# Further Discussions of the Complex Dynamics of a 2D Logistic Map: Basins of Attraction and Fractal Dimensions

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. 2D Logistic Map

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

- A transcritical or fold bifurcation if ${\mu}_{1}=1$ or ${\mu}_{1}=\frac{{\mu}_{2}}{{\mu}_{2}-1}$.
- A flip bifurcation if ${\mu}_{1}=3$ or ${\mu}_{1}=\frac{{\mu}_{2}}{{\mu}_{2}-3}$.

**Proof.**

## 3. Local and Global Analysis

## 4. Noninvertible Map and Critical Curves

**Proposition**

**3.**

**Proof.**

**Proposition**

**4.**

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Bifurcation diagrams with respect to x and y on varying ${\mu}_{1}$ considering ${\mu}_{2}=4$ and ${\mu}_{2}$ considering ${\mu}_{1}=3$. (

**b**) Largest Lyapunov exponent. (

**c**–

**f**) Basins of attraction for different period cycles. The circle point refers to the fixed point ${e}_{3}$. The dark and light gray in these figures refer to the basins of the corresponding period cycles.

**Figure 2.**(

**a**–

**c**) Different chaotic behaviors of the map on varying ${\mu}_{1}$ and ${\mu}_{2}$ is fixed to $3.74$. (

**d**) Chaotic behavior of the map for different values of ${\mu}_{2}$ and ${\mu}_{1}$. (

**e**,

**f**) Transient dynamic behaviors of the map on varying ${\mu}_{2}$ and ${\mu}_{1}$ is fixed to the bifurcated value 3.

**Figure 3.**(

**a**–

**e**) Transient dynamic behaviors of the map when varying ${\mu}_{2}$ and keeping the other parameter fixed. (

**f**) 2D bifurcation diagram in the $\left({\mu}_{1},{\mu}_{2}\right)$-plane. It includes several colors that represent different period cycles. The other colors are for higher period cycles. The white color refers to divergent and nonconvergent points.

**Figure 4.**The basins of attraction of (

**a**) Period 2-cycle at ${\mu}_{1}=2.187$ and ${\mu}_{2}=5.706$. (

**b**) Period 4-cycle at ${\mu}_{1}=2.618$ and ${\mu}_{2}=5.642$. (

**c**) Period 5-cycle at ${\mu}_{1}=2.916$ and ${\mu}_{2}=5.693$. (

**d**) Period 6-cycle at ${\mu}_{1}=3.076$ and ${\mu}_{2}=5.029$. (

**e**) Period 7-cycle at ${\mu}_{1}=2.813$ and ${\mu}_{2}=5.744$. (

**f**) Period 8-cycle at ${\mu}_{1}=3.165$ and ${\mu}_{2}=4.685$. (

**g**) Period 10-cycle at ${\mu}_{1}=2.804$ and ${\mu}_{2}=5.604$. (

**h**,

**I**) Chaotic attractors at ${\mu}_{1}=3.72$, ${\mu}_{2}=2.234$ and ${\mu}_{1}=3.969$, ${\mu}_{2}=3.74$, respectively. The yellow color refers to the divergent points where the dark and light grey colors refer the basins for the corresponding periodic cycle.

**Figure 5.**$LC$ and $L{C}_{-1}$ and the corresponding region, ${Z}_{i}$, $i=0,2,4$ at the parameters’ values, ${\mu}_{1}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}3.969$ and ${\mu}_{2}=3.74$. On the x-axis the interval $(\frac{{\mu}_{1}}{4},0)$ belongs to ${Z}_{4}$ while in the y-axis the interval $(0,\frac{{\mu}_{2}}{4})$ belongs to ${Z}_{4}$. In addition, the interval $(\frac{{\mu}_{1}}{4},1)$ in the x-axis and $(\frac{{\mu}_{2}}{4},1)$ in the y-axis belong to ${Z}_{0}$.

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**MDPI and ACS Style**

Askar, S.S.; Al-Khedhairi, A.
Further Discussions of the Complex Dynamics of a 2D Logistic Map: Basins of Attraction and Fractal Dimensions. *Symmetry* **2020**, *12*, 2001.
https://doi.org/10.3390/sym12122001

**AMA Style**

Askar SS, Al-Khedhairi A.
Further Discussions of the Complex Dynamics of a 2D Logistic Map: Basins of Attraction and Fractal Dimensions. *Symmetry*. 2020; 12(12):2001.
https://doi.org/10.3390/sym12122001

**Chicago/Turabian Style**

Askar, Sameh S., and Abdulrahman Al-Khedhairi.
2020. "Further Discussions of the Complex Dynamics of a 2D Logistic Map: Basins of Attraction and Fractal Dimensions" *Symmetry* 12, no. 12: 2001.
https://doi.org/10.3390/sym12122001