# Complex Chaotic Attractor via Fractal Transformation

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## Abstract

**:**

## 1. Introduction

## 2. The Complex Chaotic Systems Based on Rotation Transformation

#### 2.1. Rotation Multiwing Chaotic System

#### 2.2. Rotation Multiscroll Chaotic System

#### 2.3. Rotation Compound Chaotic System

## 3. Chaotic Attractors with Fractal Transformation

#### 3.1. The Fractal Algorithm

#### 3.1.1. The Binary Fractal Algorithm

#### 3.1.2. The Ternary Fractal Algorithm

#### 3.2. Complex Chaotic Attractors with the Binary Fractal Transformation

#### 3.2.1. Rotation Multiwing with the Binary Fractal Transformation

#### 3.2.2. Rotation Multiscroll with the Binary Fractal Transformation

#### 3.2.3. Rotation Compound Chaotic System with the Binary Fractal Transformation

#### 3.3. Chaotic Attractors with the Ternary Fractal Transformation

## 4. Dynamics Analysis of the Complex Chaotic Systems

#### 4.1. Bifurcation Diagram

#### 4.2. Complexity Analysis

_{0}entropy, and permutation entropy (PE) [40,41,42]. Among them, PE algorithm is a proper choice to estimate the numerical series accurately and rapidly. Thus, the complexity of the complex chaotic system with fractal is analyzed by permutation entropy (PE) algorithm. The larger the PE value is, the more complex the time series are.

#### 4.3. Spectrum Distribution Characteristics

## 5. DSP Implementation

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Rotation multiwing chaotic system: (

**a**) Phase diagram on $x-z$ plane; and (

**b**) Poincaré section (y = 0).

**Figure 2.**Rotation multiscroll chaotic system: (

**a**) Phase diagram on $x-z$ plane; and (

**b**) Poincaré section (y = 0).

**Figure 3.**Rotation compound chaotic system: (

**a**) Phase diagram on $x-z$ plane; and (

**b**) Poincaré section (y = 0).

**Figure 6.**Rotation multiwing chaotic system with the binary fractal transformation: (

**a**) once fractal transformation; (

**b**) twice fractal transformation; and (

**c**) three times fractal transformation.

**Figure 7.**Rotation multiscroll chaotic system with the binary fractal transformation: (

**a**) once fractal transformation; (

**b**) twice fractal transformation; and (

**c**) three times fractal transformation.

**Figure 8.**Rotation compound chaotic system with the binary fractal transformation: (

**a**) once fractal transformation; (

**b**) twice fractal transformation; and (

**c**) three times fractal transformation.

**Figure 9.**Phase diagram of the Lorenz system with the ternary fractal transformation: (

**a**) Lorenz system; (

**b**) once fractal transformation; (

**c**) twice fractal transformation; and (

**d**) three times fractal transformation.

**Figure 10.**Rotation Lorenz system with the ternary fractal transformation: (

**a**) rotation Lorenz system; (

**b**) once fractal transformation; (

**c**) twice fractal transformation; and (

**d**) three times fractal transformation.

**Figure 11.**Rotation Chua system with the ternary fractal transformation: (

**a**) rotation Chua system; (

**b**) once fractal transformation; (

**c**) twice fractal transformation; and (

**d**) three times fractal transformation.

**Figure 12.**Compound rotation chaotic system with the ternary fractal transformation: (

**a**) rotation compound chaotic system; (

**b**) once fractal transformation; (

**c**) twice fractal transformation; and (

**d**) three times fractal transformation.

**Figure 13.**Bifurcations of the Lorenz system: (

**a**) Lorenz system; (

**b**) Lorenz system with binary fractal transformation; and (

**c**) Lorenz system with ternary fractal transformation.

**Figure 14.**Bifurcations of the rotation Lorenz system: (

**a**) rotation Lorenz system; (

**b**) rotation Lorenz system with binary fractal transformation; and (

**c**) rotation Lorenz system with ternary fractal transformation.

**Figure 15.**Bifurcations of the compound chaotic system: (

**a**) compound chaotic system; (

**b**) compound chaotic system with binary fractal transformation; and (

**c**) compound chaotic system with ternary fractal transformation.

**Figure 16.**PE complexity of chaotic system: (

**a**) Lorenz system; (

**b**) rotation Lorenz system; and (

**c**) compound chaotic system.

**Figure 17.**PE complexity of chaotic system with fractal transformation: (

**a**) Lorenz system with the binary fractal transformation; (

**b**) rotation Lorenz system with binary fractal transformation; and (

**c**) compound chaotic system with binary fractal transformation.

**Figure 18.**PE complexity of chaotic system with the fractal transformation: (

**a**) Lorenz system with ternary fractal transformation; (

**b**) rotation Lorenz system with ternary fractal transformation; and (

**c**) compound chaotic system with ternary fractal transformation.

**Figure 19.**Spectrum distribution of the rotation Lorenz system with fractal transformation: (

**a**) rotation Lorenz system; (

**b**) rotation Lorenz system with binary fractal transformation; and (

**c**) rotation Lorenz system with ternary fractal transformation.

**Figure 20.**Spectrum distribution of the rotation Chua system with fractal transformation: (

**a**) rotation Chua system; (

**b**) rotation Chua system with binary fractal transformation; and (

**c**) rotation Chua system with ternary fractal transformation.

**Figure 21.**Spectrum distribution of the compound chaotic system with fractal transformation: (

**a**) compound chaotic system; (

**b**) compound chaotic system with binary fractal transformation; and (

**c**) compound chaotic system with ternary fractal transformation.

**Figure 23.**Computer simulation results of the rotation Chua chaotic system: (

**a**) once fractal transformation; (

**b**) twice fractal transformation; and (

**c**) three times fractal transformation. DSP results of the rotation Chua chaotic system: (

**d**) once fractal transformation; (

**e**) twice fractal transformation; and (

**f**) three times fractal transformation.

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Dai, S.; Sun, K.; He, S.; Ai, W.
Complex Chaotic Attractor via Fractal Transformation. *Entropy* **2019**, *21*, 1115.
https://doi.org/10.3390/e21111115

**AMA Style**

Dai S, Sun K, He S, Ai W.
Complex Chaotic Attractor via Fractal Transformation. *Entropy*. 2019; 21(11):1115.
https://doi.org/10.3390/e21111115

**Chicago/Turabian Style**

Dai, Shengqiu, Kehui Sun, Shaobo He, and Wei Ai.
2019. "Complex Chaotic Attractor via Fractal Transformation" *Entropy* 21, no. 11: 1115.
https://doi.org/10.3390/e21111115