# Symmetries and Dynamics of Generalized Biquaternionic Julia Sets Defined by Various Polynomials

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Biquaternions and Their Properties

#### 2.2. Julia Sets within Biquaternions

#### 2.2.1. Power Polynomials

**Definition**

**1.**

**Theorem**

**1.**

**Proof**

**of Theorem 1.**

#### 2.2.2. Monic Higher-Degree Polynomials

## 3. Symmetry of Biquaternionic Julia Sets

#### 3.1. Symmetry of Biquaternionic Julia Sets Defined by Power Polynomials

**Remark**

**1.**

**Remark**

**2.**

**Remark**

**3.**

#### 3.2. Symmetry of Biquaternionic Julia Sets Defined by Monic Higher-Degree Polynomials

## 4. Stability of Biquaternionic Julia Sets

#### 4.1. Stability of Biquaternionic Julia Sets Defined by Power Polynomials

#### 4.1.1. 1-Cycle Stability

#### 4.1.2. 2-Cycle Stability

#### 4.1.3. 3-Cycle Stability

#### 4.2. Stability of Biquaternionic Julia Sets Defined by Monic Higher-Degree Polynomials

#### 4.2.1. 1-Cycle Stability

#### 4.2.2. 2-Cycle Stability

#### 4.2.3. 3-Cycle Stability

## 5. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## List of Symbols

${a}_{n}$ | complex elements of a biquaternion |

${b}_{n}$ | complex elements of a biquaternion |

$c$ | constant value of the Julia/Mandelbrot set |

$\u2102$ | complex number space |

${\u2102}_{2}$ | bicomplex number space |

${\u2102}_{n}$ | multicomplex number space |

$\u2102\otimes \u2102$ | bicomplex number space |

$\u2102\otimes \mathbb{H}$ | biquaternion number space |

${d}_{n}$ | complex elements of the constant value of the biquaternionic Julia set |

$D$ | vector part of constant value of the biquaternionic Julia set |

${g}_{n}$ | real elements of the extended representation of a biquaternion |

${h}_{n}$ | real elements of the extended representation of a biquaternion |

$\mathbb{H}$ | quaternion number space |

${i}_{n}$ | imaginary units |

$I$ | invariant unit matrix |

$j$ | imaginary unit |

${J}_{\u2102\otimes \mathbb{H}}^{p}$ | generalized biquaternionic Julia set |

$\mathbb{N}$ | natural number space |

$O$ | Landau symbol |

$\mathbb{O}$ | octonion number space |

$p$ | power of the iterated variable of the Julia/Mandelbrot set |

$\tilde{q}$ | biquaternion |

$\mathbb{R}$ | real number space |

${S}_{n}$ | scalar parts of the iterated variable of the biquaternionic Julia set |

$\mathbb{S}$ | sedenion number space |

${v}_{n}$ | complex elements of vector parts of biquaternions |

${V}_{n}$ | vector parts of the iterated variable of the biquaternionic Julia set |

${w}_{n}$ | complex elements of vector parts of the iterated variable of the biquaternionic Julia set |

${x}_{n}$ | complex elements of scalar parts of the iterated variable of the biquaternionic Julia set |

${y}_{n}$ | complex elements of vector parts of the iterated variable of the biquaternionic Julia set |

$z$ | iterated variable of the Julia/Mandelbrot set |

$\epsilon $ | small perturbation parameter |

${\lambda}_{n}$ | eigenvalues |

$\xi $ | biquaternionic root of −1 |

${\mathsf{\sigma}}_{n}$ | Pauli matrices |

${\omega}_{n}$ | arbitrary complex numbers |

${\Im}_{1}{\Im}_{2}$ | symmetry plane along the axes of imaginary values ${i}_{1}$ and ${i}_{2}$ |

$\Re {\Im}_{1}$ | symmetry plane along the axes of reals and imaginary ${i}_{1}$ values |

$\Re {\Im}_{2}$ | symmetry plane along the axes of reals and imaginary ${i}_{2}$ values |

## Appendix A

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**Figure 1.**Three-dimensional projections of biquaternionic J sets obtained from (8) for $c=0$ and various values of $p$, a perspective view. (

**a**) $p=2$, (

**b**) $p=3$, (

**c**) $p=5$, (

**d**) $p=20$, (

**e**) $p=100$, (

**f**) $p=1000$.

**Figure 2.**Three-dimensional projections of various of biquaternionic J sets obtained from (8) for $c=0$ and $p=2$: (

**a**) $\Re {\Im}_{1}$ plane, (

**b**) $\Re {\Im}_{2}$ plane, (

**c**) ${\Im}_{1}{\Im}_{2}$ plane.

**Figure 3.**Three-dimensional projections of various biquaternionic J sets obtained from (8) for $c=0$ and $p=1000$: (

**a**) $\Re {\Im}_{1}$ plane, (

**b**) $\Re {\Im}_{2}$ plane, (

**c**) ${\Im}_{1}{\Im}_{2}$ plane.

**Figure 4.**Three-dimensional projections of characteristic biquaternionic J sets obtained from (8) for Dendrite $p=2$, $c={i}_{1}$ (

**a**), San Marco $p=2$, $c=-0.75$ in isometric (

**b**), top (

**c**), and side (

**d**) views, respectively.

**Figure 5.**Three-dimensional projections of biquaternionic J sets generated with monic higher-degree polynomials (12) for $c=0$ and various values of $p$, a perspective view. (

**a**) $p=2$, (

**b**) $p=3$, (

**c**) $p=5$, (

**d**) $p=20$, (

**e**) $p=100$, (

**f**) $p=1000$.

**Figure 6.**Three-dimensioanl projections of biquaternionic J sets generated with monic higher-degree polynomials (12) for Dendrite $p=2$, $c={i}_{1}$ (

**a**), and San Marco $p=2$, $c=-0.75$ (

**b**), in isometric views.

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Katunin, A.
Symmetries and Dynamics of Generalized Biquaternionic Julia Sets Defined by Various Polynomials. *Symmetry* **2023**, *15*, 43.
https://doi.org/10.3390/sym15010043

**AMA Style**

Katunin A.
Symmetries and Dynamics of Generalized Biquaternionic Julia Sets Defined by Various Polynomials. *Symmetry*. 2023; 15(1):43.
https://doi.org/10.3390/sym15010043

**Chicago/Turabian Style**

Katunin, Andrzej.
2023. "Symmetries and Dynamics of Generalized Biquaternionic Julia Sets Defined by Various Polynomials" *Symmetry* 15, no. 1: 43.
https://doi.org/10.3390/sym15010043