Complexity

Volume 2017, Article ID 7871467, 11 pages

https://doi.org/10.1155/2017/7871467

## A Novel Chaotic System without Equilibrium: Dynamics, Synchronization, and Circuit Realization

^{1}Faculty of Computers and Information, Benha University, Benha, Egypt^{2}Nanoelectronics Integrated Systems Center (NISC), Nile University, Giza, Egypt^{3}Department of Physics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece^{4}Faculty of Physics, Department of Electronics, Computers, Telecommunications and Control, National and Kapodistrian University of Athens, 15784 Athens, Greece^{5}School of Electronics and Telecommunications, Hanoi University of Science and Technology, 01 Dai Co Viet, Hanoi, Vietnam^{6}Engineering Mathematics and Physics, Cairo University, Giza, Egypt^{7}Research and Development Center, Vel Tech University, Avadi, Chennai, Tamil Nadu 600062, India^{8}Laboratory of Mathematics, Informatics and Systems (LAMIS), University of Larbi Tebessi, 12002 Tebessa, Algeria^{9}University of Puebla, Puebla, PUE, Mexico

Correspondence should be addressed to Christos Volos; moc.liamg@solovhc

Received 21 October 2016; Revised 12 December 2016; Accepted 22 December 2016; Published 2 February 2017

Academic Editor: Carlos Gershenson

Copyright © 2017 Ahmad Taher Azar 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

A few special chaotic systems without unstable equilibrium points have been investigated recently. It is worth noting that these special systems are different from normal chaotic ones because the classical Shilnikov criterion cannot be used to prove chaos of such systems. A novel unusual chaotic system without equilibrium is proposed in this work. We discover dynamical properties as well as the synchronization of the new system. Furthermore, a physical realization of the system without equilibrium is also implemented to illustrate its feasibility.

#### 1. Introduction

A considerable amount of literature has been published on chaotic systems in last decades, for example, Lorenz’s system [1], Rössler’s system [2], Chen and Ueta’s system [3], simple chaotic flows [4, 5], memristive chaotic system with heart-shaped attractors [6], chaotic circuit based on memristor [7, 8], MOS-transistors based oscillators [9, 10], mixed analog-digital designs [11], fully digital realization of chaotic systems [12, 13], or electromechanical oscillator [14]. Complexity of chaotic systems has been used in various engineering applications from asymmetric color pathological image encryption [15, 16], control and synchronization [17, 18], a chaotic video communication scheme via WAN remote transmission [19], and image encryption with avalanche effects [20] to audio encryption scheme [21] and so on.

It is now well established from a variety of studies that equilibrium points play a vital role in our understanding of chaos in nonlinear systems [22–24]. In general, conventional chaotic systems have unstable equilibria and we are able to verify chaos in such systems with the Shilnikov criterion [25, 26]. However, recent researches have consistently shown that chaotic behavior can be observed in three-dimensional (3D) systems with no equilibrium [27].

The study of systems without equilibria has a long history, describing various electromechanical models with rotation and electrical circuits with cylindrical phase space. One of the first such examples has been described by Arnold Sommerfeld in 1902 [28], by studying the oscillations caused by a motor driving an unbalanced weight and discovered the resonance capture, which is called “Sommerfeld effect.” This phenomenon represents the failure of a rotating mechanical system to be spun up by a torque-limited rotor to a desired rotational velocity due to its resonant interaction with another part of the system [29, 30]. Many decades later, in 1984-85, Nosé [31] and Hoover [32] have led the study with their proposed dynamical system without equilibria and its various modifications, where hidden chaotic oscillations can be found [4, 33–36].

Systematic search routine was developed by Jafari et al. to determine simple quadratic flows with no equilibria [24, 27]. Wang and Chen found a new system without equilibrium while studying a chaotic system with any number of equilibria [24]. Wei discovered dynamical properties of a no-equilibrium chaotic system by applying a constant to the Sprott D system [37]. Multiple attractors in a three-dimensional system with no-equilibrium point were reported in [38]. Akgul et al. designed a random number generator with a 3D chaotic system without equilibrium point [39]. In addition, 4D no-equilibrium systems with hyperchaos were presented in [40–42]. It is interesting to note that chaotic systems without equilibrium display “hidden attractors” [43–46]. There has been considerable interest in discovering hidden attractors because they cannot be localized by applying common computational procedures [47–52].

This study makes a contribution to research on systems with hidden attractors by exploring a new chaotic system without equilibrium. In the next section, the description and dynamics of the no-equilibrium system are presented. Synchronization of two new chaotic systems without equilibrium is studied in Section 3. The theoretical system has been realized by an electronic circuit as reported in Section 4. Finally, conclusion remarks are drawn in the last section.

#### 2. Description and Dynamics of the System without Equilibrium

Jafari et al. have introduced an effective approach for investigating potential systems without equilibrium [27]. Authors constructed general models and applied a systematical search routine to obtain seventeen simple flows with no equilibrium [27]. Motivated by Jafari et al.’s systems, in this work we consider a general form as follows:in which three state variables of the general form are , , and , while nine parameters are () with . An absolute nonlinearity has been included in (1) because it is a potential term for designing nonlinear systems with special characteristics [53, 54].

In order to find the equilibrium of system (1), we solve the three following equations:By substituting (2), (3) into (4), we haveIt is easy to verify that the equation is inconsistent forIn other words, in this case the general model (1) has no equilibrium.

By applying a systematic search procedure [27] into (1), a simple three-dimensional system is obtained in the following form:in which three state variables are , , and while two positive parameters are . According to condition (6), it is trivial to verify that there is no equilibrium in the new system (7).

It is interesting that system (7) can generate chaotic signals although there is the absence of equilibrium. For , and the initial conditions , system (7) generates chaotic behavior as shown in Figure 1. As can be seen in Figure 1, chaotic waveforms and broadband spectra indicate the chaoticity of system (7). In addition, chaotic phase portraits of system (7) are illustrated in Figure 2. Calculated Lyapunov exponents and Kaplan-York dimension of the system without equilibrium (7) are , , , and , respectively. In other words, system (7) has hidden attractors, which is important for a wide range of scientific and engineering processes [55–58]. In our work, the well-known algorithm of Wolf et al. [59] has been applied to calculate Lyapunov exponents. The time of the computation is 10,000. It is noted that, due to the different values of the finite-time local Lyapunov exponents and Lyapunov dimension for different points, the maximum of the finite-time local Lyapunov dimensions on the grid of point has to be considered [60–62].