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

Cecile Monthus

TL;DR

This work analyzes the Pelikan random map, a random mixture of a chaotic doubling map and a non-chaotic half-contraction, by developing two complementary frameworks: a closed density-subspace perspective leading to a Markov chain with resetting and a binary-decomposition perspective yielding two global variables $z_t$ and $F_t$. It derives explicit stationary states, finite-time propagators, first-passage-time distributions, and spectral decompositions, with precise chaotic-region and critical-point scalings, including large-deviation rate functions for additive observables. The results illuminate how resetting and conditioning shape relaxation, rare events, and time correlations across chaotic, non-chaotic, and critical regimes, enabling exact characterizations of convergence, excursion dynamics, and density fluctuations in this paradigmatic random-dynamics model. The insights have broad relevance for non-equilibrium statistical physics, intermittent dynamics, and stochastic processes with resetting, offering exact benchmarks for similar systems and informing the design of analytic techniques for complex dynamical maps.

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 $π_t(n)$ 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{σ_l (t)}{2^l} $, where the dynamics for the half-infinite lattice $l=1,2,..$ of the binary variables $σ_l(t) \in \{0,1\}$ 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 $τ\in [0,t-1]$ where $z_{τ+1}=0=z_τ$ 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 $0<p<\frac{1}{2}$.

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 work analyzes the Pelikan random map, a random mixture of a chaotic doubling map and a non-chaotic half-contraction, by developing two complementary frameworks: a closed density-subspace perspective leading to a Markov chain with resetting and a binary-decomposition perspective yielding two global variables and . It derives explicit stationary states, finite-time propagators, first-passage-time distributions, and spectral decompositions, with precise chaotic-region and critical-point scalings, including large-deviation rate functions for additive observables. The results illuminate how resetting and conditioning shape relaxation, rare events, and time correlations across chaotic, non-chaotic, and critical regimes, enabling exact characterizations of convergence, excursion dynamics, and density fluctuations in this paradigmatic random-dynamics model. The insights have broad relevance for non-equilibrium statistical physics, intermittent dynamics, and stochastic processes with resetting, offering exact benchmarks for similar systems and informing the design of analytic techniques for complex dynamical maps.

Abstract

The Pelikan random trajectories are generated by choosing the chaotic doubling map with probability and the non-chaotic half-contracting map with probability . We compute various dynamical observables as a function of the parameter 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 partitioning the interval : the dynamics for the weights of these intervals corresponds to a biased random walk on the half-infinite lattice with resetting occurring with probability from the origin towards any site drawn with the distribution . In the second perspective, we study the Pelikan dynamics for any initial condition via the binary decomposition , where the dynamics for the half-infinite lattice of the binary variables can be reformulated in terms of two global variables : corresponds to a biased random walk on the half-infinite lattice that may remain at the origin with probability , while counts the number of time-steps where 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 as well as at the intermittent critical point towards the non-chaotic region .
Paper Structure (71 sections, 310 equations, 1 figure)

This paper contains 71 sections, 310 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Reminder on the Pelikan random map on the interval $x \in [0,1[$
  3. Pelikan random trajectories $x_{t=0,1,2,..}$ on the interval $x \in [0,1[$
  4. Distribution of the finite-time Lyapunov exponent on the time-window $[0,T]$
  5. Dynamics for the probability density $\rho_t( x)$ on the real space interval $x \in [0,1[$
  6. Markov chain dynamics in the simplest subspace of probability densities
  7. Closed dynamics for the weights $\pi_t(n)$ of the binary intervals $x \in [ 2^{-n-1}, 2^{-n}[$
  8. Steady state $\pi_*(n)$ in the chaotic region $\frac{1}{2} <p \leq 1$
  9. Derivation as the non-equilibrium steady state $\pi_*(n)$ of the biased random walk with resetting from the origin
  10. Reminder on the properties of the steady state $\rho_*(x)$ as a function of the parameter $\frac{1}{2} < p \leq 1$
  11. Discussion
  12. Properties of the finite-time propagator $\pi_t(n \vert n_0)$ for the weights
  13. Explicit time-Laplace-transform ${\tilde{\pi}}_s(n \vert n_0)$ of the propagator $\pi_t(n \vert n_0)$
  14. Asymptotic analysis in the chaotic region $\frac{1}{2} < p \leq 1$
  15. Asymptotic analysis in the non-chaotic region $0<p < \frac{1}{2}$
  16. ...and 56 more sections

Figures (1)

  • Figure 1: The Markov chain for the weights $\pi_t(n)$ on the half-infinite lattice $n=0,1,2,...+\infty$ corresponds to a biased random walk that jumps to the left with probability $p$ and to the right with probability $(1-p)$, while at the origin $n=0$, the impossible jump to the left is replaced by a reset towards any site $n=0,1,..,+\infty$ with the probability $p 2^{-n-1}$.