# Unveiling Dynamical Symmetries in 2D Chaotic Iterative Maps with Ordinal-Patterns-Based Complexity Quantifiers

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## Abstract

**:**

## 1. Introduction

## 2. Ordinal Patterns and Complexity Quantifiers

#### 2.1. Ordinal Patterns (OPs)

#### 2.2. Permutation Entropy (PE)

#### 2.3. Fisher Information Measure (FIM)

#### 2.4. Temporal and Reversible DYnamical Symmetries (TARDYS)

## 3. Hénon Map

#### 3.1. 1D Analysis: $b=0.3$, $1.0\le a\le 1.4$

#### 3.2. Two-Parameters Analysis

#### 3.3. Initial Conditions Analysis

## 4. The Standard Map

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Figure A1.**PE (

**a**,

**c**) and ${T}_{\delta}$ (

**b**,

**d**) versus the control parameters a and b for the Burger map (

**a**,

**b**) and the Lozi map (

**c**,

**d**).

## References

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**Figure 1.**(

**a**) Bifurcation diagram of the Hénon map for $b=0.3$, scanning between $1.0\le a\le 1.4$. (

**b**) Words’ probabilities of dimension $D=3$ ($w1=\u2018012\u2019$, $w2=\u2018021\u2019$, $w3=\u2018102\u2019$, $w4=\u2018201\u2019$, $w5=\u2018201\u2019$, and $w6=\u2018210\u2019$) for the range of parameters as (

**a**). Words’ probabilities capture the different dynamical regimes, from chaos to regularity, but can also capture different kinds of periodicity, as well as capture dynamical constraints in the chaotic regions, distinguishing among different families of chaos.

**Figure 2.**(

**a**) TARDYS quantifiers (see Equation (3)) for the Hénon map. The regions between the vertical dotted lines indicate regions with different dynamics. The region between the red dashed lines is zoomed-in in (

**c**). (

**b**) Bifurcation diagram for the zoomed-in region, which presents a small window of periodicity anticipated by ${T}_{\alpha}\ne {T}_{\beta}$.

**Figure 3.**Permutation entropy (PE) for the Hénon map for the range $0\le a\le 2$ and $-0.5\le b\le 0.5$ with initial conditions as in previous figures. Differentiated regions (I to V) identify regions with different dynamics and temporal constraints.

**Figure 4.**TARDYS quantifiers for the Hénon map for the same region as Figure 3. (

**a**) ${T}_{\alpha}$, (

**b**) ${T}_{\beta}$, (

**c**) ${T}_{\delta}$, and (

**d**) ${T}_{\rho}$.

**Figure 5.**(

**a**) FIM versus PE for the Hénon map. $0\le a\le 2$; $-0.5\le b\le 0.5$, $[{x}_{1}=0.1,{y}_{1}=0.3]$. (

**c**) Reversibility ${T}_{\rho}$ versus PE. Red dots indicate a narrow region: $1\le a\le 1.4$; $0.299\le b\le 0.301$, as seen in Figure 1. (

**b**) FIM versus PE for the Hénon (blue), Burger (red), and Lozi (black) maps. (

**d**) ${T}_{\rho}$ versus PE. They all show clear overlap, indicating that some regions in parameter space present the same dynamics and the same underlying symmetries.

**Figure 6.**PE for the Henón map for the position space $-2\le x,y\le 2$ for four different combinations of the control parameters a and b. (

**a**) $a=0.5;\phantom{\rule{3.33333pt}{0ex}}b=-0.25$, (

**b**) $a=0.5;\phantom{\rule{3.33333pt}{0ex}}b=0.0$, (

**c**) $a=1.5;\phantom{\rule{3.33333pt}{0ex}}b=0.15$, (

**d**) $a=1.4;\phantom{\rule{3.33333pt}{0ex}}b=0.3$.

**Figure 7.**FIM versus PE plane for the Standard map for different values of the control parameter K. The figure is computed scanning ${10}^{4}$ states in the whole $[\theta ,p]$ phase space between $K={10}^{-6}$ and $K=10$. (

**a**) $K={10}^{-6}$. (

**b**) $K={10}^{-3}$. (

**c**) $K=0.1$. (

**e**) $K=0.5$. (

**f**) $K=1$. (

**d**) $K={10}^{-6}$ to $K=10$ on the same FIM-PE plane.

**Figure 8.**Landscapes of PE on phase space for the standard map for different values of K. (

**a**) $K={10}^{-6}$. (

**b**) $K={10}^{-3}$. (

**c**) $K=0.1$. (

**d**) $K=0.5$. (

**e**) $K=1$. (

**f**) $K=5$.

Pattern | Word |
---|---|

‘123’ | 1 |

‘132’ | 2 |

‘213’ | 3 |

‘231’ | 4 |

‘312’ | 5 |

‘321’ | 6 |

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**MDPI and ACS Style**

Novak, B.S.; Aragoneses, A.
Unveiling Dynamical Symmetries in 2D Chaotic Iterative Maps with Ordinal-Patterns-Based Complexity Quantifiers. *Dynamics* **2023**, *3*, 750-763.
https://doi.org/10.3390/dynamics3040040

**AMA Style**

Novak BS, Aragoneses A.
Unveiling Dynamical Symmetries in 2D Chaotic Iterative Maps with Ordinal-Patterns-Based Complexity Quantifiers. *Dynamics*. 2023; 3(4):750-763.
https://doi.org/10.3390/dynamics3040040

**Chicago/Turabian Style**

Novak, Benjamin S., and Andrés Aragoneses.
2023. "Unveiling Dynamical Symmetries in 2D Chaotic Iterative Maps with Ordinal-Patterns-Based Complexity Quantifiers" *Dynamics* 3, no. 4: 750-763.
https://doi.org/10.3390/dynamics3040040