{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T13:31:12Z","timestamp":1787232672033,"version":"build-2736575974"},"reference-count":61,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","funder":[{"DOI":"10.13039\/501100004895","name":"European Social Fund","doi-asserted-by":"crossref","award":["E48"],"award-info":[{"award-number":["E48"]}],"id":[{"id":"10.13039\/501100004895","id-type":"DOI","asserted-by":"crossref"}]},{"name":"Spanish Research","award":["MTM2012-31883"],"award-info":[{"award-number":["MTM2012-31883"]}]},{"name":"Spanish Research","award":["MTM2015-64095-P"],"award-info":[{"award-number":["MTM2015-64095-P"]}]},{"DOI":"10.13039\/501100004281","name":"Narodowe Centrum Nauki","doi-asserted-by":"publisher","award":["2014\/14\/A\/ST1\/00453"],"award-info":[{"award-number":["2014\/14\/A\/ST1\/00453"]}],"id":[{"id":"10.13039\/501100004281","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100007041","name":"Universidad de Zaragoza","doi-asserted-by":"publisher","award":["UZCUD2015-CIE-05"],"award-info":[{"award-number":["UZCUD2015-CIE-05"]}],"id":[{"id":"10.13039\/501100007041","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Appl. Dyn. Syst."],"published-print":{"date-parts":[[2016,1]]},"abstract":"<jats:p>It has recently been reported [P. C. Reich, Neurocomputing, 74 (2011), pp. 3361--3364] that it is quite difficult to distinguish between chaos and hyperchaos in numerical simulations which are frequently \u201cnoisy.\u201d For the classical four-dimensional (4D) R\u00f6ssler model [O. E. R\u00f6ssler, Phys. Lett. A, 71 (1979), pp. 155--157] we show that the coexistence of two invariant sets with different nature (a global hyperchaotic invariant set and a chaotic attractor) and heteroclinic connections between them give rise to long hyperchaotic transient behavior, and therefore it provides a mechanism for noisy simulations. The same phenomena is expected in other 4D and higher-dimensional systems. The proof combines topological and smooth methods with rigorous numerical computations. The existence of (hyper)chaotic sets is proved by the method of covering relations [P. Zgliczy\u0144ski and M. Gidea, J. Differential Equations, 202 (2004), pp. 32--58]. We extend this method to the case of a nonincreasing number of unstable directions which is necessary to study hyperchaos to chaos transport. The cone condition [H. Kokubu, D. Wilczak, and P. Zgliczy\u0144ski, Nonlinearity, 20 (2007), pp. 2147--2174] is used to prove the existence of homoclinic and heteroclinic orbits between some periodic orbits which belong to both hyperchaotic and chaotic invariant sets. In particular, the existence of a countable infinity of heteroclinic orbits linking hyperchaos with chaos justifies the presence of long transient behavior.<\/jats:p>","DOI":"10.1137\/15m1039201","type":"journal-article","created":{"date-parts":[[2016,2,18]],"date-time":"2016-02-18T10:19:09Z","timestamp":1455790749000},"page":"356-390","source":"Crossref","is-referenced-by-count":20,"title":["Coexistence and Dynamical Connections between Hyperchaos and Chaos in the 4D R\u00f6ssler System: A Computer-Assisted Proof"],"prefix":"10.1137","volume":"15","author":[{"given":"Daniel","family":"Wilczak","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sergio","family":"Serrano","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Roberto","family":"Barrio","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2016,2,18]]},"reference":[{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1145\/2382585.2382590"},{"key":"atypb3","first-page":"253","volume":"22","author":"Afraimovic V. 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