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Article

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

1
Instituto Politécnico Nacional, IPN-CITEDI, Tijuana 22435, Mexico
2
Departamento de Posgrado, Universidad Politécnica de Lázaro Cárdenas (UPLC), Lázaro Cárdenas 60950, Mexico
3
Tecnológico Nacional de México, Instituto Tecnológico de Celaya, Celaya 38010, Mexico
4
Instituto Nacional de Astrofísica, Óptica y Electrónica, INAOE, San Andres Cholula 72840, Mexico
*
Authors to whom correspondence should be addressed.
These authors contributed equally to this work.
Academic Editors: Árpád Bűrmen and Tadej Tuma
Mathematics 2021, 9(11), 1194; https://doi.org/10.3390/math9111194
Received: 16 March 2021 / Revised: 8 May 2021 / Accepted: 10 May 2021 / Published: 25 May 2021
(This article belongs to the Special Issue Optimization Theory and Applications)
This paper presents the application of three optimization algorithms to increase the chaotic behavior of the fractional order chaotic Chen system. This is achieved by optimizing the maximum Lyapunov exponent (MLE). The applied optimization techniques are evolutionary algorithms (EAs), namely: differential evolution (DE), particle swarm optimization (PSO), and invasive weed optimization (IWO). In each algorithm, the optimization process is performed using 100 individuals and generations from 50 to 500, with a step of 50, which makes a total of ten independent runs. The results show that the optimized fractional order chaotic Chen systems have higher maximum Lyapunov exponents than the non-optimized system, with the DE giving the highest MLE. Additionally, the results indicate that the chaotic behavior of the fractional order Chen system is multifaceted with respect to the parameter and fractional order values. The dynamical behavior and complexity of the optimized systems are verified using properties, such as bifurcation, LE spectrum, equilibrium point, eigenvalue, and sample entropy. Moreover, the optimized systems are compared with a hyper-chaotic Chen system on the basis of their prediction times. The results show that the optimized systems have a shorter prediction time than the hyper-chaotic system. The optimized results are suitable for developing a secure communication system and a random number generator. Finally, the Halstead parameters measure the complexity of the three optimization algorithms that were implemented in MATLAB. The results reveal that the invasive weed optimization has the simplest implementation. View Full-Text
Keywords: Chen system; evolutionary algorithm; fractional order system; global optimization; Maximum Lyapunov exponent Chen system; evolutionary algorithm; fractional order system; global optimization; Maximum Lyapunov exponent
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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

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