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Invariant graphs and dynamics of a family of continuous piecewise linear planar maps

Anna Cima, Armengol Gasull, Víctor Mañosa, Francesc Mañosas

TL;DR

The paper analyzes the two-parameter family F_{a,b}(x,y)=(|x|-y+a, x-|y|+b) of continuous piecewise-linear maps, revealing a sharp dichotomy: for a≥0 orbits are eventually periodic with highly constrained periodic behavior, while for a<0 all dynamics concentrates on a finite invariant graph Γ, reducing the study to one dimension. It proves the existence of an invariant graph Γ for a<0 (via conjugation to a=-1) and provides an explicit, case-by-case construction of Γ across parameter regimes; the long-term dynamics is then analyzed via entropy on Γ and rotation-number theory when Γ is a circle. A central contribution is a detailed catalog of invariant graphs (up to 37 topologically distinct forms) and precise entropy calculations h(F|_Γ) in many b-intervals, including discontinuities and Cantor ω-limit phenomena; the results explain the prevalence of simple dynamics in numerical simulations and establish a unified framework for generalized Lozi-type maps. The work blends topological entropy theory on graphs, Markov-partition methods, and unimodal/trapezoidal dynamics to characterize complex one-dimensional behavior arising from a two-dimensional dissipative system, with potential impact on the study of other piecewise-linear maps with rank-1 collapse regions.

Abstract

We consider the family of piecewise linear maps $$F_{a,b}(x,y)=\left(|x| - y + a, x - |y| + b\right),$$ where $(a,b)\in \mathbb{R}^2$. This family belongs to a wider one that has deserved some interest in the recent years as it provides a framework for generalized Lozi-type maps. Among our results, we prove that for $a\ge 0$ all the orbits are eventually periodic and moreover that there are at most three different periodic behaviors formed by at most seven points. For $a<0$ we prove that for each $b\in\mathbb{R}$ there exists a compact graph $Γ,$ which is invariant under the map $F$, such that for each $(x,y)\in \mathbb{R}^2$ there exists $n\in\mathbb{N}$ (that may depend on $x$) such that $F_{a,b}^n(x,y)\in Γ.$ We give explicitly all these invariant graphs and we characterize the dynamics of the map restricted to the corresponding graph for all $(a,b)\in\mathbb{R}^2$ obtaining, among other results, a full characterization of when $F_{a,b}|_Γ$ has positive or zero entropy.

Invariant graphs and dynamics of a family of continuous piecewise linear planar maps

TL;DR

The paper analyzes the two-parameter family F_{a,b}(x,y)=(|x|-y+a, x-|y|+b) of continuous piecewise-linear maps, revealing a sharp dichotomy: for a≥0 orbits are eventually periodic with highly constrained periodic behavior, while for a<0 all dynamics concentrates on a finite invariant graph Γ, reducing the study to one dimension. It proves the existence of an invariant graph Γ for a<0 (via conjugation to a=-1) and provides an explicit, case-by-case construction of Γ across parameter regimes; the long-term dynamics is then analyzed via entropy on Γ and rotation-number theory when Γ is a circle. A central contribution is a detailed catalog of invariant graphs (up to 37 topologically distinct forms) and precise entropy calculations h(F|_Γ) in many b-intervals, including discontinuities and Cantor ω-limit phenomena; the results explain the prevalence of simple dynamics in numerical simulations and establish a unified framework for generalized Lozi-type maps. The work blends topological entropy theory on graphs, Markov-partition methods, and unimodal/trapezoidal dynamics to characterize complex one-dimensional behavior arising from a two-dimensional dissipative system, with potential impact on the study of other piecewise-linear maps with rank-1 collapse regions.

Abstract

We consider the family of piecewise linear maps where . This family belongs to a wider one that has deserved some interest in the recent years as it provides a framework for generalized Lozi-type maps. Among our results, we prove that for all the orbits are eventually periodic and moreover that there are at most three different periodic behaviors formed by at most seven points. For we prove that for each there exists a compact graph which is invariant under the map , such that for each there exists (that may depend on ) such that We give explicitly all these invariant graphs and we characterize the dynamics of the map restricted to the corresponding graph for all obtaining, among other results, a full characterization of when has positive or zero entropy.
Paper Structure (27 sections, 38 theorems, 42 equations, 61 figures, 1 table)

This paper contains 27 sections, 38 theorems, 42 equations, 61 figures, 1 table.

Table of Contents

  1. Introduction and main results
  2. Preliminary results
  3. Topological entropy
  4. Rotation numbers and set of periods of a rotation interval
  5. Additional notation
  6. The case $a\ge 0$
  7. The case $a=1$
  8. The case $a=0.$ Proof of Theorem \ref{['t:teoA']}
  9. The case $a<0$ (I). Proofs of Theorems \ref{['t:teoB']} and \ref{['t:teoC']}
  10. Proof of Theorem \ref{['t:teoB']}
  11. Structure of the $\omega$-limits and proof of Theorem \ref{['t:teoC']}
  12. The case $a<0$ (II). Case analysis and proof of Theorem \ref{['t:teoD']}
  13. The case $a=-1$ and $b\leq -2$
  14. The case $a=-1$ and $-2< b\leq -1$
  15. The case $a=-1$ and $-1< b\leq -3/4$
  16. ...and 12 more sections

Key Result

Theorem A

If $a\ge 0,$ for each $\mathbf{x}\in{\mathbb R}^2$ there exists $n\ge 0,$ that may depend on $\mathbf{x},$ such that $F^n(\mathbf{x})\in \operatorname{Per}(F).$ Moreover, the set $\operatorname{Per}(F)$ is formed by a fixed point and, depending on $a$ and $b,$ either two or none 3 periodic orbits.

Figures (61)

  • Figure 1: Dynamics of $F^3$ on $Q_1$ when $b>2.$$(a)$ The polygonal $L=F^3(Q_1)$, left; $(b)$ The graphic of $F^3$ restricted to $L$, right.
  • Figure 2: Dynamics of $F^3$ on $Q_1.$$(a)$ The polygonal $K=F^3(Q_1)$, left. $(b)$ Graphic of $F^3$ restricted to $K$, right.
  • Figure 3: Images of $F(Q_1)$ and $F(Q_3)$ (in the figure $b<0$).
  • Figure 4: The evolution of $K$ under the action of $F$ when $-1<b\leq 0.$
  • Figure 5: The evolution of $K$ under the action of $F$ when $0<b\leq {1}/{2}.$
  • ...and 56 more figures

Theorems & Definitions (73)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Lemma 1
  • Remark 2
  • Lemma 3
  • Remark 4
  • Definition 5
  • Theorem 6
  • ...and 63 more