Complexity

Volume 2018, Article ID 3151840, 17 pages

https://doi.org/10.1155/2018/3151840

## Experimental Verification of Optimized Multiscroll Chaotic Oscillators Based on Irregular Saturated Functions

^{1}Faculty of Electronics Sciences, Autonomous University of Puebla, 72570 Puebla, PUE, Mexico^{2}Department of Electronics, National Institute of Astrophysics, Optics and Electronics, 72840 Tonantzintla, PUE, Mexico^{3}Faculty of Physics and Mathematics Sciences, Autonomous University of Puebla, 72570 Puebla, PUE, Mexico^{4}Laboratory of Nonlinear Systems, Circuits & Complexity, Department of Physics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece

Correspondence should be addressed to J. M. Muñoz-Pacheco; xm.paub.oerroc@ocehcap.msusej

Received 1 October 2017; Revised 25 December 2017; Accepted 30 January 2018; Published 6 March 2018

Academic Editor: Kevin Wong

Copyright © 2018 J. M. Muñoz-Pacheco et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Multiscroll chaotic attractors generated by irregular saturated nonlinear functions with optimized positive Lyapunov exponent are designed and implemented. The saturated nonlinear functions are designed in an irregular way by modifying their parameters such as slopes, delays between slopes, and breakpoints. Then, the positive Lyapunov exponent is optimized using the differential evolution algorithm to obtain chaotic attractors with 2 to 5 scrolls. We observed that the resulting chaotic attractors present more complex dynamics when different patterns of irregular saturated nonlinear functions are considered. After that, the optimized chaotic oscillators are physically implemented with an analog discrete circuit to validate the use of proposed irregular saturated functions. Experimental results are consistent with MATLAB™ and SPICE circuit simulator. Finally, the synchronization between optimized and nonoptimized chaotic oscillators is demonstrated.

#### 1. Introduction

The chaotic behavior has attracted a lot of attention for scientific community due to extreme sensitivity to its initial conditions and the broadband nature of its chaotic signals [1–28]. Therefore, in the last years, literature is vast in papers oriented to study new chaotic systems [1–5], propose novel applications [6–9], increase the degree of chaos (hyperchaotic systems [8, 10, 11]), get fractional order chaotic systems [7, 12, 13], synchronize the chaotic behavior [14–18], optimize chaotic systems [19–22], and implement chaotic oscillators using electronic circuits [23–28]. In all these studies, chaos behavior is analyzed and verified by using different approaches, for example, frequency spectrum, Poincaré maps, bifurcation diagrams, Lyapunov exponents, and stability of equilibrium points. Among them, Lyapunov exponents provide a direct measure of the sensitive dependence on initial conditions by quantifying the exponential rates at which neighboring orbits on an attractor diverge as the system evolves in time [29–31].

For an -dimensional nonlinear system, if the system has at least one positive Lyapunov exponent (LE) and is purely deterministic, then it is chaotic. Indeed, a tool commonly used to determine the presence of chaos in several numerical and experimental results is to compute only the positive LE [31].

Besides, the positive LE can be very useful to determine the unpredictability grade of the chaotic oscillator because its magnitude specifies the maximum average exponential rate corresponding to divergence of trajectories on an attractor and thus the maximum amount of instability along any direction [29–31]. That is, a high value of the positive LE can be taken as an indication of a high degree of chaos in the dynamical system [19–22, 32–35].

For instance, in [32] the speed effects and leg amputations on the dynamic stability of running are analyzed by computing the largest LE. The results revealed that the value of the largest LE is positive not only for unaffected patients but also for the one-leg affected patients. However, the positive LE of embedded time-series data from the affected leg was higher than for the unaffected leg indicating a more rich dynamics. In [33] a numerical scheme based on the Lattice Boltzmann method for the flow in complex mixer geometries to compute trajectories of passive tracers for the quantification of chaotic mixing was reported. They reported a better efficiency in chaotic micromixers when the value of positive LE was higher. Moreover, chaotic systems with a high value of the positive LE have also been used to improve the performance of chaos-based applications, for example, in [34] it was demonstrated that a high value of the positive LE in an optimization algorithm based on a particle swarm implies that the particles are inclined to explore different regions and find better fitness values. Therefore, the particle swarm with just a little variation in the value of positive LE usually achieved a better performance, especially for multimodal functions. In [35] the efficiency of hybrid chaotic optimization algorithms was studied by revealing effects on the search speed as a function of chaotic sequences from different chaotic maps. It was found that the higher the magnitude of positive LE, the faster the search speed in whole optimization space. Accordingly, the efficiency of global optimization was directly proportional to the value of positive LE.

