Parameter space arrangement of a model system nearby domain of existence of Plykin type attractor

TL;DR

This paper investigates the parameter space around the existence domain of Plykin-type attractors in a sphere transformation model, using numerical methods to identify regions of hyperbolic chaos and other dynamics.

Contribution

It introduces a computational approach to map the parameter space of a sphere transformation model and identify the hyperbolic Plykin attractor region.

Findings

Plykin attractor exists in a bounded parameter region with hyperbolic dynamics.

Outside this region, non-hyperbolic chaos, periodic, and quasiperiodic behaviors occur.

The study provides charts and portraits illustrating different dynamical regimes.

Abstract

For a model system defined as combination of sequentially applied continuous transformations of a sphere, the question of arrangement of the parameter space around the domain of existence of the Plykin-type attractor is considered. Results of numerical calculations are presented, including charts of dynamical regimes and Lyapunov exponents on the parameter plane, as well as portraits of attractors in characteristic regions of chaotic and regular dynamics. The Plykin attractor region is determined and depicted using the computational procedure for checking hyperbolicity, which consists in analyzing angles between expanding and contracting tangent subspaces of typical trajectories on the attractor. The Plykin attractor takes place in a bounded continuous region of the parameter plane that corresponds to roughness (structural stability) of the hyperbolic dynamics. Outside that region, various types of dynamics are observed including non-hyperbolic chaos, periodic and quasiperiodic behaviors.