On the emergence and properties of weird quasiperiodic attractors

TL;DR

This paper analyzes a simplified 2D discontinuous piecewise linear map $F$ with a single discontinuity line to study weird quasiperiodic attractors (WQAs). It derives the bifurcation structure in parameter spaces, identifies divergence regions $D_{1/n}^R$ and $D_{1/n}^L$ linked to basic and complementary symbolic sequences, and provides explicit boundary conditions $B_\sigma$ (along with admissibility criteria) that mark where WQAs can appear. The authors show how WQAs arise near these divergence boundaries via mechanism involving invariant halflines and their images crossing the discontinuity, and they illustrate coexistence with fixed points and with divergent dynamics. The work clarifies that, although intricate, WQAs are not chaotic since $F$ cannot support hyperbolic cycles, and it lays out a framework for understanding WQAs in similar discontinuous, homogeneous maps, raising questions about their occurrence in higher dimensions and under nonlinearity.

Abstract

We recently described a specific type of attractors of two-dimensional discontinuous piecewise linear maps, characterized by two discontinuity lines dividing the phase plane into three partitions, related to economic applications. To our knowledge, this type of attractor, which we call a weird quasiperiodic attractor, has not yet been studied in detail. They have a rather complex geometric structure and other interesting properties that are worth understanding better. To this end, we consider a simpler map that can also possess weird quasiperiodic attractors, namely, a 2D discontinuous piecewise linear map $F$ with a single discontinuity line dividing the phase plane into two partitions, where two different homogeneous linear maps are defined. Map $F$ depends on four parameters -- the traces and determinants of the two Jacobian matrices. In the parameter space of map $F$, we obtain specific regions associated with the existence of weird quasiperiodic attractors; describe some characteristic properties of these attractors; and explain one of the possible mechanisms of their appearance.

Figures (13)

• Figure 1: Attractors of map $F$ for (a) $\delta_{L}=0.75$, $\delta_{R}=1.2$, $\tau_{L}=-0.7$, $\tau_{R}=-2.5$; (b) $\delta_{L}=0.7$, $\delta_{R}=1.001$, $\tau_{L}=0.3$, $\tau_{R}=0.71$; (c) $\delta_{L}=0.9$, $\delta_{R}=1.1$, $\tau_{L}=-2.5$, $\tau_{R}=-0.7$; (d) $\delta_{L}=0.9$, $\delta_{R}=1.1$, $\tau_{L}=-2.5$, $\tau_{R}=-1.2$; (e) $\delta_{L}=0.84$, $\delta_{R}=1.15$, $\tau_{L}=-1$, $\tau_{R}=-1.9$; (f) $\delta_{L}=1.05$, $\delta_{R}=0.7$, $\tau_{L}=-0.75$, $\tau_{R}=-1.6$. Discontinuity line $x=-1$ and its images, $y=\delta_{L}$ and $y=\delta_{R},$ by maps $F_{L}$ and $F_{R},$ respectively, are also shown.
• Figure 2: Two coexisting weird quasiperiodic attractors (shown in black and dark blue) of map $F$ and their basins (in light blue and yellow) for (a) $\delta_{L}=0.9$, $\delta_{R}=1.1$, $\tau_{L}=0.3$, $\tau_{R}=0.71;$ (b) $\delta_{L}=0.9$, $\delta_{R}=1.11$, $\tau_{L}=-2$, $\tau_{R}=-1.91$.
• Figure 3: Bifurcation structure of map $F$ (a) in the $(\tau_{L},\tau_{R})$-parameter plane for $\delta_{L}=0.9,$$\delta_{R}=0.7$ , and (b) in the $(\delta_{R},\tau_{R})$-parameter plane for $\delta_{L}=0.9,$$\tau_{L}=-2.5$. The boundaries $B_{LR^{n-1}}$ and $B_{L^{2}R^{n-2}}$ of the divergence region $D_{1/n}^{R},$ defined in (\ref{['B_LRn-1']}) and (\ref{['B_L2Rn-2']}), respectively, and the boundaries $B_{RL^{n-1}}$ and $B_{R^{2}L^{n-2}}$ of the divergence region $D_{1/n}^{L}$ defined in (\ref{['B_RLn-1']}) and (\ref{['B_R2Ln-2']}), respectively, are shown for $n=2,...,9.$ Blue and yellow parameter regions indicate convergence to the fixed point $O$ and to a WQA, respectively. Gray regions indicate divergence.
• Figure 4: A 1D bifurcation diagram of map $F$ showing $x$ versus $\tau_{L}$ in (a), and $y$ versus $\tau_{L}$ in (b) for $\delta_{L}=0.9,$$\delta_{R}=0.7,$ and $\tau_{R}=-2.$
• Figure 5: (a) The existence regions (in yellow) of the WQAs near the divergence region $D_{1/5}^{R}$ (in gray) bounded by curves $B_{LR^{4}}$ and $B_{L^{2}R^{3}}$ in the $(\tau_{L},\tau_{R})$-parameter plane for $\delta_{L}=0.9,$$\delta_{R}=0.7;$ (b) 1D bifurcation diagram $x$ versus $\tau_{R}$ for $\tau_{L}=-2$. The phase portrait of map $F$ at parameter points marked in (a) by $a,$$b,$ ..., $f$ is shown in Fig. \ref{['D5Rexam']}(a),(b),...,(f), respectively, and by $g$ in Fig. \ref{['D5Rex3']}(a).