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Abundance of weird quasiperiodic attractors in piecewise linear discontinuous maps

Laura Gardini, Davide Radi, Noemi Schmitt, Iryna Sushko, Frank Westerhoff

TL;DR

This paper characterizes a broad class of $n$-dimensional piecewise linear discontinuous maps that share a common fixed point and can host weird quasiperiodic attractors (WQAs). It proves that hyperbolic chaos cannot occur and that, in 2D nongeneric reductions, the dynamics on invariant segments reduce to $1$D PWL circle maps, while the generic nonfixed attractor is a WQA that may coexist with other invariant sets. The authors provide extensive 2D and 3D examples with varied discontinuity sets (lines and circles) and demonstrate mechanisms by which WQAs arise, including first-return maps and invariant polygons; they also generalize the framework to $\mathbb{R}^{n}$ and outline several open questions regarding stability, Lyapunov spectra, and transitions to chaos. The work offers a unified approach to understanding nonchaotic but richly structured attractors in discontinuous piecewise linear systems, with potential implications for modeling in economics and engineering. Overall, the results establish WQAs as a robust, generic attractor class in this setting and pave the way for further theoretical and applied exploration of higher-dimensional discontinuous dynamics.

Abstract

In this work, we consider a class of $n$-dimensional, $n\geq2$, piecewise linear discontinuous maps that can exhibit a new type of attractor, called a weird quasiperiodic attractor. While the dynamics associated with these attractors may appear chaotic, we prove that chaos cannot occur. The considered class of $n$-dimensional maps allows for any finite number of partitions, separated by various types of discontinuity sets. The key characteristic, beyond discontinuity, is that all functions defining the map have the same real fixed point. These maps cannot have hyperbolic cycles other than the fixed point itself. We consider the two-dimensional case in detail. We prove that in nongeneric cases, the restriction, or the first return, of the map to a segment of straight line is reducible to a piecewise linear circle map. The generic attractor, different from the fixed point, is a weird quasiperiodic attractor, which may coexist with other attractors or attracting sets. We illustrate the existence of these attractors through numerous examples, using functions with different types of Jacobian matrices, as well as with different types of discontinuity sets. In some cases, we describe possible mechanisms leading to the appearance of these attractors. We also give examples in the three-dimensional space. Several properties of this new type of attractor remain open for further investigation.

Abundance of weird quasiperiodic attractors in piecewise linear discontinuous maps

TL;DR

This paper characterizes a broad class of -dimensional piecewise linear discontinuous maps that share a common fixed point and can host weird quasiperiodic attractors (WQAs). It proves that hyperbolic chaos cannot occur and that, in 2D nongeneric reductions, the dynamics on invariant segments reduce to D PWL circle maps, while the generic nonfixed attractor is a WQA that may coexist with other invariant sets. The authors provide extensive 2D and 3D examples with varied discontinuity sets (lines and circles) and demonstrate mechanisms by which WQAs arise, including first-return maps and invariant polygons; they also generalize the framework to and outline several open questions regarding stability, Lyapunov spectra, and transitions to chaos. The work offers a unified approach to understanding nonchaotic but richly structured attractors in discontinuous piecewise linear systems, with potential implications for modeling in economics and engineering. Overall, the results establish WQAs as a robust, generic attractor class in this setting and pave the way for further theoretical and applied exploration of higher-dimensional discontinuous dynamics.

Abstract

In this work, we consider a class of -dimensional, , piecewise linear discontinuous maps that can exhibit a new type of attractor, called a weird quasiperiodic attractor. While the dynamics associated with these attractors may appear chaotic, we prove that chaos cannot occur. The considered class of -dimensional maps allows for any finite number of partitions, separated by various types of discontinuity sets. The key characteristic, beyond discontinuity, is that all functions defining the map have the same real fixed point. These maps cannot have hyperbolic cycles other than the fixed point itself. We consider the two-dimensional case in detail. We prove that in nongeneric cases, the restriction, or the first return, of the map to a segment of straight line is reducible to a piecewise linear circle map. The generic attractor, different from the fixed point, is a weird quasiperiodic attractor, which may coexist with other attractors or attracting sets. We illustrate the existence of these attractors through numerous examples, using functions with different types of Jacobian matrices, as well as with different types of discontinuity sets. In some cases, we describe possible mechanisms leading to the appearance of these attractors. We also give examples in the three-dimensional space. Several properties of this new type of attractor remain open for further investigation.

Paper Structure

This paper contains 15 sections, 4 theorems, 17 equations, 25 figures.

