Stability index, uncertainty exponent, and thermodynamic formalism for intermingled basins of chaotic attractors

TL;DR

This paper introduces a rigorous method using thermodynamic formalism to evaluate the stability index and uncertainty exponent, which quantify the intermingled basins in skew product systems with chaotic attractors.

Contribution

It provides a novel approach to rigorously estimate the stability index and uncertainty exponent for systems with intermingled basins using thermodynamic formalism.

Findings

Rigorous evaluation of the stability index and uncertainty exponent.

Application of thermodynamic formalism to skew product systems.

Enhanced understanding of intermingled basins in chaotic attractors.

Abstract

Skew product systems with monotone one-dimensional fibre maps driven by piecewise expanding Markov interval maps may show the phenomenon of intermingled basins. To quantify the degree of intermingledness the uncertainty exponent and the stability index were suggested by various authors and characterized (partially). Here we present an approach to evaluate/estimate these two quantities rigorously using thermodynamic formalism for the driving Markov map.