Maximizing Lyapunov Exponents in a Chaotic Oscillator by Applying Differential Evolution

Abstract

This paper shows the application of the heuristic Differential Evolution (DE) algorithm to maximize the positive Lyapunov exponent in a third-order multi-scroll chaotic oscillator. The case of study is the saturated nonlinear function series-based chaotic oscillator. The positive Lyapunov exponent is computed for a 4-scrolls chaotic oscillator by varying the coefficients of the dynamical system (a, b, c, d₁), in the range [0.001..1.000]. The experiments are performed and compared executing DE and a simple Genetic algorithm. The results show that DE algorithm is quite suitable to maximize the positive Lyapunov exponent of truncated coefficients over the continuous parameter spaces, because statistical studies show a small standard deviation. The comparison of the phase space diagrams of non-optimized and optimized chaotic oscillators show that for a low value of the positive Lyapunov exponent the attractor is well defined, while for its maximum value the attractor is not well appreciated, but the higher value of the exponent increases the unpredictability grade of the chaotic system.