# Maximizing the Chaotic Behavior of Fractional Order Chen System by Evolutionary Algorithms

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## Abstract

**:**

## 1. Introduction

- (i)
- the optimization of the MLE of the fractional order chaotic Chen system by DE, PSO, and IWO, whereby the system’s fractional order q is not fixed, but it is considered to be a design variable and optimized alongside the conventional design variables, which are the system parameters. This was done because a slight change in the value of the fractional order q can also lead to a big change in the LEs; and,
- (ii)
- optimizing the system parameters and fractional order q gives up to an 80% increase in the value of the MLE over the non-optimized system. The highest optimized MLE is obtained from the DE optimization. Consequently, the optimized fractional order chaotic Chen systems are more complex and unpredictable, which makes them suitable for developing random number generators and secure communication systems for cryptographic applications.

## 2. Theoretical Framework

#### 2.1. Fractional Order Chaotic Chen Oscillator

#### 2.2. Computation of Lyapunov Exponents

**Definition**

**1.**

**Definition**

**2.**

#### 2.3. Description of the Evolutionary Algorithms

#### 2.3.1. Differential Evolution

**Definition**

**3.**

#### 2.3.2. Particle Swarm Optimization

**Definition**

**4.**

#### 2.3.3. Invasive Weed Optimization

#### 2.4. Complexity Analysis and Instability of Equilibria

**Theorem**

**1.**

**Remark**

**1.**

**Remark**

**2.**

## 3. Results

- (i)
**Computer configuration:**Intel(R) Core(TM) i7-4790, 3.60GHz; RAM: 12 GB; Operating System: Windows 10;- (ii)
**DE:**Crossover probability = 0.3;- (iii)
**PSO:**Constriction coefficient $K=2/\Phi -2+\sqrt{{\Phi}^{2}-4\Phi}$; $\Phi ={c}_{1}+{c}_{2}$; ${c}_{1}=2.05$; ${c}_{2}=2.05$; Damping ratio = 1;- (iv)
**IWO:**Minimum number of seeds = 0; Maximum number of seeds = 5; Variance reduction exponent = 4; Initial value of standard deviation = $0.75$; Final value of standard deviation = $1\times {10}^{-6}$.

#### 3.1. Comparison with Hyper-Chaotic Fractional Order System

#### 3.2. Complexity of Optimization Codes

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Phase diagrams in 3D plane for the optimized fractional order Chen system for each EA. (

**a**) DE: equilibrium points $(E{P}_{0},E{P}_{1},E{P}_{2})=([0,0,0],[10.0612,10.0612,25.9081],[-10.0612,-10.0612,25.9081])$, (

**b**) PSO: equilibrium points $(E{P}_{0},E{P}_{1},E{P}_{2})=([0,0,0],[10.1513,10.1513,25.7959],[-10.1513,-10.1513,25.7959])$, and (

**c**) IWO: equilibrium points $(E{P}_{0},E{P}_{1},E{P}_{2})=([0,0,0],[10.0749,10.0749,25.8950],[-10.0749,-10.0749,25.8950])$.

**Figure 2.**Timeseries of the optimized fractional order chaotic Chen systems. Each state of the optimized system is superimposed in the respective graph, as indicated by the legend. (

**a**) State x (

**b**) State y (

**c**) State z.

**Figure 3.**Bifurcation diagram and LE spectra of non-optimized fractional order Chen system. For the LE spectra, ${L}_{1}$ is in blue color, ${L}_{2}$ in red, and ${L}_{3}$ in yellow. $a=35$, $b=3$, $c=28$, $q=0.9800$ and MLE $\phantom{\rule{0.166667em}{0ex}}=2.0293$. (

**a**) Parameter a (

**b**) Parameter b (

**c**) Parameter c (

**d**) Fractional order q.

**Figure 4.**Bifurcation diagram and LE spectra of fractional order Chen system for DE best result at 400 to 500 generations. For the LE spectra, ${L}_{1}$ is in blue color, ${L}_{2}$ in red, and ${L}_{3}$ in yellow. $a=34.0919$, $b=3.9072$, $c=30.0000$, $q=0.7923$ and MLE $\phantom{\rule{0.166667em}{0ex}}=3.6451$. (

