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

^{1}

^{2}

^{3}

^{4}

^{*}

^{†}

## 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

- Sujarani, R.; Manivannan, D.; Manikandan, R.; Vidhyacharan, B. Lightweight Bio-Chaos Crypt to Enhance the Security of Biometric Images in Internet of Things Applications. Wirel. Pers. Commun.
**2021**. [Google Scholar] [CrossRef] - Chen, H.; Ji, Q.; Wang, H.; Yang, Q.; Cao, Q.; Gong, Q.; Yi, X.; Xiao, Y. Chaos-assisted two-octave-spanning microcombs. Nat. Commun.
**2020**, 11. [Google Scholar] [CrossRef] [PubMed] - Peng, Z.; Yu, W.; Wang, J.; Wang, J.; Chen, Y.; He, X.; Jiang, D. Dynamic analysis of seven-dimensional fractional-order chaotic system and its application in encrypted communication. J. Ambient Intell. Humaniz. Comput.
**2020**, 11, 5399–5417. [Google Scholar] [CrossRef] - Freitas, V.; Yanchuk, S.; Zaks, M.; Macau, E. Synchronization-based symmetric circular formations of mobile agents and the generation of chaotic trajectories. Commun. Nonlinear Sci. Numer. Simul.
**2021**, 94, 105543. [Google Scholar] [CrossRef] - Lian, J.; Yu, W.; Xiao, K.; Liu, W. Cubic Spline Interpolation-Based Robot Path Planning Using a Chaotic Adaptive Particle Swarm Optimization Algorithm. Math. Probl. Eng.
**2020**, 2020, 849240. [Google Scholar] [CrossRef] - Sridharan, K.; Ahmadabadi, Z. A Multi-System Chaotic Path Planner for Fast and Unpredictable Online Coverage of Terrains. IEEE Robot. Autom. Lett.
**2020**, 5, 5268–5275. [Google Scholar] [CrossRef] - Liu, L.; Zhang, Q.; Wei, D.; Li, G.; Wu, H.; Wang, Z.; Guo, B.; Zhang, J. Chaotic Ensemble of Online Recurrent Extreme Learning Machine for Temperature Prediction of Control Moment Gyroscopes. Sensors
**2020**, 20, 4786. [Google Scholar] [CrossRef] [PubMed] - Shirzhiyan, Z.; Keihani, A.; Farahi, M.; Shamsi, E.; GolMohammadi, M.; Mahnam, A.; Haidari, M.; Jafari, A. Introducing chaotic codes for the modulation of code modulated visual evoked potentials (c-VEP) in normal adults for visual fatigue reduction. PLoS ONE
**2019**, 14, e0213197. [Google Scholar] [CrossRef] - Nobukawa, S.; Shibata, N. Controlling Chaotic Resonance using External Feedback Signals in Neural Systems. Sci. Rep.
**2019**, 9, 4990. [Google Scholar] [CrossRef] [Green Version] - Belmiloudi, A. Dynamical behavior of nonlinear impulsive abstract partial differential equations on networks with multiple time-varying delays and mixed boundary conditions involving time-varying delays. J. Dyn. Control Syst.
**2015**, 21, 95–146. [Google Scholar] [CrossRef] - Demina, M.V.; Kuznetsov, N.S. Liouvillian integrability and the Poincaré problem for nonlinear oscillators with quadratic damping and polynomial forces. J. Dyn. Control. Syst.
**2020**, 27, 1–13. [Google Scholar] [CrossRef] - Dawidowicz, A.L.; Poskrobko, A. On chaos behaviour of nonlinear Lasota equation in Lebesgue spaces. J. Dyn. Control. Syst.
**2020**, 27, 1–8. [Google Scholar] [CrossRef] - Carbajal-Gómez, V.H.; Tlelo-Cuautle, E.; Fernández, F.V. Optimizing the positive Lyapunov exponent in multi-scroll chaotic oscillators with differential evolution algorithm. Appl. Math. Comput.
