# The Emergence of the Normal Distribution in Deterministic Chaotic Maps

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## Abstract

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## 1. Introduction

## 2. Central Limit Theorem for Deterministic Maps

## 3. The Bernoulli Map

## 4. The Logistic Map: Full Chaos and Intermittency

## 5. A Fat-Tailed Invariant Distribution

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Calculation of the Variances of N-Term Sums, Equation (6)

#### Appendix A.1. The Bernoulli Map

#### Appendix A.2. The Logistic Map in the Fully Chaotic Regime

## Appendix B. Kullback–Leibler Divergence for the Distribution of Random Sampling Sums

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**Figure 1.**

**Left**, dark line: Numerical results for the distribution ${\rho}_{{s}_{N}}$ of the sums ${s}_{N}$ defined in Equation (3), in the case the Bernoulli (10) map with $m=2$, for three small values of N. The light curve is the Gaussian expected for $N\to \infty $, and the dashed curve is a Gaussian with the same variance as predicted for ${\rho}_{{s}_{N}}$.

**Right**, dark curve: The distribution ${\rho}_{{s}_{N}}^{\mathrm{random}}$ for sums of N values of x randomly sampled from ${\rho}_{x}\left(x\right)$ is a normalized version of the Irwin–Hall distribution [15,16], which can be obtained analytically through the successive self-convolution of ${\rho}_{x}$. The light curve is the Gaussian expected for $N\to \infty $. Note the different scales on the left and right columns.

**Figure 2.**Main panel: The Kullback–Leibler divergences ${D}_{G}$, ${D}_{{G}_{N}}$, and ${D}_{\mathrm{random}}$, defined in the text, as functions of the number of terms in the sums ${s}_{N}$ of Equation (3), for the Bernoulli map (10) with $m=2$. The straight lines in this log–log plot have a slope $-2$. The inset shows, as dots, numerical results for the variance ${\sigma}_{{s}_{N}}^{2}$ over the distribution ${\rho}_{{s}_{N}}\left({s}_{N}\right)$. The dashed line joins the analytical values predicted from Equation (12).

**Figure 3.**The Kullback–Leibler divergence ${D}_{G}$ for the Bernoulli map (10) with various values of m, and ${D}_{\mathrm{random}}$ (which is the same for all m). The straight lines have slope $-2$.

**Figure 5.**The Kullback–Leibler divergence ${D}_{G}$ for the logistic map of Equation (16) in the regime of full chaos, $\lambda =4$, and ${D}_{\mathrm{random}}$, as a functions of N. In this case, ${D}_{{G}_{N}}$ coincides with ${D}_{G}$. The full and dashed straight lines have slopes $-2$ and $-1$, respectively.

**Figure 6.**

**Left**: 900 successive iterations of the logistic map, Equation (16), in the intermittent regime, $\lambda =3.828$. The arrows at $t=300$ and 500 point at “turbulent” and period-3 “laminar” intervals, respectively.

**Right**: The correlation ${c}_{k}=\overline{[x\left(t\right)-\overline{x}][x(t+k)-\overline{x}]}$ as a function of k in the same intermittent regime, calculated numerically from sequences of ${10}^{7}$ iterations of $x\left(t\right)$. Symbols are connected by lines to facilitate visualization.

**Figure 7.**The Kullback–Leibler divergences ${D}_{G}$ for the logistic map (16) in the intermittent regime, $\lambda =3.828$, and ${D}_{\mathrm{random}}$, as functions of N. For the former, triangles correspond to values of N which are multiples of 3. The slope of the dashed straight line is $-1$. Inset: Numerical results for the variance ${\sigma}_{{s}_{N}}^{2}$ of the sums ${s}_{N}$, as a function of N. The arrow to the right indicates the variance obtained for large N. Symbols are connected by dashed lines to facilitate visualization.

**Figure 9.**The Kullback–Leibler divergences ${D}_{C}$ and ${D}_{\mathrm{random}}$ for the distributions of the sums of Equation (18), with the values of x obtained from the map (20) and the distribution of Equation (22), respectively. The full straight line has slope $-1$, and the dashed line, with slope $-0.68$, is a linear fitting of ${D}_{C}$ for $N\ge 2$.

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**MDPI and ACS Style**

Zanette, D.H.; Samengo, I.
The Emergence of the Normal Distribution in Deterministic Chaotic Maps. *Entropy* **2024**, *26*, 51.
https://doi.org/10.3390/e26010051

**AMA Style**

Zanette DH, Samengo I.
The Emergence of the Normal Distribution in Deterministic Chaotic Maps. *Entropy*. 2024; 26(1):51.
https://doi.org/10.3390/e26010051

**Chicago/Turabian Style**

Zanette, Damián H., and Inés Samengo.
2024. "The Emergence of the Normal Distribution in Deterministic Chaotic Maps" *Entropy* 26, no. 1: 51.
https://doi.org/10.3390/e26010051