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Interval maps mimicking circle rotations

Jakub Bielawski, Thiparat Chotibut, Fryderyk Falniowski, Michał Misiurewicz, Georgios Piliouras

Abstract

We investigate the dynamics of maps of the real line whose behavior on an invariant interval is close to a rational rotation on the circle. We concentrate on a specific two-parameter family, describing the dynamics arising from models in game theory, mathematical biology and machine learning. If one parameter is a rational number, $k/n$, with $k,n$ coprime, and the second one is large enough, we prove that there is a periodic orbit of period $n$. It behaves like an orbit of the circle rotation by an angle $2πk/n$ and attracts trajectories of Lebesgue almost all starting points. We also discover numerically other interesting phenomena. While we do not give rigorous proofs for them, we provide convincing explanations.

Interval maps mimicking circle rotations

Abstract

We investigate the dynamics of maps of the real line whose behavior on an invariant interval is close to a rational rotation on the circle. We concentrate on a specific two-parameter family, describing the dynamics arising from models in game theory, mathematical biology and machine learning. If one parameter is a rational number, , with coprime, and the second one is large enough, we prove that there is a periodic orbit of period . It behaves like an orbit of the circle rotation by an angle and attracts trajectories of Lebesgue almost all starting points. We also discover numerically other interesting phenomena. While we do not give rigorous proofs for them, we provide convincing explanations.

Paper Structure

This paper contains 5 sections, 3 theorems, 22 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Notation and assumptions
  3. Rotation-like behavior
  4. EOS maps
  5. Bifurcations for a rational $b$ for EOS maps

Key Result

Lemma 3.1

For every $F\in \mathfrak{F}$ there exists an attracting periodic orbit of $F$ of period $n$, that lies in first and third lap. Both critical points of $F$ are in the immediate basin of attraction of this periodic orbit.

Figures (6)

  • Figure 1: The maps $F(x)=x+b-\frac{1}{e^{-90x}+1}$ (black) and $G$ (cyan) for $b=3/11$. The map $F$ belongs to $\mathfrak{F}$. The yellow region is the set $[b-1,b]^2$.
  • Figure 2: The maps $F$ (green) and $F^3$ (black) defined by \ref{['e2']} for $b=1/3$ and $a=40$. Horizontal and vertical red lines are at the levels $b-1$, $b-2/3$, $b-1/3$ and $b$.
  • Figure 3: Bifurcation diagrams for EOS maps $b=k/11$, where $k=1,2,3,4,5$. On the horizontal axis there is $a$, from $100$ to $180$; the vertical cyan lines are every $10$. Horizontal red lines indicate the positions of the critical points.
  • Figure 4: Graphs of an EOS map $F$ for $a=110$ on $[b-1,b]$. In blue we see the graphs of $F^{r_j}$ on $J_j$, and in magenta the graphs of $F^n$ on $K_j$.
  • Figure 5: Bifurcation diagrams for $b=k/11$, where $k=1,2,3,4,5$. On the horizontal axis there is $a$, from $80$ to $89$; the vertical cyan lines are every $1$. Horizontal red lines indicate the positions of the critical points.
  • ...and 1 more figures

Theorems & Definitions (8)

  • Remark 2.1
  • Remark 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • Remark 4.1