**2013**, 219, 8163–8168. [Google Scholar] [CrossRef] - Xu, G.; Shekofteh, Y.; Akgül, A.; Li, C.; Panahi, S. A new chaotic system with a self-excited attractor: Entropy measurement, signal encryption, and parameter estimation. Entropy
**2018**, 20, 86. [Google Scholar] [CrossRef] [Green Version] - Balibrea, F.; Caballero, M.V. Stability of orbits via Lyapunov exponents in autonomous and nonautonomous systems. Int. J. Bifurcat. Chaos.
**2013**, 23, 1350127. [Google Scholar] [CrossRef] - Zhou, S.; Wang, X. Simple estimation method for the largest Lyapunov exponent of continuous fractional-order differential equations. Phys. A Stat. Mech. Appl.
**2021**, 563, 125478. [Google Scholar] [CrossRef] - De la Fraga, L.G.; Tlelo-Cuautle, E.; Carbajal-Gómez, V.H.; Muñoz-Pacheco, J.M. On maximizing positive Lyapunov exponents in a chaotic oscillator with heuristics. Rev. Mex. Fis.
**2012**, 58, 274–281. [Google Scholar] - Carbajal-Gómez, V.H.; Tlelo-Cuautle, E.; Fernández, F.V.; de la Fraga, L.G.; Sánchez-López, C. Maximizing Lyapunov Exponents in a Chaotic Oscillator by Applying Differential Evolution. Int. J. Nonlinear Sci. Num.
**2014**, 15, 11–17. [Google Scholar] [CrossRef] - De la Fraga, L.G.; Tlelo-Cuautle, E. Optimizing the maximum Lyapunov exponent and phase space portraits in multi-scroll chaotic oscillators. Nonlinear Dynam.
**2014**, 76, 1503–1515. [Google Scholar] [CrossRef] - Hua, A.; Zhang, Y.; Zhou, Y. Two-Dimensional Modular Chaotification System for Improving Chaos Complexity. IEEE Trans. Signal Process.
**2020**, 68, 1937–1949. [Google Scholar] [CrossRef] - Liu, B.; Xiang, H.; Liu, L. Reducing the dynamical degradation of digital chaotic maps with time-delay linear feedback and parameter perturbation. Math. Probl. Eng.
**2020**, 12, 4926937. [Google Scholar] [CrossRef] [Green Version] - Sun, H.G.; Zhang, Y.; Baleanu, D.; Chen, W.; Chen, Y.Q. A new collection of real world applications of fractional calculus in science and engineering. Commun. Nonlinear Sci.
**2018**, 64, 213–231. [Google Scholar] [CrossRef] - Srivastava, H.; Saxena, R.; Parmar, R. Some Families of the Incomplete H-Functions and the Incomplete H-Functions and Associated Integral Transforms and Operators of Fractional Calculus with Applications. Russ. J. Math. Phys.
**2018**, 25, 116–138. [Google Scholar] [CrossRef] - Tolba, M.F.; AbdelAty, A.M.; Soliman, N.S.; Said, L.A.; Madian, A.H.; Azar, A.T.; Radwan, A.G. FPGA implementation of two fractional order chaotic systems. AEU-Int. J. Electron. C.
**2017**, 78, 162–172. [Google Scholar] [CrossRef] - Bouzeriba, A.; Boulkroune, A.; Bouden, T. Projective synchronization of two different fractional-order chaotic systems via adaptive fuzzy control. Neural Compt. Appl.
**2016**, 27, 1349–1360. [Google Scholar] [CrossRef] - Mescia, L.; Bia, P.; Caratelli, D. Fractional Calculus Based Electromagnetic Tool to Study Pulse Propagation in Arbitrary Dispersive Dielectrics. Phys. Status Solidi A
**2018**, 216, 1800557. [Google Scholar] [CrossRef] [Green Version] - Gonzalez, E.A.; Petrás, I.; Ortigueira, M.D. Novel Polarization Index Evaluation Formula and Fractional-Order Dynamics in Electric Motor Insulation Resistance. Fract. Calc. Appl. Anal.
