Phase transition in the singularity spectrum of an intermingled basin

A two-dimensional piecewise linear mapping is introduced as a solvable model to characterize the multifractal structure of an intermingled basin. To this end, we make use of the multifractal formalism and introduce a partition function. The singularity spectrum, which characterizes local scaling property of the intermingled basin, is then determined. We have found that if the system is not symmetric, the singularity spectrum of either basin shows a phase transition, corresponding to the existence of two phases the orbits experience in the system, i.e., local one governed by the chaotic motions on the chaotic attractor, and the other global one reflecting nonhyperbolic motions characteristic of the intermingled basin.

Figure 1

Maps of (a) Eq. (2) and (b) Eq. (3). In (b), for $y_t<\frac{2}{3}$, the upper branch is used if and only if $x_t<a$, and, for $y_t\ge \frac{2}{3}$, the lower branch is used if and only if $x_t\ge a+b$.

Figure 2

Intermingled basins for (a) $a=0.35$ and $c=0.4$, and (b) $a=0.4$ and $c=0.35$. The basins shown in black and white have $f\left(\alpha \right)$ of Figs. 5 and 5, respectively, in both (a) and (b).

Figure 3

Rectangles $R^k$ ($k=0,1,2$) and $R_n$ ($n=0,\pm 1,\pm 2,...$). The interior of $R^0\cap R_n$ ($R^2\cap R_n$) is mapped onto the interior $R_{n+1}$ ($R_{n-1}$). The interior of $R^1\cap R_n$ with $n\ge 1$ ($n\le 0$) is mapped onto the interior of $R_{n+1}$ ($R_{n-1}$). The arrows indicate the directions of mapping, e.g., the light gray region is mapped onto the gray region.

Figure 4

Corresponding Markov chain. The state “$n$” corresponds to the rectangle $R_n$, where $n=0,\pm 1,\pm 2,...$.

Figure 5

Singularity spectra for (a) $a=0.35$ and $c=0.4$, and (b) $a=0.4$ and $c=0.35$. In (b), $f\left(\alpha \right)$ has a linear slope in the range $1\le \alpha \le {\alpha }^{*}=1.12994$.

Figure 6

$\alpha \left(q\right)$ for $a=0.35$ and $c=0.4$ (solid line), and for $a=0.4$ and $c=0.35$ (dotted line) exhibiting a jump at $q=q^{*}=0.128284$.

Figure 7

Graphical expressions of (a) $M$ and (b) $\underline{M}$.