**a**) Parameter a (

**b**) Parameter b (

**c**) Parameter c (

**d**) Fractional order q.

**Figure 5.**Bifurcation diagram and LE spectra of fractional order Chen system for PSO best result at 450 and 500 generations. For the LE spectra, ${L}_{1}$ is in blue color, ${L}_{2}$ in red, and ${L}_{3}$ in yellow. $a=34.2041$, $b=3.9948$, $c=30.0000$, $q=0.7939$ and MLE $\phantom{\rule{0.166667em}{0ex}}=3.6381$. (

**a**) Parameter a (

**b**) Parameter b (

**c**) Parameter c (

**d**) Fractional order q.

**Figure 6.**Bifurcation diagram and LE spectra of fractional order Chen system for IWO best result at 450 and 500 generations. For the LE spectra, ${L}_{1}$ is in blue color, ${L}_{2}$ in red, and ${L}_{3}$ in yellow. $a=34.1050$, $b=3.9198$, $c=30.0000$, $q=0.7933$ and MLE $\phantom{\rule{0.166667em}{0ex}}=3.6070$. (

**a**) Parameter a (

**b**) Parameter b (

**c**) Parameter c (

**d**) Fractional order q.

**Figure 7.**The LE spectrum of hyper-chaotic fractional order Chen system. ${L}_{1}$ is in blue color, ${L}_{2}$ in purple, ${L}_{3}$ in red, and ${L}_{4}$ yellow color.

**Table 1.**The optimization results showing the optimized fractional order chaotic Chen system by DE, PSO, and IWO against the non-optimized system.

Parameter | LEs | Equilibrium Point | Eigenvalue | Sample Entropy | Instability |
---|---|---|---|---|---|

$({\mathbf{\lambda}}_{\mathbf{1}}$, ${\mathbf{\lambda}}_{\mathbf{2}}$, ${\mathbf{\lambda}}_{\mathbf{3}})$ | |||||

Non-optimized Chen | $\{2.0293,$ | $[0,0,0]$ | $\{-30.83,23.83,-3\}$ | $\forall \phantom{\rule{0.166667em}{0ex}}q\phantom{\rule{0.166667em}{0ex}}\u03f5\phantom{\rule{0.166667em}{0ex}}(0,1)$ | |

$a=35.0000$, $b=3.0000$, | $0,$ | $[7.9373,7.9373,21]$ | $\{-18.42,4.21+14.88i,4.21-14.88i\}$ | 0.00943 | $q>0.8244$ |

$c=28.0000$, $q=0.9800$ | $-1.3973\}$ | $[-7.9373,-7.9373,21]$ | $\{-18.42,4.21+14.88i,4.21-14.88i\}$ | $q>0.8244$ | |

DE-Chen | $\{3.6451,$ | $[0,0,0]$ | $\{-31.83,27.74,-3.90\}$ | $\forall \phantom{\rule{0.166667em}{0ex}}q\phantom{\rule{0.166667em}{0ex}}\u03f5\phantom{\rule{0.166667em}{0ex}}(0,1)$ | |

$a=34.0919$, $b=3.9072$, | $0,$ | $[10.0612,10.0612,25.9081]$ | $\{-19.75,5.87+17.74i,5.87-17.74i\}$ | 0.01938 | $q>0.7964$ |

$c=30.0000$, $q=0.7923$ | $-0.2569\}$ | $[-10.0612,-10.0612,25.9081]$ | $\{-19.75,5.87+17.74i,5.87-17.74i\}$ | $q>0.7964$ | |

PSO-Chen | $\{3.6381,$ | $[0,0,0]$ | $\{-31.880,27.67,-3.99\}$ | $\forall \phantom{\rule{0.166667em}{0ex}}q\phantom{\rule{0.166667em}{0ex}}\u03f5\phantom{\rule{0.166667em}{0ex}}(0,1)$ | |

$a=34.2041$, $b=3.9948$, | $0,$ | $[10.1513,10.1513,25.7959]$ | $\{-19.93,5.86+17.86i,5.86-17.86i\}$ | 0.01937 | $q>0.7981$ |

$c=30.0000$, $q=0.7939$ | $-0.4457\}$ | $[-10.1513,-10.1513,25.7959]$ | $\{-19.93,5.86+17.86i,5.86-17.86i\}$ | $q>0.7981$ | |

IWO-Chen | $\{3.6070,$ | $[0,0,0]$ | $\{-31.841,27.736,-3.919\}$ | ||

$a=34.1050$, $b=3.9198$, | $0,$ | $[10.0749,10.0749,25.8950]$ | $\{-19.77,5.87+17.76i,5.87-17.76i\}$ | 0.01935 | $q>0.7966$ |

$c=30.0000$, $q=0.7933$ | $-0.9951\}$ | $[-10.0749,-10.0749,25.8950]$ | $\{-19.77,5.87+17.76i,5.87-17.76i\}$ | $q>0.7966$ |

**Table 2.**LEs of selected states of hyper-chaotic fractional order Chen system (the best hyper-chaotic state is italicized).