**2018**, 21, 613–627. [Google Scholar] [CrossRef] [Green Version] - Ray, S.S.; Sahoo, S.; Das, S. Formulation and solutions of fractional continuously variable order mass–spring–damper systems controlled by viscoelastic and viscous–viscoelastic dampers. Adv. Mech. Eng.
**2016**, 8, 1–13. [Google Scholar] [CrossRef] [Green Version] - He, S.; Sun, K.; Peng, Y. Detecting chaos in fractional-order nonlinear systems using the smaller alignment index. Phys. Lett. A
**2019**, 383, 2267–2271. [Google Scholar] [CrossRef] - Boulkroune, A.; Bouzeriba, A.; Bouden, T. Fuzzy generalized projective synchronization of incommensurate fractional-order chaotic systems. Neurocomputing
**2016**, 173, 606–614. [Google Scholar] [CrossRef] - Faieghi, M.R.; Delavari, H. Chaos in fractional-order Genesio–Tesi system and its synchronization. Commun. Nonlinear Sci. Numer. Simul.
**2012**, 17, 731–741. [Google Scholar] [CrossRef] - Danca, M.F.; Garrapa, R. Suppressing chaos in discontinuous systems of fractional order by active control. Appl. Math. Comput.
**2015**, 257, 89–102. [Google Scholar] [CrossRef] [Green Version] - He, J.M.; Chen, F.Q. A new fractional order hyperchaotic Rabinovich system and its dynamical behaviors. Int. J. Nonlinear Mech.
**2017**, 95, 73–81. [Google Scholar] [CrossRef] - Tlelo-Cuautle, E.; Pano-Azucena, A.D.; Guillen-Fernandez, O.; Silva-Juarez, A. Analog Implementations of Fractional-Order Chaotic Systems. In Analog/Digital Implementation of Fractional Order Chaotic Circuits and Applications; Springer Nature: Cham, Switzerland, 2020; Chapter 4; pp. 93–114. [Google Scholar]
- Tlelo-Cuautle, E.; Pano-Azucena, A.D.; Guillen-Fernandez, O.; Silva-Juarez, A. Synchronization and Applications of Fractional-Order Chaotic Systems. In Analog/Digital Implementation of Fractional Order Chaotic Circuits and Applications; Springer Nature: Cham, Switzerland, 2020; Chapter 6; pp. 175–201. [Google Scholar]
- Pano-Azucena, A.; Ovilla-Martinez, B.; Tlelo-Cuautle, E.; Muñoz-Pacheco, J.; de la Fraga, L. FPGA-based implementation of different families of fractional-order chaotic oscillators applying Grünwald–Letnikov method. Commun. Nonlinear Sci. Numer. Simul.
**2019**, 72, 516–527. [Google Scholar] [CrossRef] - Núñez Pérez, J.; Adeyemi, V.; Sandoval-Ibarra, Y.; Serrato-Andrade, R.; Cárdenas, J.; Tlelo-Cuautle, E. Mathematical and numerical analysis of the dynamical behavior of Chen oscillator. Int. J. Dyn. Control
**2020**, 8, 386–395. [Google Scholar] [CrossRef] - Platas-Garza, M.; Zambrano-Serrano, E.; Rodríguez-Cruz, J.; Posadas-Castillo, C. Implementation of an encrypted-compressed image wireless transmission scheme based on chaotic fractional-order systems. Chin. J. Phys.
**2021**, 71, 22–37. [Google Scholar] [CrossRef] - Liu, T.; Yan, H.; Banerjee, S.; Mou, J. A fractional-order chaotic system with hidden attractor and self-excited attractor and its DSP implementation. Chaos Solitons Fractals
**2021**, 145, 110791. [Google Scholar] [CrossRef] - Soleimanizadeh, A.; Nekoui, M. Optimal type-2 fuzzy synchronization of two different fractional-order chaotic systems with variable orders with an application to secure communication. Soft Comput.
