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

Abstract

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.

Chaos-Enhanced Archimede Algorithm for Global Optimization of Real-World Engineering Problems and Signal Feature Extraction

“…These unique properties have enabled chaotic maps to be incorporated into several renowned optimization algorithms, including moth flame optimization (MFO), firefly algorithm (FA), artificial bee colony (ABC), biogeographybased optimization (BBO), particle swarm optimization (PSO), and gray wolf optimizer (GWO). This fusion of chaos theory with optimization algorithms has paved the way for more efficient solutions and better performance, both in solving complex problems and in improving engineering processes [31][32][33]. Chaotic maps have added an extra dimension to the search for solutions by introducing an element of dynamics and unpredictability, which has proved beneficial in escaping local minima and improving the quality of the solutions found.…”

Chaotic States of Transistor-Based Tuned-Collector Oscillator

“…As pointed out in papers [16][17][18][19][20], searching for the chaos process can be transformed into an optimization problem. In doing so, robust chaos was detected within the analyzed model of the tuned collector oscillator for the following mutual inductance and hypothetical bias point of the transistor modeled by impedance parameters…”

Chaotic Steady States of the Reinartz Oscillator: Mathematical Evidence and Experimental Confirmation

“…Of course, the topology can differ slightly for specific applications. Several recent papers, for example, [18][19][20][21][22], utilize a multi-objective fitness function to find a robust chaotic motion within the lower-order deterministic dynamical systems. The proposed methods are often general, such that both autonomous and driven systems can be investigated.…”