On Hyperbolic Attractors in Complex Shimizu -- Morioka Model

TL;DR

This paper introduces a modified complex Shimizu-Morioka system exhibiting a uniformly hyperbolic attractor similar to the Smale-Williams solenoid, confirmed through numerical simulations and transversality tests.

Contribution

The study presents a new complex-valued system with a hyperbolic attractor, demonstrating its properties and numerical evidence of hyperbolicity, advancing understanding of hyperbolic chaos in complex systems.

Findings

Numerical evidence of a hyperbolic attractor close to Smale-Williams solenoid.

Demonstration of transversality of tangent subspaces confirming hyperbolicity.

The attractor's topological structure is confirmed through Poincaré cross-section analysis.

Abstract

We present a modified complex-valued Shimizu -- Morioka system with uniformly hyperbolic attractor. The numerically observed attractor in Poincar\'{e} 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.