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

TL;DR

This work analyzes a two-parameter family of planar skew-product maps with two invariant subspaces that each host chaotic attractors, uncovering open regions where basins are intermingled and, alternatively, locally riddled. Using a thermodynamic formalism with stability indices based on Keller's framework, it quantifies transverse stability and derives a separating invariant graph $\phi^*$ that delineates basins; it further establishes blowout bifurcations as parameters cross critical curves. A multifractal analysis of the stability-index level sets yields Legendre-transform relations that describe the Hausdorff dimensions of these level sets, providing a fine-grained picture of basin geometry. The results advance understanding of complex basin structures in skew-product dynamics and offer a rigorous toolkit for measuring intermingling and fractal boundaries via Lyapunov spectra and thermodynamic quantities.

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.

Figures (5)

• Figure 1: This plot represents the fiber maps $g_a$ (solid) and $g_b$ (dashed) for $a=b=2/5$.
• Figure 2: The dashed curve represents the function $a\mapsto{a}/({1+a})$ and the solid black curve represents the function $a\mapsto{a}/{(1-a)}$. The area above the lower and below the upper curve are hatched differently and define the regions $\Gamma_0$ and $\Gamma_1$ in the $ab$-plane. The intersection of these areas are denoted by $\Gamma$. For each $(a,b)\in \Gamma$, both invariant sets $A_0$ and $A_1$ of the skew product system $F_{a,b}$ are Milnor attractors with locally riddled basins.
• Figure 3: The intermingled basins for the chaotic attractors $A_0$ and $A_1$ are shown for the parameters $a=0.45$ and $b=0.33$ and with respect to the SRB measure $\nu_{\mathrm{ac}}$. The black points corresponds to the basin of attraction $\mathcal{B}(A_0)$, while the white region corresponds to the basin of attraction $\mathcal{B}(A_1)$.
• Figure 4: The intermingled basins for the chaotic attractors $A_0$ and $A_1$ are shown for the parameters $a=0.45$ and $b=0.33$, but in this case, we consider a Gibbs measure $\nu$ that assigns weights of 1/3 and 2/3 to the intervals [0,1/2] and [1/2,1], respectively. The black points corresponds to the basin of attraction $\mathcal{B}(A_0)$, while the white region corresponds to the basin of attraction $\mathcal{B}(A_1)$.
• Figure 5: The separating graph $\phi^*$ for the parameters $a=0.49$, $b=0.49$.

Theorems & Definitions (46)

• Remark 2.1
• Definition 2.2: Invariant graph
• Remark 2.3
• Remark 2.4
• Definition 2.5
• Remark 2.6
• Definition 2.7
• Remark 2.8
• Proposition 2.9
• Theorem 1