Coexistence and Dynamical Connections between Hyperchaos and Chaos in the 4D Rössler System: A Computer-Assisted Proof

Abstract

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. Zgliczyński 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. Zgliczyński, 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.

When chaos meets hyperchaos: 4D Rössler model

“…1 still shows "noise", so that higher transient and/or integration time is necessary to get an absolutely clear picture. In a more detailed study [34], it has been found that a . This observation permits to perform better simulations of Lyapunov exponents, but in any case it does not provide a complete explanation of the "noisy" patterns and why we need such a long transient time in this system.…”

Periodic solutions and invariant torus in the Rössler System

“…Since then, the chaotic behavior of the Rössler System has been addressed in several works. We may cite, for instance, [3,24,26] and the references therein. Detecting periodic orbits in the Rössler System (1) has also been a subject of interest of many authors.…”

Computer assisted proofs of two-dimensional attracting invariant tori for ODEs

“…The next theorem is used to establish homoclinic connections between such two points. Computer assisted methods of proof for more general configurations are discussed in [48,49,50,51]. 1.…”