# Optimizing the Maximum Lyapunov Exponent of Fractional Order Chaotic Spherical System by Evolutionary Algorithms

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## Abstract

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## 1. Introduction

- (i)
- Optimizing the chaotic behavior of the 3-D spherical-attractor-generating system by maximizing the MLE with the application of DE, PSO and IWO metaheuristics. The results showed a significantly larger MLEs than the non-optimized system, which is evident in the phase space portraits, time series and the entropy of the optimized systems.
- (ii)
- Construction of a hyperchaotic system of 4-D. The hyperchaotic system was created with a state feedback controller added to the second equation of the original 3-D system. The analyses of the hyperchaotic system revealed that it possesses rich dynamics, exhibiting three different states, namely, hyperchaotic, chaotic, and periodic.

## 2. Chaotic Dynamical System Considered

## 3. Novel Hyperchaotic System

## 4. Optimization Algorithms

#### 4.1. Differential Evolution

#### 4.2. Particle Swarm Optimization

#### 4.3. Invasive Weed Optimization

## 5. Evaluation of the Lyapunov Exponents

- (i)
- A variational system ${\dot{D}}_{*}^{q}y\left(t\right)=Jf\left(x\right)y\left(t\right)$ of the original dynamical system ${\dot{D}}_{*}^{q}x=f\left(x\right)$ is formed using the $n\times n$ Jacobian matrix J of f.
- (ii)
- The original dynamical system is given the initial condition ${X}_{0}=(-0.04,-15.8,-1.4)$, while the initial condition of the variational system is set to I, an $n\times n$ identity matrix.
- (iii)
- The integration of the original and variational systems are done until the orthonormalization period K is reached.
- (iv)
- The variational system is then orthonormalized using the Continuous Gram–Schmidt orthogonalization.
- (v)
- Next, the algorithm obtains and gathers in time the logarithm of the norm of each Lyapunov vector in the variational system.
- (vi)
- The next integration begins with the new orthonormalized vectors as the initial conditions.
- (vii)
- Steps (iii) to (vi) are repeated until the integration period T is reached.
- (viii)
- The n Lyapunov exponents are obtained by evaluating:$${L}_{m}\approx \frac{1}{T}\sum _{i=1}^{K}ln\Vert {\omega}_{m}\Vert ,$$

## 6. Results

#### 6.1. MLE Optimization

- (i)
- Computer configuration: Intel(R) Core(TM) i7-4790, 3.60 GHz; RAM: 12 GB; Operating System: Windows 10;
- (ii)
- DE: Crossover probability = 0.3;
- (iii)
- PSO: Constriction coefficient $K=\frac{2}{\varphi -2+\sqrt{{\varphi}^{2}-4\varphi}}\phantom{\rule{0.166667em}{0ex}};\phantom{\rule{0.166667em}{0ex}}\varphi ={c}_{1}+{c}_{2};\phantom{\rule{0.166667em}{0ex}}{c}_{1}=2.05;\phantom{\rule{0.166667em}{0ex}}{c}_{2}=2.05$; Damping ratio $=\phantom{\rule{0.166667em}{0ex}}1$;
- (iv)
- IWO: Minimum number of seeds = 0; Maximum number of seeds $=\phantom{\rule{0.166667em}{0ex}}5$; Initial value of standard deviation $=\phantom{\rule{0.166667em}{0ex}}0.75$; Variance reduction exponent $=\phantom{\rule{0.166667em}{0ex}}4$; Final value of standard deviation $=\phantom{\rule{0.166667em}{0ex}}1\times {10}^{-6}$.

#### 6.2. Optimized Systems against Hyperchaotic System

## 7. Discussion

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

3-D | Three-Dimension |

4-D | Four-dimensional |

ABM | Adams–Bashforth–Moulton |

CGSO | Continuous Gram–Schmidt Orthogonalization |

DE | Differential Evolution |

EA | Evolutionary Algorithm |

${h}_{KS}$ | Kolmogorov–Sinai entropy |

IWO | Invasive Weed Optimization |

LE | Lyapunov Exponent |

MLE | Maximum Lyapunov Exponent |

ODE | Ordinary Differential Equation |

PSO | Particle Swarm Optimization |

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**Figure 4.**Time series of states x, y and z of the non-optimized and optimized fractional-order chaotic systems presented in Table 1: (

**a**) non-optimized; (

**b**) DE-optimized; (

**c**) PSO-optimized; (

**d**) IWO-optimized.

**Figure 5.**Bifurcation diagrams and Lyapunov exponents spectra of the optimized fractional-order system (2), using parameters ${a}_{1}$, ${a}_{2}$, ${c}_{1}$, d and fractional-order q, for the non-optimized system.

