On maximizing the positive Lyapunov exponent of chaotic oscillators applying DE and PSO

Abstract

Lyapunov exponents are related to the exponentially fast divergence or convergence of nearby orbits in phase space, and they can be used to evaluate the Kaplan–Yorke dimension. In this manner, since the existence of a positive Lyapunov exponent (LE+) is taken as an indication that chaotic behavior exists, and due to the huge search spaces of the design variables of chaotic oscillators, we show the application of differential evolution (DE) and particle swarm optimization (PSO) algorithms to maximize LE+. Four chaotic oscillators are optimized herein, for which we detail the evaluation of their equilibrium points and their eigenvalues that are used to estimate the step-size h to perform appropriate numerical simulation. Both DE and PSO are calibrated to perform different number of generations with three different sizes of individuals in the populations, and with search spaces around the values already published for the four chaotic oscillators. As a result, we show that both DE and PSO algorithms provide higher values of LE+ and Kaplan–Yorke dimension compared to the ones already published in the literature.