TY - JOUR
T1 - Optimizing the Maximum Lyapunov Exponent of Fractional Order Chaotic Spherical System by Evolutionary Algorithms
AU - Adeyemi, Vincent Ademola
AU - Tlelo-Cuautle, Esteban
AU - Perez-Pinal, Francisco Javier
AU - Nuñez-Perez, Jose Cruz
N1 - Publisher Copyright:
© 2022 by the authors.
PY - 2022/8
Y1 - 2022/8
N2 - The main goal of this work is to optimize the chaotic behavior of a three-dimensional chaotic-spherical-attractor-generating fractional-order system and compare the results with its novel hyperchaotic counterpart. The fractional-order chaotic system is a smooth system perturbed with a hyperbolic tangent function. There are two major contributions in this investigation. First, the maximum Lyapunov exponent of the chaotic system was optimized by applying evolutionary algorithms, which are meta-heuristics search algorithms, namely, the differential evolution, particle swarm optimization, and invasive weed optimization. Each of the algorithms was populated with one hundred individuals, the maximum generation was five hundred, and the total number of design variables was eleven. The results show a massive increase of over 5000% in the value of the maximum Lyapunov exponent, thereby leading to an increase in the chaotic behavior of the system. Second, a hyperchaotic system of four dimensions was constructed from the inital chaotic system. The dynamics of the optimized chaotic and the new hyperchaotic systems were analyzed using phase portraits, time series, bifurcation diagrams, and Lyapunov exponent spectra. Finally, comparison between the optimized chaotic systems and the hyperchaotic states shows an evidence of more complexity, ergodicity, internal randomness, and unpredictability in the optimized systems than its hyperchaotic counterpart according to the analysis of their information entropies and prediction times.
AB - The main goal of this work is to optimize the chaotic behavior of a three-dimensional chaotic-spherical-attractor-generating fractional-order system and compare the results with its novel hyperchaotic counterpart. The fractional-order chaotic system is a smooth system perturbed with a hyperbolic tangent function. There are two major contributions in this investigation. First, the maximum Lyapunov exponent of the chaotic system was optimized by applying evolutionary algorithms, which are meta-heuristics search algorithms, namely, the differential evolution, particle swarm optimization, and invasive weed optimization. Each of the algorithms was populated with one hundred individuals, the maximum generation was five hundred, and the total number of design variables was eleven. The results show a massive increase of over 5000% in the value of the maximum Lyapunov exponent, thereby leading to an increase in the chaotic behavior of the system. Second, a hyperchaotic system of four dimensions was constructed from the inital chaotic system. The dynamics of the optimized chaotic and the new hyperchaotic systems were analyzed using phase portraits, time series, bifurcation diagrams, and Lyapunov exponent spectra. Finally, comparison between the optimized chaotic systems and the hyperchaotic states shows an evidence of more complexity, ergodicity, internal randomness, and unpredictability in the optimized systems than its hyperchaotic counterpart according to the analysis of their information entropies and prediction times.
KW - chaos
KW - evolutionary algorithms
KW - fractional order
KW - hyperchaos
KW - optimization
UR - http://www.scopus.com/inward/record.url?scp=85136821711&partnerID=8YFLogxK
U2 - 10.3390/fractalfract6080448
DO - 10.3390/fractalfract6080448
M3 - Artículo
AN - SCOPUS:85136821711
SN - 2504-3110
VL - 6
JO - Fractal and Fractional
JF - Fractal and Fractional
IS - 8
M1 - 448
ER -