Coexistence and dynamical connections between hyperchaos and chaos in the 4D Rössler system: A computer-assisted proof
Wilczak, D. ; Serrano Pastor, Sergio (Universidad de Zaragoza) ; Barrio Gil, Roberto (Universidad de Zaragoza)
Resumen: 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 "noisy." For the classical four-dimensional (4D) Rössler model O. E. Rössler, 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. Zgliczynski 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. Zgliczynski, 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.
Idioma: Inglés
DOI: 10.1137/15M1039201
Año: 2016
Publicado en: SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS 15, 1 (2016), 356-390
ISSN: 1536-0040
Financiación: info:eu-repo/grantAgreement/ES/DGA/E48
Financiación: info:eu-repo/grantAgreement/ES/MICINN/MTM2012-31883
Financiación: info:eu-repo/grantAgreement/ES/MICINN/MTM2015-64095-P
Financiación: info:eu-repo/grantAgreement/ES/UZ/CUD2015-CIE-05
Tipo y forma: Article (Published version)
Área (Departamento): Matemática Aplicada (Departamento de Matemática Aplicada)
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Exportado de SIDERAL (2016-11-28-12:09:32)
Idioma: Inglés
DOI: 10.1137/15M1039201
Año: 2016
Publicado en: SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS 15, 1 (2016), 356-390
ISSN: 1536-0040
Financiación: info:eu-repo/grantAgreement/ES/DGA/E48
Financiación: info:eu-repo/grantAgreement/ES/MICINN/MTM2012-31883
Financiación: info:eu-repo/grantAgreement/ES/MICINN/MTM2015-64095-P
Financiación: info:eu-repo/grantAgreement/ES/UZ/CUD2015-CIE-05
Tipo y forma: Article (Published version)
Área (Departamento): Matemática Aplicada (Departamento de Matemática Aplicada)
0
Exportado de SIDERAL (2016-11-28-12:09:32)
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Notice créée le 2016-07-29, modifiée le 2016-11-28
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