**2021**, 25, 6415–6426. [Google Scholar] [CrossRef] - Tlelo-Cuautle, E.; Pano-Azucena, A.D.; Guillen-Fernandez, O.; Silva-Juarez, A. Characterization and Optimization of Fractional-Order Chaotic Systems. In Analog/Digital Implementation of Fractional Order Chaotic Circuits and Applications; Springer Nature: Cham, Switzerland, 2020; Chapter 3; pp. 75–91. [Google Scholar]
- Goodarzi, M.; Mohades, A.; Forghani-elahabad, M. Improving the Gridshells’ Regularity by Using Evolutionary Techniques. Mathematics
**2021**, 9, 440. [Google Scholar] [CrossRef] - Lu, N.; Yang, Y. Application of evolutionary algorithm in performance optimization of embedded network firewall. Microprocess. Microsyst.
**2020**, 76, 103087. [Google Scholar] [CrossRef] - Turgut, M.; Sağban, H.; Turgut, O.; Özmen, T. Whale optimization and sine–cosine optimization algorithms with cellular topology for parameter identification of chaotic systems and Schottky barrier diode models. Soft Comput.
**2021**, 25, 1365–1409. [Google Scholar] [CrossRef] - Tlelo-Cuautle, E.; Carbajal-Gomez, V.; Obeso-Rodelo, P.; Rangel-Magdaleno, J.; Núñez-Pérez, J.C. FPGA realization of a chaotic communication system applied to image processing. Nonlinear Dynam.
**2015**, 82, 1879–1892. [Google Scholar] [CrossRef] - Al-Saidi, N.M.G.; Younus, D.; Natiq, H.; Ariffin, M.R.K.; Asbullah, M.A.; Mahad, Z. A New Hyperchaotic Map for a Secure Communication Scheme with an Experimental Realization. Symmetry
**2020**, 12, 1881. [Google Scholar] [CrossRef] - Mahmoud, E.E.; Higazy, M.; Althagafi, O.A. A Novel Strategy for Complete and Phase Robust Synchronizations of Chaotic Nonlinear Systems. Symmetry
**2020**, 12, 1765. [Google Scholar] [CrossRef] - Anees, A.; Hussain, I. A Novel Method to Identify Initial Values of Chaotic Maps in Cybersecurity. J. Dyn. Control Syst.
**2020**, 11, 140. [Google Scholar] [CrossRef] [Green Version] - Nuñez-Perez, J.; Adeyemi, V.; Sandoval-Ibarra, Y.; Pérez-Pinal, F.; Tlelo-Cuautle, E. FPGA Realization of Spherical Chaotic System with Application in Image Transmission. Math. Probl. Eng.
**2021**, 2021, 5532106. [Google Scholar] [CrossRef] - Chen, G.; Ueta, T. Yet another chaotic attractor. Int. J. Bifurcat. Chaos.
**1999**, 9, 1465–1466. [Google Scholar] [CrossRef] - Wang, X.; Chen, G. Generating Lorenz-like and Chen-like attractors from a simple algebraic structure. Sci. China Inform. Sci.
**2014**, 57, 1–7. [Google Scholar] [CrossRef] [Green Version] - Benettin, G.; Galgani, L.; Giorgilli, A.; Strelcyn, J.M. Lyapunov characteristic exponents for smooth dynamical systems and for hamiltonian systems. A method for computing all of them. Part II: Numerical application. Meccanica
**1980**, 15, 21–30. [Google Scholar] [CrossRef] - Benettin, G.; Pasquali, S.; Ponno, A. The Fermi-Pasta-Ulam problem and its underlying integrable dynamics: An approach through Lyapunov exponents. J. Stat. Phys.
**2018**, 171, 521–542. [Google Scholar] [CrossRef] [Green Version] - Storn, R.; Price, K. Differential Evolution: A Simple and Efficient Adaptive Scheme for Global Optimization Over Continuous Spaces. J. Glob. Optim.