**Figure 6.**Bifurcation diagrams and Lyapunov exponents spectra of the optimized fractional-order system (2), using parameters ${a}_{1}$, ${a}_{2}$, ${c}_{1}$, d and fractional-order q, for the DE-optimized system.

**Figure 7.**Bifurcation diagrams and Lyapunov exponents spectra of the optimized fractional-order system (2), using parameters ${a}_{1}$, ${a}_{2}$, ${c}_{1}$, d and fractional-order q, for the PSO-optimized system.

**Figure 8.**Bifurcation diagrams and Lyapunov exponents spectra of the optimized fractional-order system (2), using parameters ${a}_{1}$, ${a}_{2}$, ${c}_{1}$, d and fractional-order q, for the IWO-optimized system.

**Figure 9.**Dynamics of fractional-order hyperchaotic system (4) showing the bifurcation diagram (

**left**) and Lyapunov exponent spectrum (

**right**), when (

**a**) $g=5$ and r varies; (

**b**) $r=15$ and g varies.

**Figure 10.**Phase portraits of the hyperchaotic state when $r=15.25$ and $g=5$ (

**a**) w-z-x plane (

**b**) x-y-z plane (

**c**) x-z-w plane (

**d**) y-z plane.

**Table 1.**Optimization results for fractional-order chaotic system (2), showing the DE, PSO, and IWO-optimized systems against the non-optimized system.

Parameter | MLE | Equilibrium Point | Eigenvalue $({\mathit{\lambda}}_{1}$, ${\mathit{\lambda}}_{2}$, ${\mathit{\lambda}}_{3})$ | Information Entropy | Instability Condition |
---|---|---|---|---|---|

Non-optimized Spherical | |||||

${a}_{1}=-4.1000$, ${a}_{2}=1.2000$, ${a}_{3}=13.4500$, | $0.0183$ | $[0.7217,$ | $\{-0.3150,$ | 0.9772 | $q>0.6378$ |

$b=0.1610$, $c=3.5031$, ${c}_{1}=2.7600$, | $-2.5698,$ | $3.9016+6.1032i,$ | |||

${c}_{2}=0.6000$, ${c}_{3}=13.1300$, $d=1.6000$, | $0.1394]$ | $3.9016-6.1032i\left)\right\}$ | |||

$e=0.0000$, $q=0.9999$ | |||||

DE-optimized Spherical | |||||

${a}_{1}=-4.9206$, ${a}_{2}=14.5710$, ${a}_{3}=18.0771$, | $1.0808$ | $[5.1098,$ | $\{-5.9584,$ | 6.93343 | $q>0.9150$ |

$b=1.3100$, $c=4.4500$, ${c}_{1}=19.9999$, | $-0.0798,$ | $3.6694+27.3886i,$ | |||

${c}_{2}=4.9999$, ${c}_{3}=6.7000$, $d=1.4089$, | $1.3264]$ | $3.6694-27.3886i\}$ | |||

$e=1.6126$, $q=0.9417$ | |||||

PSO-optimized Spherical | |||||

${a}_{1}=-4.9500$, ${a}_{2}=17.1430$, ${a}_{3}=18.0771$, | $1.0775$ | $[5.1000,$ | $\{-6.0344,$ | 6.8620 | $q>0.9145$ |

$b=1.3100$, $c=4.4500$, ${c}_{1}=19.9999$, | $-0.0800,$ | $3.6969+27.3943i,$ | |||

${c}_{2}=5.0000$, ${c}_{3}=6.7093$, $d=1.4156$, | $1.3206]$ | $3.6969-27.3943i\}$ | |||

$e=1.6126$, $q=0.9408$ | |||||

IWO-optimized Spherical | |||||

${a}_{1}=-4.9589$, ${a}_{2}=27.4290$, ${a}_{3}=18.0771$, | $1.0662$ | $[5.2151,$ | $\{-6.1480,$ | 6.5619 | $q>0.9150$ |

$b=1.2720$, $c=4.4900$, ${c}_{1}=19.9999$, | $-0.0785,$ | $3.7533+28.0013i,$ | |||

${c}_{2}=5.0000$, ${c}_{3}=6.7100$, $d=1.4156$, | $1.3114]$ | $3.7533-28.0013i\}$ | |||

$e=1.6126$, $q=0.9408$ |

**Table 2.**Selected Lyapunov exponents for the hyperchaotic fractional-order system of (4) when (i) $g=5$ was fixed and r varies, and (ii) when $r=15$ was fixed and g varies.

