Multifractal analysis of intermingled basins and blowout bifurcations in a parametetric family of skew product maps

TL;DR

This paper investigates the complex intermingled basins and blowout bifurcations in a two-parameter family of skew product maps, revealing fractal basin boundaries and multifractal structures through detailed analysis.

Contribution

It introduces a multifractal analysis approach to characterize riddled basins and blowout bifurcations in a novel class of planar maps with invariant subspaces.

Findings

Intermingled basins of attraction are identified in the parameter space.

Fractal boundary curves separate the basins of chaotic attractors.

Multifractal analysis of stability index level sets is performed.

Abstract

In this paper we study a two-parameter family of planar maps characterized by two distinct invariant subspaces. The model reveals the existence of two chaotic attractors within these subspaces. We identify parameter values at which these attractors either exhibits a locally riddled basin of attraction or transitions into a chaotic saddle. In particular, we demonstrate that, for an open region in the parameter plane, their basins are intermingled. It is shown that a fractal boundary curve separates the basins of attraction of these two chaotic attractors, providing a detailed characterization of the riddled basin structure. Additionally, we show that the model undergoes a blowout bifurcation. An estimation of the stability index is examined using thermodynamic formalism. We also perform a multifractal analysis of the level sets of the stability index.