Complexity

Volume 2018, Article ID 6794791, 9 pages

https://doi.org/10.1155/2018/6794791

## Family of Bistable Attractors Contained in an Unstable Dissipative Switching System Associated to a SNLF

Dynamical Systems Laboratory, CULagos, University of Guadalajara, 1144 Enrique Díaz de León St., Paseos de la Montaña, Lagos de Moreno, Jalisco 47460, Mexico

Correspondence should be addressed to J. L. Echenausía-Monroy; moc.liamg@aisuanehce.siul.esoj and G. Huerta-Cuellar; xm.gdu.sogal@atreuh.g

Received 27 April 2018; Revised 13 July 2018; Accepted 30 July 2018; Published 15 October 2018

Academic Editor: Viet-Thanh Pham

Copyright © 2018 J. L. Echenausía-Monroy 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

This work presents a multiscroll generator system, which addresses the issue by the implementation of 9-level saturated nonlinear function, SNLF, being modified with a new control parameter that acts as a bifurcation parameter. By means of the modification of the newly introduced parameter, it is possible to control the number of scrolls to generate. The proposed system has richer dynamics than the original, not only presenting the generation of a global attractor; it is capable of generating monostable and bistable multiscrolls. The study of the basin of attraction for the natural attractor generation (9-scroll SNLF) shows the restrictions in the initial conditions space where the system is capable of presenting dynamical responses, limiting its possible electronic implementations.

#### 1. Introduction

Over the last few years, the development and implementation of chaotic oscillators have been extensively studied, taking a special interest in the generation of systems with multiscrolls in their phase space, such as the Lorenz [1] and Chua [2] systems, which present a double-scroll attractor. There are several methods to obtain multiscroll behavior, for example, by adding breakpoints to Chua’s function [3, 4], by using a system with hysteresis [5, 6], implementing step functions, using sine/cosine functions, or by generating piecewise linear functions [7–10]. The disadvantage of these methods is that the systems have more fixed points than scrolls generated, so increasing the number of scrolls in the phase space turns into a more complex task to address. As an alternative to this dilemma arises, the conception of a saturated nonlinear function, SNLF, which is based on the operational amplifiers performance [11, 12], which guarantees to find as many scrolls as segmentation points the function possesses, being a simpler way to approach the topic in the scrolls generation.

An example of this kind of systems can be described within the theory of unstable dissipative systems, UDS [13, 14], which characterizes the systems of three differential equations based on the location of the eigenvalues that it possesses. UDS’s are classified as type I or type II, where the order of the type represents the number of eigenvalues with negative real part. In general, any three-dimensional dynamic system is considered an UDS if and only if it has a combination of eigenvalues that coincide with the definition of a hyperbolic saddle-node, and the sum of these components is negative, i.e., the dissipation condition is fulfilled [15]. Examples of these systems are found in Rössler [16], Lorenz [1], and Chua [2], among some other systems [17–19]. This kind of combination in the eigenvalues favors the appearance of multiscroll behavior, by means of the implementation of the appropriated nonlinear function.

In recent years, the design and control of systems with multiple scrolls have been a subject of interest for the scientific community, having a great impact in their application, such as secure communication systems, neuronal modeling, and generation of pseudorandom systems [20–23]. In this work, the modification in a multiscroll dynamics by means of controlling the associated nonlinear function is presented, introducing a new control parameter. This new control parameter helps to generate a specific number of scrolls, generating regions of coexisting attractors, as well as the generation of attractors with double-wing and three equilibrium points.

This work is structured as follows: the first section contains an introduction that describes previous works and theoretical principles of the system. The second section shows the UDS definition and the description of the multiscroll generator system. In the third section, the methodology and results of the studied system are shown. The analysis of the bifurcation diagrams exhibits the coexistence of two attractors for fixed set parameters, bistability, which is illustrated by the construction of the corresponding basins of attractions. The main conclusions are shown in the last section.

#### 2. Theoretical Background

##### 2.1. Unstable Dissipative Systems Theory

In the same spirit as [13, 24, 25], it is considered as a system of autonomous differential equations of third order, where is the state vector, is a constant matrix, is a constant position vector, and is a nonlinear function. The behavior of the system is governed by the eigenvalues of the matrix , which generates a great variety of characteristic values, presenting special attention to those saddle-node points that have a stable and an unstable variety. This kind of eigenvalues is responsible for both stretching and successive folding in the dynamic of the system, which favors the generation of multiscrolls [14, 26].

A system can be considered as an UDS type I if their equilibrium points correspond to a hyperbolic-saddle-node, i.e., one eigenvalue is negative real (dissipative component) and the other two are complex conjugated with positive real part (unstable and oscillatory component), where the sum of the components must be less than zero. By other side, an UDS type II, eq. (1), is described in the opposite way, i.e., one eigenvalue is positive real (unstable component) and the other two are complex conjugated with negative real part (dissipative and oscillatory component), and the sum of them must be less than zero [13].

##### 2.2. Multiscroll Generator

The multiscroll generator system studied is described by a set of three coupled differential equations that makes use of the definition of a saturated nonlineal function, SNLF, as a method for the scroll generation [12, 27], eq. (2). where are the state variables, is the SNLF, is the upper limit of scrolls to generate, and are the system parameters that define the behavior of the dynamic. This work is focused on the region for which the system is defined as a UDS I. Within this proposition, the multiscroll appearance is possible, by generating a conservative component that causes the oscillation of the system over an equilibrium point, while the other two dissipative components favor the visit to other fixed points, resulting in the dynamics of a multiscroll system. The operation region of the system is defined by the combination of the system parameters, in this case, the following consideration is contemplated: .

Considering the previous condition, it is possible to examine the behavior of the equilibrium points by sweeping the control parameter and finding the operation zone where their eigenvalues are consistent with an UDS I definition. The control parameter variation is developed by means of the characteristic polynomial of the system described in eq. (2), , and are plotted by considering the split of the real and imaginary component of each eigenvalue, , , being the real part, and the imaginary part. Figure 1 shows the analysis of the eigenvalues over a range value defined as , Figure 1(a) shows the operation zone delimited by . Figures 1(b) and 1(c) confirm the conditions for the UDS I definition. the real negative eigenvalue, and the complex conjugated eigenvalues with positive real part.