{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T14:59:51Z","timestamp":1774364391896,"version":"3.50.1"},"reference-count":101,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2020,8,27]],"date-time":"2020-08-27T00:00:00Z","timestamp":1598486400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100000038","name":"Natural Sciences and Engineering Research Council of Canada","doi-asserted-by":"publisher","award":["RGPIN-2018-06418"],"award-info":[{"award-number":["RGPIN-2018-06418"]}],"id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>Delay-differential equations belong to the class of infinite-dimensional dynamical systems. However, it is often observed that the solutions are rapidly attracted to smooth manifolds embedded in the finite-dimensional state space, called inertial manifolds. The computation of an inertial manifold yields an ordinary differential equation (ODE) model representing the long-term dynamics of the system. Note in particular that any attractors must be embedded in the inertial manifold when one exists, therefore reducing the study of these attractors to the ODE context, for which methods of analysis are well developed. This contribution presents a study of a previously developed method for constructing inertial manifolds based on an expansion of the delayed term in small powers of the delay, and subsequent solution of the invariance equation by the Fraser functional iteration method. The combined perturbative-iterative method is applied to several variations of a model for the expression of an inducible enzyme, where the delay represents the time required to transcribe messenger RNA and to translate that RNA into the protein. It is shown that inertial manifolds of different dimensions can be computed. Qualitatively correct inertial manifolds are obtained. Among other things, the dynamics confined to computed inertial manifolds display Andronov\u2013Hopf bifurcations at similar parameter values as the original DDE model.<\/jats:p>","DOI":"10.3390\/a13090209","type":"journal-article","created":{"date-parts":[[2020,8,27]],"date-time":"2020-08-27T08:05:18Z","timestamp":1598515518000},"page":"209","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Perturbative-Iterative Computation of Inertial Manifolds of Systems of Delay-Differential Equations with Small Delays"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7795-4756","authenticated-orcid":false,"given":"Marc R.","family":"Roussel","sequence":"first","affiliation":[{"name":"Alberta RNA Research and Training Institute, Department of Chemistry and Biochemistry, University of Lethbridge, Lethbridge, AB T1K 3M4, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,8,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"7106","DOI":"10.1063\/1.460194","article-title":"On the Geometry of Transient Relaxation","volume":"94","author":"Roussel","year":"1991","journal-title":"J. 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