Bernoulli mapping with hole and a saddle-node scenario of the birth of hyperbolic Smale--Williams attractor

TL;DR

This paper models the emergence of hyperbolic chaotic attractors using a Bernoulli map with a hole, revealing cascade bifurcations and their persistence in multidimensional systems.

Contribution

It introduces a Bernoulli map with a hole to describe the saddle-node scenario of hyperbolic attractor formation, extending understanding of chaotic set emergence.

Findings

Chaotic sets arise via cascade of period-adding bifurcations.

Regularities are preserved in multidimensional models.

Limits of the 1D model's applicability are discussed.

Abstract

One-dimensional Bernoulli mapping with hole is suggested to describe the regularities of the appearance of a chaotic set under the saddle-node scenario of the birth of the Smale--Williams hyperbolic attractor. In such a mapping, a non-trivial chaotic set (with non-zero Hausdorff dimension) arises in the general case as a result of a cascade of period-adding bifurcations characterized by geometric scaling both in the phase space and in the parameter space. Numerical analysis of the behavior of models demonstrating the saddle-node scenario of birth of a hyperbolic chaotic Smale--Williams attractor shows that these regularities are preserved in the case of multidimensional systems. Limits of applicability of the approximate 1D model are discussed.