/C520416788","wikidata":"https://www.wikidata.org/wiki/Q5419229","display_name":"Exact solutions in general relativity","level":2,"score":0.28679999709129333},{"id":"https://openalex.org/C185184677","wikidata":"https://www.wikidata.org/wiki/Q1366795","display_name":"Exact differential equation","level":4,"score":0.2815999984741211},{"id":"https://openalex.org/C8272713","wikidata":"https://www.wikidata.org/wiki/Q176737","display_name":"Stochastic process","level":2,"score":0.27730000019073486},{"id":"https://openalex.org/C44280652","wikidata":"https://www.wikidata.org/wiki/Q104837","display_name":"Phase (matter)","level":2,"score":0.2759999930858612},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.2694999873638153},{"id":"https://openalex.org/C33962884","wikidata":"https://www.wikidata.org/wiki/Q378637","display_name":"Dynamical system (definition)","level":3,"score":0.2655999958515167},{"id":"https://openalex.org/C53622274","wikidata":"https://www.wikidata.org/wiki/Q2496679","display_name":"Correlation function (quantum field theory)","level":3,"score":0.2531000077724457}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.3390/e28030350","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e28030350","pdf_url":"https://www.mdpi.com/1099-4300/28/3/350/pdf?version=1773994853","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:41900001","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/41900001","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:33f526b9b4b4429b83803820b30e9fa7","is_oa":true,"landing_page_url":"https://doaj.org/article/33f526b9b4b4429b83803820b30e9fa7","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 3, p 350 (2026)","raw_type":"article"},{"id":"pmh:oai:pubmedcentral.nih.gov:13025812","is_oa":true,"landing_page_url":"https://pmc.ncbi.nlm.nih.gov/articles/PMC13025812/","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/e28030350","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e28030350","pdf_url":"https://www.mdpi.com/1099-4300/28/3/350/pdf?version=1773994853","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":[],"awards":[{"id":"https://openalex.org/G8323133587","display_name":null,"funder_award_id":"ECS00000036","funder_id":"https://openalex.org/F4320311030","funder_display_name":"Istituto Nazionale di Alta Matematica \"Francesco Severi\""},{"id":"https://openalex.org/G8968660317","display_name":null,"funder_award_id":"ECS00000036","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"}],"funders":[{"id":"https://openalex.org/F4320311030","display_name":"Istituto Nazionale di Alta Matematica \"Francesco Severi\"","ror":"https://ror.org/01vx64p53"},{"id":"https://openalex.org/F4320320300","display_name":"European Commission","ror":"https://ror.org/00k4n6c32"},{"id":"https://openalex.org/F4320334080","display_name":"Gruppo Nazionale per la Fisica Matematica","ror":null},{"id":"https://openalex.org/F4320337512","display_name":"Nuclear Physics","ror":"https://ror.org/02atag894"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7138891025.pdf","grobid_xml":"https://content.openalex.org/works/W7138891025.grobid-xml"},"referenced_works_count":36,"referenced_works":["https://openalex.org/W1548618691","https://openalex.org/W1916001099","https://openalex.org/W1972926598","https://openalex.org/W1988388205","https://openalex.org/W1990204968","https://openalex.org/W2003940353","https://openalex.org/W2016755336","https://openalex.org/W2030114330","https://openalex.org/W2034092645","https://openalex.org/W2074847124","https://openalex.org/W2087098125","https://openalex.org/W2093022216","https://openalex.org/W2097139575","https://openalex.org/W2124015892","https://openalex.org/W2134312352","https://openalex.org/W2142816172","https://openalex.org/W2275036067","https://openalex.org/W2296537081","https://openalex.org/W2345649492","https://openalex.org/W2464896617","https://openalex.org/W2476901185","https://openalex.org/W2488246962","https://openalex.org/W2963030804","https://openalex.org/W3006903434","https://openalex.org/W3046602060","https://openalex.org/W3089107780","https://openalex.org/W3208903225","https://openalex.org/W4213373779","https://openalex.org/W4221162351","https://openalex.org/W4243336679","https://openalex.org/W4293200410","https://openalex.org/W4390048786","https://openalex.org/W4404533868","https://openalex.org/W4409911928","https://openalex.org/W4417210030","https://openalex.org/W7084500764"],"related_works":[],"abstract_inverted_index":{"The":[0],"exact":[1,124,138],"response":[2,36],"theory,":[3],"also":[4],"known":[5],"as":[6,84,86],"Transient":[7],"Time":[8],"Correlation":[9],"Function":[10],"formalism,":[11],"is":[12,46,92,105,115],"a":[13,21,95,141],"powerful":[14],"method":[15],"concerning":[16],"how":[17],"observables":[18],"respond":[19],"to":[20,38,117,126],"given":[22],"perturbation":[23],"of":[24,27,30,69],"the":[25,28,47,77,90,102,112,123,133],"dynamics":[26,91,104,114],"systems":[29],"interest,":[31],"and":[32,61,120,137],"it":[33],"extends":[34],"linear":[35,136],"theory":[37,57],"generic":[39],"(autonomous)":[40],"dynamical":[41],"systems.":[42],"Its":[43],"main":[44],"ingredient":[45],"so-called":[48],"dissipation":[49],"function.":[50],"In":[51,129],"this":[52,56],"paper,":[53],"we":[54,62,131],"adapt":[55],"for":[58,81,122,140],"time-lagged":[59],"systems,":[60],"illustrate":[63],"its":[64],"applicability":[65],"considering":[66],"simple":[67],"examples":[68],"delay":[70],"equations,":[71],"with":[72],"different":[73],"memory":[74],"terms.":[75],"Adopting":[76],"technique":[78],"already":[79],"used":[80],"time":[82],"deterministic":[83],"well":[85],"stochastic":[87],"time-dependent":[88],"perturbations,":[89],"described":[93],"in":[94,100],"higher":[96],"dimensional":[97],"phase":[98,110],"space,":[99],"which":[101],"delay-dependent":[103],"mapped":[106],"into":[107],"an":[108],"augmented":[109],"space:":[111],"new":[113],"proven":[116],"be":[118,127],"autonomous":[119],"suitable":[121],"responses":[125],"computed.":[128],"addition,":[130],"explore":[132],"comparison":[134],"between":[135],"approaches":[139],"specific":[142],"kernel":[143],"choice.":[144]},"counts_by_year":[],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2026-03-20T00:00:00"}
