Coexistence and dynamical connections between hyperchaos and chaos in the 4D Rössler system: A computer-assisted proof

Wilczak, D. ; Serrano, S. (Universidad de Zaragoza) ; Barrio, R. (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

Factor impacto JCR: 1.761 (2016)

Categ. JCR: MATHEMATICS, APPLIED rank: 34 / 255 = 0.133 (2016) - Q1 - T1
Categ. JCR: PHYSICS, MATHEMATICAL rank: 16 / 55 = 0.291 (2016) - Q2 - T1

Factor impacto SCIMAGO: 1.288 - Modeling and Simulation (Q1) - Analysis (Q1)

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): Área Matemática Aplicada (Dpto. Matemática Aplicada)