Advances in Mathematical Physics

Volume 2016, Article ID 4024836, 8 pages

http://dx.doi.org/10.1155/2016/4024836

## A Chaotic System with an Infinite Number of Equilibrium Points: Dynamics, Horseshoe, and Synchronization

^{1}School of Electronics and Telecommunications, Hanoi University of Science and Technology, 01 Dai Co Viet, Hanoi, Vietnam^{2}Department of Physics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece^{3}Research and Development Centre, Vel Tech University, Avadi, Chennai 600062, India^{4}Institute for Advanced Study, Shenzhen University, Shenzhen, Guangdong 518060, China

Received 27 September 2016; Accepted 14 November 2016

Academic Editor: Zhi-Yuan Sun

Copyright © 2016 Viet-Thanh Pham 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

Discovering systems with hidden attractors is a challenging topic which has received considerable interest of the scientific community recently. This work introduces a new chaotic system having hidden chaotic attractors with an infinite number of equilibrium points. We have studied dynamical properties of such special system via equilibrium analysis, bifurcation diagram, and maximal Lyapunov exponents. In order to confirm the system’s chaotic behavior, the findings of topological horseshoes for the system are presented. In addition, the possibility of synchronization of two new chaotic systems with infinite equilibria is investigated by using adaptive control.

#### 1. Introduction

Nonlinear systems with chaotic behavior have been exploited since the 1960s [1–4]. Their applications have been witnessed in numerous areas, for example, secure digital communication systems [5], multiple input multiple output radar [6], image encryption with random bit sequence [7], or optimization algorithms [8]. Although almost normal chaotic systems have a countable number of equilibrium points, few unusual systems with infinite number of equilibria have been investigated in the last five years [9]. Chaotic system with line equilibrium was reported in [9–11]. A new class of chaotic systems with circle and square equilibrium was discovered by using predefined general forms [12, 13]. In addition, hyperchaotic behavior was observed in a four-dimensional system with a curve of equilibria [14] or four-dimensional systems with a line of equilibria [15–17].

Remarkably, systems with an infinite number of equilibrium points are considered as systems with “hidden attractors” based on the view point of computation [18–21]. Hidden attractors cause unexpected effects for engineering systems [22–25]. However, the characteristics of hidden attractors are not well understood [26]. The community has raised some concerns about discovering hidden attractors in known systems [27, 28], finding new systems with hidden attractors [29, 30], studying synchronization schemes for systems with hidden attractors [31], or verifying chaotic dynamics in systems with hidden attractors with topological horseshoes [32, 33].

Motivated by special features of systems with hidden attractors, we introduce a new system with an open curve of equilibrium points in this work. In the next section, the model of the new system is described and its dynamics are discovered through different nonlinear tools. Chaotic dynamics of the proposed system are studied through topological horseshoes in Section 3. A possible synchronization of two new identical systems is discussed in Section 4. Finally, Section 5 concludes our work.

#### 2. New System with an Infinite Number of Equilibrium Points and Its Properties

The new system proposed in the present work is a three-dimensional continuous system described asin which three state variables are , , and . It is worth noting that there is only one positive parameter () in system (1).

It is straightforward to find the equilibrium points of the proposed system by setting the right hand side of (1) to equal zero, that is,Equation (2) reveals that . By substituting into (3) and (4) we haveIn other words, system (1) has an infinite number of equilibrium points:

For the equilibrium , the Jacobian matrix of system (1) is given byOn combining this result with , we obtain its characteristic equationIt is easy to verify that the characteristic equation (8) has one zero eigenvalue () and two nonzero eigenvalues () which depend on the sign of the discriminant:

For , we get positive eigenvalues . Two nonzero eigenvalues are for the positive discriminant. When the discriminant (9) is negative, a pair of complex conjugate eigenvalues is . These eigenvalues state that the equilibrium point is unstable for and .

It is interesting that system (1) with uncountable equilibria is chaotic for and the initial condition . Chaotic attractors of system (1) are presented in Figure 1. Its Lyapunov exponents and Kaplan–Yorke dimension are , , , and , respectively. The well-known Wolf’s method [34] has been applied to calculate the Lyapunov exponents in our work. The time of computation is 10,000. It is worth noted that, in general, in numerical experiments one cannot expect to get the same values of the finite-time local Lyapunov exponents and the Lyapunov dimension for different points [35–37]. Therefore, the maximum of the finite-time local Lyapunov dimensions on the grid of points has to be considered [35–37].