In this framework, optimized chaotic systems with a high value of the positive LE can be extremely suitable to enhance the existing chaos-based applications. In electronics, a great variety of multiscroll chaotic oscillators has been implemented with commercially available electronic devices, as well as with integrated circuits technology [5, 10, 23, 26–28, 36, 37]. However, those experimental realizations are not optimized to provide a high value of the positive Lyapunov exponent (PLE). Although some authors have already used optimization algorithms based on evolutionary computation to get optimized multiscrolls chaotic systems [19, 22], they were obtained by using piecewise-linear (PWL) functions in the form of saturated nonlinear functions (SNLF) with symmetry properties. In addition, the experimental verification of those approaches is lacking.

This paper is motivated by the aforementioned discussion. In that scenario, we design irregular SNLF to obtain multiscroll chaotic attractors with optimized values of the positive LE. Additionally, we also demonstrate its practical feasibility by the physical implementation of the resulting multiscroll chaotic oscillators. Two cases were considered to design the irregular SNLFs. The first one consists of changing the breakpoints of SNLF to get different slopes, whereas the second one modifies the delay between slopes in different sections of SNLF. In both cases, once the parameters of SNLF are defined, we apply the evolutionary algorithms reported in [19, 22] to find the optimal value for system’s parameters which maximizes the magnitude of positive LE. As a result, multiscroll chaotic oscillators with a more complex dynamics are generated. Experimental results for 2-, 3-, 4-, and 5-scroll chaotic attractors were obtained with the aim of an analog discrete circuit based on commercial operational amplifiers (OpAmps). Further, we show the synchronization of those optimized chaotic oscillators by using generalized Hamiltonian forms because they can enhance the synchronization and realization of secure communication systems, for instance.

The paper is organized as follows. Section 2 describes the multiscroll chaotic oscillator under study; Section 3 outlines the steps to obtain optimized values of positive LE as well as the electronic design. Sections 4 and 5 present the experimental confirmation of the proposed approach for the two cases: slopes varying and different delays between slopes, respectively. Section 6 demonstrates the synchronization between optimized and nonoptimized chaotic oscillators. Finally, conclusions are given in Section 7.

#### 2. SNLF-Based Multiscroll Chaotic Oscillator

The case of study in this work is the chaotic oscillator described bywhere is the state variables, is the SNLF, and is the system’s parameters. To maximize the value of positive LE requires varying the coefficients of chaotic oscillator, leading to a huge number of combinations. Herein, system’s parameters are varied within the range . So, we define four variables where each one can have = possible combinations. This result justifies the application of heuristics like the ones already introduced in [19, 22]. For all cases analyzed in this work, the phase-space plots show the state variables versus .

The first step consists of manipulating with the goal of incrementing the complexity of multiscroll chaotic oscillator. To generate 2 scrolls, the SNFL description is given bywhere represents the value of saturated regions, is the slope between two saturated regions, is a break point connecting a saturated region with a slope, and is the delay, as shown in Figure 1. In general, the number of scrolls to be generated equals the number of saturated regions . Therefore, (2) can be augmented to generate -scrolls as shown in [38]. It means that by augmenting segments in a symmetric way as shown in Figures 1(a) and 1(b), respectively, even and odd number of scrolls are generated. However, in this work we show how to use irregular SNLF functions, that is, nonsymmetric, to obtain multiscrolls as the ones shown in Figure 2. The irregular SNLF functions are herein designed using different values for breakpoints with where is the number of breakpoints, slopes , and saturated levels .