Advances and Applications in Chaotic Systems by Sundarapandian Vaidyanathan & Christos Volos

Author:Sundarapandian Vaidyanathan & Christos Volos
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Keywords

Multi-scroll chaotic oscillatorMaximum Lyapunov exponent EntropyEvolutionary algorithmOperational amplifier

1 Introduction

In Electronics, a great variety of chaotic oscillators has been implemented with different kinds of electronic devices [7, 15, 19], using integrated circuits technology [20], and more recently by using field programmable gate arrays [17]. In addition, recent works show a relationship between the number of scrolls and the value of the maximum Lyapunov exponent (MLE) [2, 18], and also a relationship between the number of scrolls and the associated entropy [23]. Both characteristics associated to MLE and entropy are quite desirable to improve for the development of enhanced applications in nonlinear systems, like for example: the implementation of random number generators [1, 4, 10, 23] that are quite useful in robotics [16, 21].

In this chapter we show the experimental realization of multi-scroll chaotic attractors that are optimized to provide a high value of the MLE, and for each attractor it is also guaranteed to have a good distribution of the trajectories that is visualized in the phase-space portraits. As concluded in [23], the next sections show that the MLE increases by increasing the number of scrolls, indicating a better unpredictability of the dynamical system due to the increment of its associated entropy. In addition, we also show the experimental realization of multi-scroll chaotic attractors having a uniform distribution of its trajectories in the phase-space portrait, because when the phases are not well distributed among all the scrolls, some scrolls cannot be formed, thus leading to a pretty difficult problem for the electronic implementation.

The optimization algorithm proposed in [2], is applied herein. It is based on the evolutionary algorithm known as non-dominated sorting genetic algorithm (NSGA-II) [3], and it optimizes two characteristics, namely: (a) Maximizing the positive Lyapunov exponent, and (b) minimizing the dispersions of the phase transitions among all scrolls in an attractor. The results of the optimization algorithm show that both characteristics are in conflict, so that a feasible set of solutions is provided to select the best one according to the problem at hand.

From the set of feasible solutions provided by applying [2], we select some multi-scroll chaotic attractors having high MLE, and then we realize experiments by implementing the chaotic oscillators with commercially available operational amplifiers. The purpose of the experiments are oriented to verify the relationship on the number of scrolls, their MLE value and the associated entropy.

The case of study is the multi-scroll chaotic oscillator based on saturated nonlinear function series, already described in [7]. It is a third order continuous-time dynamical system, and then it has three Lyapunov exponents, one being positive, which is known as MLE and its value indicates the degree of chaotic behavior. It is optimized by applying the evolutionary algorithm given in [2], and its associated entropy is evaluated from both numerical simulations and experimental data for generating 2-, 5- and 10-scrolls. The goal of performing the experiments is to verify that the higher the number of scrolls the higher the MLE and its associated entropy [9]. Further, optimized chaotic oscillators

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