Stability index, uncertainty exponent, and thermodynamic formalism for intermingled basins of 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 [1,5,16,30]. To quantify the degree of intermingledness the uncertainty exponent [23] and the stability index [29,20] were suggested and characterized (partially). Here we present an approach to evaluate/estimate these two quantities rigorously using thermodynamic formalism for the driving Markov map.

Chaotically driven sigmoidal maps

“…The result is that ϕ + from [9] is the lower bounding graph of our Gϕ + , and similar for ϕ − . Having this in mind, it is obvious that all examples from [9] belong to our class A). Indeed, Example 2.3 and Figure 1 of [9] illustrate situations, where all invariant measures lead to a) ε = 0.018: The lhs plot shows two trajectories, one starting at y = −1 and one starting at y = 1.…”

“…Having this in mind, it is obvious that all examples from [9] belong to our class A). Indeed, Example 2.3 and Figure 1 of [9] illustrate situations, where all invariant measures lead to a) ε = 0.018: The lhs plot shows two trajectories, one starting at y = −1 and one starting at y = 1. Each orbit starting above the dashed line (at y ε = 0.3) in the rhs plot will stay above it, and one starting below the dotted line (at −y ε = −0.3) will stay below it.…”

“…Extending the fibres and fibre maps in the examples from [9] slightly, puts us in the context of the present paper. The result is that ϕ + from [9] is the lower bounding graph of our Gϕ + , and similar for ϕ − . Having this in mind, it is obvious that all examples from [9] belong to our class A).…”

“…(Intermingled basins) Nearly the same model as in this note was studied in [9], where the focus was on quantifying the phenomenon of intermingled basins using thermodynamic formalism. The formal difference to the present setting is that f x (I) = I is assumed in [9] (so that the end points of I describe constant invariant graphs given a priori), while here we require f x (I) ⊆ I • (Hypothesis 2). Extending the fibres and fibre maps in the examples from [9] slightly, puts us in the context of the present paper.…”

“…The formal difference to the present setting is that f x (I) = I is assumed in [9] (so that the end points of I describe constant invariant graphs given a priori), while here we require f x (I) ⊆ I • (Hypothesis 2). Extending the fibres and fibre maps in the examples from [9] slightly, puts us in the context of the present paper. The result is that ϕ + from [9] is the lower bounding graph of our Gϕ + , and similar for ϕ − .…”