On Hyperbolic Attractors in Complex Shimizu -- Morioka Model

Abstract

We present a modified complex-valued Shimizu -- Morioka system with uniformly hyperbolic attractor. The numerically observed attractor in Poincaré cross-section is topologically close to Smale -- Williams solenoid. The arguments of the complex variables undergo Bernoulli-type map, essential for Smale -- Williams attractor, due to the geometrical arrangement of the phase space and an additional perturbation term. The transformation of the phase space near the saddle equilibrium "scatters" trajectories to new angles, then trajectories run from the saddle and return to it for the next "scatter". We provide the results of numerical simulations of the model and demonstrate typical features of the appearing hyperbolic attractor of Smale -- Williams type. Importantly, we show in numerical tests the transversality of tangent subspaces -- a pivotal property of uniformly hyperbolic attractor.