Explicit dynamical properties of the Pelikan random map in the chaotic region and at the intermittent critical point towards the non-chaotic region

TL;DR

This paper analyzes the dynamical properties of the Pelikan random map, exploring its behavior in chaotic and intermittent regimes through probabilistic and binary decomposition perspectives.

Contribution

It introduces a detailed probabilistic framework for understanding Pelikan map dynamics, including density evolution and binary variable analysis, across different parameter regimes.

Findings

Chaotic region exhibits typical and large deviation behaviors.

At the critical point, dynamics show intermittent properties.

In the non-chaotic regime, the system's behavior significantly differs from chaos.

Abstract

The Pelikan random trajectories $x_t \in [0,1[$ are generated by choosing the chaotic doubling map $x_{t+1}=2 x_t [mod 1]$ with probability $p$ and the non-chaotic half-contracting map $x_{t+1}=\frac{x_t}{2}$ with probability $(1-p)$. We compute various dynamical observables as a function of the parameter $p$ via two perspectives. In the first perspective, we focus on the closed dynamics within the subspace of probability densities that remain constant on the binary-intervals $x \in [ 2^{-n-1}, 2^{-n}[$ partitioning the interval $x \in [0,1[$ : the dynamics for the weights $\pi_t(n)$ of these intervals corresponds to a biased random walk on the half-infinite lattice $n \in \{0,1,2,..+\infty\}$ with resetting occurring with probability $p$ from the origin $n=0$ towards any site $n$ drawn with the distribution $2^{-n-1}$. In the second perspective, we study the Pelikan dynamics for any initial condition $x_0$ via the binary decomposition $x_t = \sum_{l=1}^{+\infty} \frac{\sigma_l (t)}{2^l} $, where the dynamics for the half-infinite lattice $l=1,2,..$ of the binary variables $\sigma_l(t) \in \{0,1\}$ can be reformulated in terms of two global variables : $z_t$ corresponds to a biased random walk on the half-infinite lattice $z \in \{0,1,2,..+\infty\}$ that may remain at the origin $z=0$ with probability $p$, while $F_t \in \{0,1,2,..t\}$ counts the number of time-steps $\tau \in [0,t-1]$ where $z_{\tau+1}=0=z_{\tau}$ and represents the number of the binary coefficients of the initial condition that have been erased. We discuss typical and large deviations properties in the chaotic region $\frac{1}{2}<p<1 $ as well as at the intermittent critical point $p_c=\frac{1}{2}$ towards the non-chaotic region $0<p<\frac{1}{2}$.