# Escape Criteria Using Hybrid Picard S-Iteration Leading to a Comparative Analysis of Fractal Mandelbrot Sets Generated with S-Iteration

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Basic Definitions and Preliminaries

**Definition 1.**

**Definition 2.**

**Definition 3.**

**Definition 4.**

**Definition 5.**

**Theorem 1.**

**Theorem 2.**

**Corollary 1.**

## 3. Escape Criterion

**Theorem 3.**

**Proof.**

**Corollary 2.**

**Corollary 3.**

## 4. Rich and Exquisite Patterns of the Fractal Mandelbrot Sets

Algorithm 1: Creation of the Mandelbrot sets using the hybrid Picard S-orbit |

Algorithm 2: Creation of the Mandelbrot sets using the S-orbit |

#### 4.1. Rich and Exquisite Patterns of the Mandelbrot Sets Using Hybrid Picard S-Iteration vs. S-Iteration and Quadratic Functions

#### 4.2. Rich and Exquisite Patterns of the Mandelbrot Sets Using Hybrid Picard S-Iteration vs. S-Iteration and Cubic Functions

## 5. Conclusions and Discussion

- Similarly in Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19, Figure 20, Figure 21, Figure 22 and Figure 23, we fixed parameter ${\alpha}_{1}=0.7$ and varied parameter ${\alpha}_{2}$ to obtain quadratic Mandelbrot sets. Mandelbrot sets created using the hybrid Picard S-iteration procedure are quite spontaneous from those created via S-iteration.
- The cubic Mandelbrot sets created in the hybrid Picard S-orbit and S-orbit with fixed parameter ${\alpha}_{2}=0.5$ also have an obvious variation in figures for varying parameter ${\alpha}_{1}$.
- Quadratic Mandelbrot sets are symmetrical along the x-axis whereas cubic Mandelbrot sets are symmetrical along both the x-axis and y-axis.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

MS | Mandelbrot set |

MSs | Mandelbrot sets |

SO | S Orbit |

PSO | Picard S-Orbit |

PSIP | Picard S-Iteration Process |

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**Figure 11.**Dependence of the Mandelbrot set creation time (in seconds) in the hybrid Picard S-orbit and S-orbit on ${\alpha}_{1}$.

**Figure 24.**Dependence on time (in seconds) in the hybrid Picard S-orbit and S-orbit for ${\alpha}_{2}$.

${\mathit{\alpha}}_{1}$ | Time for the Hybrid Picard S-Iteration | Time for S-Iteration |
---|---|---|

$0.5$ | $6.81250$ | $7.42188$ |

$0.6$ | $5.78125$ | $5.96875$ |

$0.7$ | $5.09375$ | $5.37500$ |

$0.8$ | $4.37500$ | $5.00000$ |

$0.9$ | $3.78125$ | $5.06250$ |

${\mathit{\alpha}}_{2}$ | Time for the Hybrid Picard S-Iteration | Time for S-Iteration |
---|---|---|

$0.2$ | $8.50000$ | $9.14063$ |

$0.3$ | $4.92188$ | $5.76563$ |

$0.4$ | $4.14063$ | $4.39063$ |

$0.5$ | $3.62500$ | $3.95313$ |

$0.7$ | $3.79688$ | $4.07813$ |

$0.8$ | $4.03125$ | $4.18750$ |

${\mathit{\alpha}}_{1}$ | Time for Picard S-Iteration | Time for S-Iteration |
---|---|---|

$0.1$ | $4.09375$ | $5.14063$ |

$0.2$ | $4.37500$ | $5.84375$ |

$0.4$ | $5.37500$ | $5.54688$ |

$0.5$ | $5.71875$ | $4.96875$ |

$0.6$ | $4.85938$ | $4.39063$ |

$0.7$ | $4.45313$ | $4.07813$ |

$0.8$ | $4.32813$ | $4.12500$ |

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Srivastava, R.; Tassaddiq, A.; Kasmani, R.M.
Escape Criteria Using Hybrid Picard S-Iteration Leading to a Comparative Analysis of Fractal Mandelbrot Sets Generated with S-Iteration. *Fractal Fract.* **2024**, *8*, 116.
https://doi.org/10.3390/fractalfract8020116

**AMA Style**

Srivastava R, Tassaddiq A, Kasmani RM.
Escape Criteria Using Hybrid Picard S-Iteration Leading to a Comparative Analysis of Fractal Mandelbrot Sets Generated with S-Iteration. *Fractal and Fractional*. 2024; 8(2):116.
https://doi.org/10.3390/fractalfract8020116

**Chicago/Turabian Style**

Srivastava, Rekha, Asifa Tassaddiq, and Ruhaila Md Kasmani.
2024. "Escape Criteria Using Hybrid Picard S-Iteration Leading to a Comparative Analysis of Fractal Mandelbrot Sets Generated with S-Iteration" *Fractal and Fractional* 8, no. 2: 116.
https://doi.org/10.3390/fractalfract8020116