{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,20]],"date-time":"2026-03-20T12:03:05Z","timestamp":1774008185078,"version":"3.50.1"},"reference-count":43,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2026,3,20]],"date-time":"2026-03-20T00:00:00Z","timestamp":1773964800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Cascade funding"},{"name":"European Union\u2014NextGenerationEU","award":["ECS00000036"],"award-info":[{"award-number":["ECS00000036"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The exact response theory, also known as Transient Time Correlation Function formalism, is a powerful method concerning how observables respond to a given perturbation of the dynamics of the systems of interest, and it extends linear response theory to generic (autonomous) dynamical systems. Its main ingredient is the so-called dissipation function. In this paper, we adapt this theory for time-lagged systems, and we illustrate its applicability considering simple examples of delay equations, with different memory terms. Adopting the technique already used for time deterministic as well as stochastic time-dependent perturbations, the dynamics is described in a higher dimensional phase space, in which the delay-dependent dynamics is mapped into an augmented phase space: the new dynamics is proven to be autonomous and suitable for the exact responses to be computed. In addition, we explore the comparison between linear and exact approaches for a specific kernel choice.<\/jats:p>","DOI":"10.3390\/e28030350","type":"journal-article","created":{"date-parts":[[2026,3,20]],"date-time":"2026-03-20T09:44:41Z","timestamp":1773999881000},"page":"350","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Exact Response Theory for Delay Equations"],"prefix":"10.3390","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0009-0003-5664-2248","authenticated-orcid":false,"given":"Federico","family":"Gollinucci","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences \u201cGiuseppe Luigi Lagrange\u201d, Politecnico di Torino, 10129 Torino, Italy"},{"name":"Istituto Nazionale di Fisica Nucleare, Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0002-9298-5603","authenticated-orcid":false,"given":"Enrico","family":"Ortu","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences \u201cGiuseppe Luigi Lagrange\u201d, Politecnico di Torino, 10129 Torino, Italy"},{"name":"Istituto Nazionale di Fisica Nucleare, Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy"},{"name":"CONCEPT Lab, Fondazione Istituto Italiano di Tecnologia, Via E. Melen 83, 16152 Genova, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4223-6279","authenticated-orcid":false,"given":"Lamberto","family":"Rondoni","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences \u201cGiuseppe Luigi Lagrange\u201d, Politecnico di Torino, 10129 Torino, Italy"},{"name":"Istituto Nazionale di Fisica Nucleare, Sezione di Torino, Via P. 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