Table of Contents

  1. Introduction
  2. Discontinuous PWL homogeneous maps
  3. 1D discontinuous PWL homogeneous maps
  4. 2D discontinuous PWL homogeneous maps
  5. 2D PWL maps with discontinuity set $x=-1$
  6. PWL map $T_{1}$
  7. PWL map $T_{1}$ with $\delta_{L}=0$
  8. Further 2D PWL maps
  9. 2D PWL homogeneous maps with different discontinuity sets
  10. Discontinuity sets I: straight lines
  11. Discontinuity sets II: circles
  12. 2D PWL homogeneous map with the fixed point $O$ in two partitions
  13. Generalization to $\mathbb{R}^{n}$
  14. Further research
  15. Conclusions

Key Result

Theorem 1

Let $G$ be a 1D discontinuous PWL homogeneous map as in Definition 1. Then:

Figures (25)

  • Figure 1: In (a), 2D bifurcation diagram in the parameter plane ($\tau_{L},\tau_{R})$ for map $T_{1}$ in (\ref{['MapT']}), with $\delta_{R}=1.11$ and $\delta_{L}=0.9.$ The origin is unstable for $T_{R}$, while for $T_{L}$ the origin is a virtual attractor in the strip between the two vertical lines, at $\tau_{L}=\pm(1+\delta_{L})$. In (b), 1D bifurcation diagram as a function of $\tau_{L}$ at $\tau_{R}=-1.5.$ In (c), phase plane at $\tau_{L}=-2$ and $\tau_{R}=-1.449.$ Map $T_{1}$ has a region Z2 between $y=0.9$ ($=\delta_{L})$ and $y=1.11$ ($=\delta_{R})$.
  • Figure 2: Qualitative representation of the first return map on a segment. In (a), at the right side of the fixed point $O$. In (b), at the left side of the fixed point $O$.
  • Figure 3: In (a), 2D bifurcation diagram in the ($\delta_{L},\tau_{L})$ parameter plane for map $T_{1}$ in (\ref{['MapT']}), with $\delta_{R}=0.8$ and $\tau_{R}=-0.5.$ The black lines denote the standard stability triangle for the linear map $T_{L},$ bounded by segments of the lines of equation $\tau_{L}=\pm(1+\delta_{L})$ and $\delta_{L}=1.$ In (b), phase plane at $\delta_{L}=1.1$ and $\tau_{L}=1$ (black dot in (a)). Map $T_{1}$ has a region $Z_{0}$ between $y=\delta_{R}$ and $y=\delta_{L}.$ In (c), phase plane at $\delta_{R}=1,$$\tau_{R}=-0.5,$$\delta_{L}=0.98$ and $\tau_{L}=0.8.$ Map $T_{1}$ has a region $Z_{2}$ between $y=\delta_{L}$ and $y=\delta_{R}$. The fixed point $O$ is a center, with an irrational rotation number; the large white region in the $R$ partition is filled with invariant ellipses, bounded by the ellipse tangent to the discontinuity line and all its images.
  • Figure 4: In (a), 2D bifurcation diagram in the ($\tau_{L},\tau_{R})$ parameter plane for map $T_{1}$ in (\ref{['MapT']}), with $\delta_{R}=0.85$ and $\delta_{L}=0.$ The left partition is mapped by map $T_{1}$ onto the critical line $y=0$. In (b), phase plane at $\tau_{L}=0.9$ and $\tau_{R}=-1.8$ (black dot in (a)), the attracting fixed point $O$ coexists with another attractor having a segment on $y=0$. In (c), first return map in the segment of the attractor belonging to the line $y=0,$ that is a PWL circle map.
  • Figure 5: In (a), 2D bifurcation diagram in the $(\delta_{L},\tau_{L})$ parameter plane for map $T_{1}$ in (\ref{['MapT']}), with $\delta_{R}=1.5$ and $\tau_{R}=0.8.$ The stability triangle of the virtual fixed point is highlighted. At the black dot, $(\delta_{L},\tau_{L})=(0,0.4),$ the left partition is mapped by map $T_{1}$ onto the critical line $y=0$. The attractor existing in the phase plane is shown in (b). In (c), first return map in the segment of the attractor belonging to line $y=0,$ that is a PWL circle map.
  • ...and 20 more figures

Theorems & Definitions (6)

  • Definition 1: class of maps
  • Theorem 1: from GRSSW-25b
  • Lemma 1
  • Theorem 2
  • Theorem 3
  • Conjecture 1