Parameter g | Parameter r | ${\mathit{L}}_{1}$ | ${\mathit{L}}_{2}$ | ${\mathit{L}}_{3}$ | ${\mathit{L}}_{4}$ | Dynamical State |
---|---|---|---|---|---|---|

$5.00$ | $-1.50$ | $-0.0016$ | $-0.0085$ | 0 | $-0.0118$ | Periodic |

$5.00$ | $-9.25$ | $0.0536$ | $-0.0232$ | 0 | $-0.0727$ | Chaotic |

$5.00$ | $-3.25$ | $0.0756$ | $-0.0391$ | 0 | $-0.0631$ | Chaotic |

$5.00$ | $15.00$ | $0.0375$ | $0.0307$ | 0 | $-0.1507$ | Hyperchaotic |

$5.00$ | $15.25$ | $0.0452$ | $0.0438$ | 0 | $-0.2146$ | Hyperchaotic |

$5.00$ | $15.50$ | $0.0206$ | $0.0200$ | 0 | $-0.1071$ | Hyperchaotic |

$5.00$ | $17.00$ | $-0.0091$ | $-0.0121$ | 0 | $-0.0232$ | Periodic |

$-10.00$ | $15.00$ | $0.0935$ | $-0.0312$ | 0 | $-0.0701$ | Chaotic |

$-5.00$ | $15.00$ | $0.0354$ | $0.0316$ | 0 | $-0.1549$ | Hyperchaotic |

$-4.75$ | $15.00$ | $-0.0500$ | $-0.0500$ | 0 | $-0.3764$ | Periodic |

$1.75$ | $15.00$ | $0.0484$ | $0.0124$ | 0 | $-0.3185$ | Hyperchaotic |

$5.00$ | $15.00$ | $0.0360$ | $0.0344$ | 0 | $-0.1518$ | Hyperchaotic |

$12.50$ | $15.00$ | $-0.0128$ | $-0.4866$ | 0 | $-0.4861$ | Periodic |

$19.25$ | $15.00$ | $0.0433$ | $-0.0321$ | 0 | $-0.0608$ | Chaotic |

**Table 3.**Information entropies and prediction times of the non-optimized and optimized chaotic systems, and the best three hyperchaotic states of the new hyperchaotic system.

System | Entropy $\mathit{H}\left(\mathit{s}\right)$ | Prediction Time ${\mathit{\mu}}_{\mathit{p}}$ |
---|---|---|

Non-optimized | $0.9772$ | 37.8769 |

DE-optimized | $6.9334$ | 0.6413 |

PSO-optimized | $6.8620$ | 0.6433 |

IWO-optimized | $6.5619$ | 0.6501 |

Hyperchaotic 1 | ||

$(r=15.00$, $g=5.00)$ | $4.6306$ | 10.1635 |

Hyperchaotic 2 | ||

$(r=15.25$, $g=5.00)$ | $4.6047$ | 7.7882 |

Hyperchaotic 3 | ||

$(r=15.00$, $g=-5.00)$ | $4.5144$ | 10.3455 |

Reference | Maximum Population | Maximum Iteration | Implementation | Algorithms | Chaotic System |
---|---|---|---|---|---|

[6] | 40 | 80 | MATLAB | DE | SNLF |

[7] | 25 | 50 | N/A | MVO | New Chaotic |

WOA | oscillator | ||||

[15] | 40 | 60 | N/A | DE, GA | SNLF |

[16] | 100 | N/A | N/A | NSGA-II | SNLF, Chua |

[36] | 40 | 100 | MATLAB | OSOA | Lorenz, Chen |

[37] | 120 | 100 | MATLAB | TLBO | Lorenz |

This investigation | 100 | 500 | MATLAB | DE, PSO, IWO | 3-D fractional-order System |

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**MDPI and ACS Style**

Adeyemi, V.-A.; Tlelo-Cuautle, E.; Perez-Pinal, F.-J.; Nuñez-Perez, J.-C.
Optimizing the Maximum Lyapunov Exponent of Fractional Order Chaotic Spherical System by Evolutionary Algorithms. *Fractal Fract.* **2022**, *6*, 448.
https://doi.org/10.3390/fractalfract6080448

**AMA Style**

Adeyemi V-A, Tlelo-Cuautle E, Perez-Pinal F-J, Nuñez-Perez J-C.
Optimizing the Maximum Lyapunov Exponent of Fractional Order Chaotic Spherical System by Evolutionary Algorithms. *Fractal and Fractional*. 2022; 6(8):448.
https://doi.org/10.3390/fractalfract6080448

**Chicago/Turabian Style**

Adeyemi, Vincent-Ademola, Esteban Tlelo-Cuautle, Francisco-Javier Perez-Pinal, and Jose-Cruz Nuñez-Perez.
2022. "Optimizing the Maximum Lyapunov Exponent of Fractional Order Chaotic Spherical System by Evolutionary Algorithms" *Fractal and Fractional* 6, no. 8: 448.
https://doi.org/10.3390/fractalfract6080448