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

Abstract

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.

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

“…The maximization of the MLE in this investigation further shows the effectiveness of the EAs as global optimization algorithm [6,7,15,16,19,36,37]. Table 1 reveals that the three algorithms obtained system parameter and fractional order values that produced better MLEs than the non-optimized fractional-order system.…”

“…1 shows the equilibrium points, eigenvalues, information entropies, and the instability analysis results. The instability analysis, examined from the eigenvalues, is determined according to the instability theorem in [19]. To get the equilibrium point, we equate each of the three equations of the chaotic system to zero and by simplification and substitutions, we arrived at the following equations to compute z, y, and x, respectively:…”

“…Furthermore, in [16], the parameters optimization was based on the nondominated sorting genetic algorithm (NSGA) applied to two multiscroll chaotic oscillators of type SNLF and Chua's diode, considering two to six scrolls. The authors in [19] presented their work on optimizing the fractional-order Chen system using the DE, PSO, and IWO. The optimized values were the three parameters and the fractional order.…”

Chaos in Analog Electronic Circuits: Comprehensive Review, Solved Problems, Open Topics and Small Example

“…Some authors have become interested in numerical procedures leading to the optimization of chaos via maximization of the largest LE. To do this, evolutionary algorithms can be used, as shown in two new papers [184,185]. Within the frame work of the first paper mentioned, the authors utilize three metaheuristic algorithms: differential evolution, particle swarm optimization, and invasive weed optimization.…”

Dynamical Analysis and Electronic Circuit Implementation of Fractional-order Chen System

“…Some researchers have focused their studies on the fractionalorder analysis of the Chen system and its engineering applications (Li and Peng 2004;Lu and Chen 2006). In their research, Nuñez-Perez et al introduced the use of different optimization algorithms to amplify the chaotic behavior of the fractional-order chaotic Chen system (FOCHEN) (Nuñez-Perez et al 2021). The outcomes demonstrate that the optimized FOCHEN systems exhibit higher maximum Lyapunov exponents compared to the non-optimized system.…”