# Choosing the Best Members of the Optimal Eighth-Order Petković’s Family by Its Fractal Behavior

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- If there exists $w\in \mathbb{C}$ such that $f\left(w\right)=w$, then w is called a fixed point of f;
- (2)
- If there are integers p greater than 1 and $w\in \mathbb{C}$ such that ${f}^{p}\left(w\right)=w$, then w is called the periodic point of f. The smallest p such that ${f}^{p}\left(w\right)=w$ is called the period of w, and $w,f\left(w\right),\cdots ,{f}^{p}\left(w\right)$ is the period p orbit;
- (3)
- Let w be a periodic point with period p, and ${\left({f}^{p}\right)}^{\prime}\left(w\right)=\lambda $;

- If $\lambda =0$, then w is a superattracting point;
- If $0\le \left|\lambda \right|<1$, then w is an attractive point;
- If $\left|\lambda \right|=1$, then w is a neutral point;
- If $\left|\lambda \right|>1$, then w is a repulsive point.

## 2. Some Concepts and Properties Related to Riemann Sphere $\tilde{\mathit{C}}$

**Definition 1.**

**Theorem 1.**

**Proof.**

**Lemma 1.**

**Proof.**

## 3. Complex Dynamics Analysis

#### 3.1. Fixed Points and Their Stability

**Lemma 2.**

**Proof.**

**Lemma 3.**

**Corollary 1.**

**Proof.**

**Theorem 2.**

**Proof.**

**Lemma 4.**

**Theorem 3.**

**Proof.**

**Corollary 2.**

**Proof.**

**Theorem 4.**

**Proof.**

**Theorem 5.**

**Lemma 5.**

#### 3.2. Analysis of Critical Points

**Corollary 3.**

**Proof.**

**Theorem 6.**

- $C{r}_{1}=-1$;
- $C{r}_{2}=\frac{1}{2}(-2-\sqrt{-4+r}\sqrt{r}-r)$;
- $C{r}_{3}=\frac{1}{2}(-2+\sqrt{-4+r}\sqrt{r}-r)$=$\frac{1}{C{r}_{2}}$;
- $C{r}_{4}=\frac{-4-2r-3{r}^{2}-\sqrt{3}\sqrt{-16r-12{r}^{2}+4{r}^{3}+3{r}^{4}}}{4(1+2r)}$;
- $C{r}_{5}=\frac{-4-2r-3{r}^{2}+\sqrt{3}\sqrt{-16r-12{r}^{2}+4{r}^{3}+3{r}^{4}}}{4(1+2r)}$ = $\frac{1}{C{r}_{4}}$.

**Remark 1.**

- Relation $C{r}_{2}=C{r}_{3}$ holds when $r=0$ or $r=4$;
- Relation $C{r}_{4}=C{r}_{5}$ holds when $r=-2$, $r=-\frac{4}{3}$, $r=0$ or $r=2$;
- Relation $C{r}_{2}=C{r}_{4}$ holds when $r=-\frac{4}{3}$, $r=0$ or $r=4$;
- Relation $C{r}_{3}=C{r}_{5}$ holds when $r=0$;
- Relation $C{r}_{1}=C{r}_{2}=C{r}_{3}=-1$ holds when $r=0$;
- Relation $C{r}_{1}=C{r}_{4}=C{r}_{5}=-1$ holds when $r=0$ or $r=2$;
- Relation $C{r}_{2}=C{r}_{3}=C{r}_{4}=C{r}_{5}$ holds when $r=0$.

#### 3.3. Parameter Spaces and Dynamical Planes

#### 3.3.1. Parameter Spaces

#### 3.3.2. Dynamical Planes

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 9.**Local detail of Figure 8.

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**MDPI and ACS Style**

Wang, X.; Li, W.
Choosing the Best Members of the Optimal Eighth-Order Petković’s Family by Its Fractal Behavior. *Fractal Fract.* **2022**, *6*, 749.
https://doi.org/10.3390/fractalfract6120749

**AMA Style**

Wang X, Li W.
Choosing the Best Members of the Optimal Eighth-Order Petković’s Family by Its Fractal Behavior. *Fractal and Fractional*. 2022; 6(12):749.
https://doi.org/10.3390/fractalfract6120749

**Chicago/Turabian Style**

Wang, Xiaofeng, and Wenshuo Li.
2022. "Choosing the Best Members of the Optimal Eighth-Order Petković’s Family by Its Fractal Behavior" *Fractal and Fractional* 6, no. 12: 749.
https://doi.org/10.3390/fractalfract6120749