Uniformly Hyperbolic Attractor of the Smale–Williams Type for a Poincaré Map in the Kuznetsov System

Abstract

Abstract. We propose a general algorithm for computer assisted verification of uniform hyperbolicity for maps which exhibit a robust attractor.The method has been successfully applied to a Poincaré map for a system of coupled non-autonomous van der Pol oscillators. The model equation has been proposed by Kuznetsov [K] and the attractor seems to be of the SmaleWilliams type.

When chaos meets hyperchaos: 4D Rössler model

“…Moreover, the global existence of the hyperchaotic invariant set gives an explanation of the difficulty in most of the numerical simulations in literature to provide with a clear study of the behavior of the systems. As to obtain an analytical proof of this fact is not possible (or at least quite complex), we give some computerassisted proofs, thanks to the use of the CAPD library [36] of rigorous computing and the techniques developed in [37][38][39][40]. To that goal, we take the parameter values a = 0.27857, b = 3, c = 0.3 and d = 0.05, where we have numerically detected the coexistence of a hyperchaotic saddle and a chaotic attractor, and we define a Poincaré section This proves (see [40,Lemma 3.5]) that B is a trapping region for P , namely P (B) ⊂ B.…”

Hyperbolic chaotic attractor in amplitude dynamics of coupled self-oscillators with periodic parameter modulation

“…Computations based on verification of the so-called cone criterion [10][11][12][13][14][15] confirmed the hyperbolic nature of the attractors [28] and, for a particular model system considered in Refs. [24,28], an accurate analysis based on the technique of computer-assisted proof has been performed by Wilczak [29].…”

A “saddle-node” bifurcation scenario for birth or destruction of a Smale–Williams solenoid

“…In this system attractor of Smale-Williams type occurs in the stroboscopic Poincaré map. At certain parameters the uniformly hyperbolic nature of this attractor was verified in computations [10,12,19,20]. Here we consider birth or disappearance of the chaotic attractor under variation of a parameter controlling a relative duration of the stages of activity and silence.…”