Explicit dynamical properties of the Pelikan random map in the chaotic region and at the intermittent critical point towards the non-chaotic region
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866908199696728064 |
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| author | Monthus, Cecile |
| author_facet | Monthus, Cecile |
| contents | 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 $0<p<\frac{1}{2}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_19999 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Explicit dynamical properties of the Pelikan random map in the chaotic region and at the intermittent critical point towards the non-chaotic region Monthus, Cecile Statistical Mechanics Dynamical Systems Probability Chaotic Dynamics 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 $0<p<\frac{1}{2}$. |
| title | Explicit dynamical properties of the Pelikan random map in the chaotic region and at the intermittent critical point towards the non-chaotic region |
| topic | Statistical Mechanics Dynamical Systems Probability Chaotic Dynamics |
| url | https://arxiv.org/abs/2409.19999 |