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While most existing studies focus on discrete-time systems (maps), this work examines a continuous-time scenario involving global changes in the structure of an observed attractor. As a representative example, we consider the classical R\u00f6ssler system. For a specific range of parameters of the system, we prove the existence of a trapping region for a certain Poincar\u00e9 map, which contains a nonempty, compact, and connected invariant set on which the topological entropy of the Poincar\u00e9 map is positive. Additionally, we prove the existence of a sequence of periodic orbit bifurcations that lead to an increase in the topological entropy of this Poincar\u00e9 map. Our results further reveal that the topological structure of the maximal invariant set in the trapping region evolves as the parameters of the system vary. These findings are rigorously supported by computer-assisted proofs, employing interval arithmetic techniques to compute guaranteed bounds on the Poincar\u00e9 map and its derivatives.<\/jats:p>","DOI":"10.1063\/5.0284636","type":"journal-article","created":{"date-parts":[[2025,10,14]],"date-time":"2025-10-14T15:55:31Z","timestamp":1760457331000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":["A mechanism for growth of topological entropy"],"prefix":"10.1063","volume":"35","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9934-3809","authenticated-orcid":false,"given":"Daniel","family":"Wilczak","sequence":"first","affiliation":[{"name":"Faculty of Mathematics and Computer Science, Jagiellonian University 1 , \u0141ojasiewicza 6, 30-348 Krak\u00f3w,","place":["Poland"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5701-1670","authenticated-orcid":false,"given":"Sergio","family":"Serrano","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica Aplicada and IUMA and CODY, University of Zaragoza 2 , Zaragoza E-50009,","place":["Spain"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8089-343X","authenticated-orcid":false,"given":"Roberto","family":"Barrio","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica Aplicada and IUMA and CODY, University of Zaragoza 2 , Zaragoza E-50009,","place":["Spain"]}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"317","published-online":{"date-parts":[[2025,10,14]]},"reference":[{"key":"2025101411552792200_c1","doi-asserted-by":"publisher","first-page":"177","DOI":"10.1007\/BF01212280","article-title":"On the concept of attractor","volume":"99","year":"1985","journal-title":"Commun. 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