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The algorithm is based on a piecewise Taylor representation of the solutions in the phase space, and it exploits the smoothing of solutions occurring in DDEs to produce enclosures of solutions of a high order. We apply the topological techniques to prove various kinds of dynamical behaviour, for example, existence of (apparently) unstable periodic orbits in Mackey\u2013Glass equation (in the regime of parameters where chaos is numerically observed) and persistence of symbolic dynamics in a delay-perturbed chaotic ODE (the R\u00f6ssler system).<\/jats:p>","DOI":"10.1007\/s10208-023-09614-x","type":"journal-article","created":{"date-parts":[[2023,6,9]],"date-time":"2023-06-09T17:02:02Z","timestamp":1686330122000},"page":"1389-1454","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["High-Order Lohner-Type Algorithm for Rigorous Computation of Poincar\u00e9 Maps in Systems of Delay Differential Equations with Several Delays"],"prefix":"10.1007","volume":"24","author":[{"given":"Robert","family":"Szczelina","sequence":"first","affiliation":[]},{"given":"Piotr","family":"Zgliczy\u0144ski","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,6,9]]},"reference":[{"key":"9614_CR1","doi-asserted-by":"publisher","first-page":"15","DOI":"10.1016\/j.jde.2021.05.052","volume":"296","author":"FA Bartha","year":"2021","unstructured":"F.A. 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