Parameter $\mathit{\varphi}$ | Chaotic State | ${\mathit{L}}_{1}$ | ${\mathit{L}}_{2}$ | ${\mathit{L}}_{3}$ | ${\mathit{L}}_{4}$ |
---|---|---|---|---|---|

$6.5$ | Chaotic | $1.5055$ | $-0.3146$ | 0 | $-11.8294$ |

14 | Periodic | $-1.4284$ | $-1.4618$ | 0 | $-7.8273$ |

$17.3$ | Periodic | $-0.8201$ | $-4.8448$ | 0 | $-5.0879$ |

$25.8$ | Chaotic | $3.3441$ | $-1.3321$ | 0 | $-12.6013$ |

$\mathit{71}.\mathit{2}$ | Hyper-chaotic | $\mathit{2}.\mathit{346}$ | $\mathit{0}.\mathit{0222}$ | $\mathit{0}$ | -$12.953$ |

90 | Hyper-chaotic | $2.0407$ | $0.0016$ | 0 | $-12.6229$ |

Parameters | DE | PSO | IWO |
---|---|---|---|

${n}_{1}$ | 44 | 43 | 39 |

${n}_{2}$ | 60 | 67 | 69 |

${N}_{1}$ | 392 | 660 | 355 |

${N}_{2}$ | 236 | 420 | 223 |

Program | 104 | 110 | 108 |

vocabulary (n) | |||

Program length (N) | 628 | 1080 | 578 |

Volume (V) | $4207.8512$ | $7323.9120$ | $3904.3322$ |

Calculated | $594.6276$ | $639.7596$ | $627.6171$ |

program length ($\widehat{N}$) | |||

Difficulty (D) | $86.5333$ | $134.7761$ | $63.0221$ |

Effort (E) | 364,119.25023 | 987,088.2961 | 246,059.2143 |

Time (T) secs | 20,229 | 54,838 | 13,670 |

Bugs (B) | $1.4026$ | $2.4413$ | $1.3014$ |

Reference | Maximum Population | Maximum Iteration | Implementation | Algorithms | Chaotic System | Complexity Measurement Method |
---|---|---|---|---|---|---|

[13] | 40 | 80 | MATLAB | DE | SNLF | None |

[14] | 25 | 50 | N/A | MVO, | New | None |

chaotic | ||||||

WOA | oscillator | |||||

[18] | 40 | 60 | N/A | DE,GA | SNLF | None |

[19] | 100 | N/A | N/A | NSGA-II | SNLF, Chua | None |

[68] | 40 | 100 | MATLAB | OSOA | Lorenz, Chen | None |

[69] | 120 | 100 | MATLAB | TLBO | Lorenz | None |

This investigation | 100 | 500 | MATLAB | DE, PSO, IWO | Fractional order Chen | Halstead Metric |

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**MDPI and ACS Style**

Nuñez-Perez, J.-C.; Adeyemi, V.-A.; Sandoval-Ibarra, Y.; Perez-Pinal, F.-J.; Tlelo-Cuautle, E. Maximizing the Chaotic Behavior of Fractional Order Chen System by Evolutionary Algorithms. *Mathematics* **2021**, *9*, 1194.
https://doi.org/10.3390/math9111194

**AMA Style**

Nuñez-Perez J-C, Adeyemi V-A, Sandoval-Ibarra Y, Perez-Pinal F-J, Tlelo-Cuautle E. Maximizing the Chaotic Behavior of Fractional Order Chen System by Evolutionary Algorithms. *Mathematics*. 2021; 9(11):1194.
https://doi.org/10.3390/math9111194

**Chicago/Turabian Style**

Nuñez-Perez, Jose-Cruz, Vincent-Ademola Adeyemi, Yuma Sandoval-Ibarra, Francisco-Javier Perez-Pinal, and Esteban Tlelo-Cuautle. 2021. "Maximizing the Chaotic Behavior of Fractional Order Chen System by Evolutionary Algorithms" *Mathematics* 9, no. 11: 1194.
https://doi.org/10.3390/math9111194