**1997**, 11, 341–359. [Google Scholar] [CrossRef] - Hamza, N.; Sarker, R.; Essam, D. Differential evolution with multi-constraint consensus methods for constrained optimization. J. Glob. Optim.
**2013**, 57, 583–611. [Google Scholar] [CrossRef] - Kennedy, J.; Eberhart, R.C.; Shi, Y. Swarm Intelligence, 1st ed.; Morgan Kaufmann Publishers: Burlington, VT, USA, 2001; p. 512. [Google Scholar]
- Baiquan, L.; Gaiqin, G.; Zeyu, L. The block diagram method for designing the particle swarm optimization algorithm. J. Glob. Optim.
**2012**, 52, 689–710. [Google Scholar] [CrossRef] - Zhou, Y.; Chen, H.; Zhou, G. Invasive weed optimization algorithm for optimization no-idle flow shop scheduling problem. Neurocomputing
**2017**, 137, 285–292. [Google Scholar] [CrossRef] - He, S.; Chen, H.; Lei, T.; Lu, S.; Dai, W.; Qiu, L.; Zhong, L. Dynamics and Complexity Analysis of Fractional-Order Chaotic Systems with Line Equilibrium Based on Adomian Decomposition. Complexity
**2020**, 2020, 5710765. [Google Scholar] - Delgado-Bonal, A.; Marshak, A. Approximate Entropy and Sample Entropy: A Comprehensive Tutorial. Entropy
**2019**, 21, 541. [Google Scholar] [CrossRef] [Green Version] - Danca, M.F. Hidden chaotic attractors in fractional-order systems. Nonlinear Dynam.
**2017**, 89, 577–586. [Google Scholar] [CrossRef] [Green Version] - Danca, M.F.; Kuznetsov, N. Matlab code for Lyapunov exponents of fractional order systems. Int. J. Bifurcat. Chaos.
**2018**, 28, 1850067. [Google Scholar] [CrossRef] [Green Version] - Khan, A.; Kumar, S. T–S fuzzy observed based design and synchronization of chaotic and hyper-chaotic dynamical systems. Int. J. Dyn. Control
**2018**, 6, 1409–1419. [Google Scholar] [CrossRef] - Pesin, Y.B. Characteristic Lyapunov exponents and smooth ergodic theory. Russ. Math. Surv.
**1977**, 32, 55–114. [Google Scholar] [CrossRef] - Zhu, Z.l.; Zhang, Q.; Yu, H.; Gao, J. A new hyper-chaos generated from Chen´s system via an external periodic perturbation. In Proceedings of the 2009 International Workshop on Chaos-Fractals Theories and Applications, Shenyang, China, 6–8 November 2009; pp. 260–266. [Google Scholar]
- Zhou, N.; Pan, S.; Cheng, S.; Zhou, Z. Image compression-encryption scheme based on hyper-chaotic system and 2D compressive sensing. Opt. Laser Technol.
**2016**, 82, 121–133. [Google Scholar] [CrossRef] - Garner, D.M.; Ling, B.W.K. Measuring and locating zones of chaos and irregularity. J. Syst. Sci. Complex.
**2014**, 27, 494–506. [Google Scholar] [CrossRef] - Lin, J.; Chen, C. Parameter estimation of chaotic systems by an oppositional seeker optimization algorithm. Nonlinear Dynam.
**2014**, 76, 509–517. [Google Scholar] [CrossRef] - Zhang, H.; Li, B.; Zhang, J.; Qin, Y.; Feng, X.; Liu, B. Parameter estimation of nonlinear chaotic system by improved TLBO strategy. Soft. Comput.
**2016**, 20, 4965–4980. [Google Scholar] [CrossRef] - Halstead, M.H. Elements of Software Science; Operating and Programming Systems Series; Elsevier: Amsterdam, The Netherlands, 1977; p. 127. [Google Scholar]

**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 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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