**1. Introduction**

Chaotic systems have been widely studied and used in various practical fields by mathematicians, physicists, scientists, and engineers in the past four decades; see [1–4] and the references therein. Many chaotic systems with different shapes of attractors have been reported, such as chaotic systems with butterfly attractors (see, e.g., [5]) and systems with multiscroll chaotic attractors (see, e.g., [6]). Recent developments include some different types of chaotic systems with no equilibrium points (see, e.g., [7]), with a single stable equilibrium (see, e.g., [8]), with a line of equilibrium points (see, e.g., [9]), with a circular equilibrium (see, e.g., [10]), with circle and square equilibrium (see, e.g., [11]), with rounded square loop equilibrium (see, e.g., [12]), and with different closed curve equilibrium (see, e.g., [13]). Furthermore, it has also been applied in many different areas including information processing (see, e.g., [14]) and chaotic masking communication (see, e.g., [15]).

According to a new classification of chaotic dynamics [16], there are two kinds of attractors: self-excited attractors and hidden attractors. Recall that an attractor is referred to as being *self-excited* if its basin of attraction intersects any arbitrarily small open neighborhood of an equilibrium, otherwise it is called a *hidden attractor*. The basin of attraction for a hidden attractor is not connected with any unstable fixed point. For example, hidden attractors are observed in the systems without fixed points, with no unstable fixed points, or with one stable fixed point. A system with infinitely many equilibrium points can be classified as one system with hidden attractors, for the reason that we do not know which part of the equilibria may be used to localize the hidden attractors, which should be treated in detail (see, e.g., [17]). Recent important investigations and developments in the study of chaotic dynamical systems with practical problems and challenges have been asked to satisfy at least one of the following criteria as Sprott mentioned in [18]: *(S1) The system should credibly model some important unsolved problem in nature and shed light on that problem; (S2) the system should exhibit some behavior previously unobserved; (S3) the system should be simpler than all other known examples exhibiting the observed behavior.* An important ongoing research topic is dedicated to discovering and developing new and novel chaotic systems with different shapes of closed curve equilibrium.

*Symmetry* **2019**, *11*, 951

The main goal of this work is to present a new system with infinitely many equilibrium points arranged on two closed curves passing the same point, which extends the general knowledge about such systems. Our new chaotic system (see Section 2 below for details) is meaningful for satisfying two of the three conditions, (S1), (S2), and (S3), as well as there being a certain novelty value in this work. In Section 2, some dynamical properties of the proposed system, which have been studied using a bifurcation diagram, phase portrait, Poincaré section, maximal Lyapunov exponents, and Kaplan–Yorke dimension, are presented. The ability of anti-synchronization of the new system is also discussed in Section 3.

## **2. A New Family with Two Closed Curve Equilibrium**

In this work, motivated by the known dynamic systems mentioned above, we study the following general model given by

$$\begin{aligned} \dot{u} &= w, \\ \dot{v} &= -wf(u,v,w), \\ \dot{w} &= g(u,v), \end{aligned} \tag{1}$$

where *u*, *v* and *w* are three state variables, *f*(*u*, *v*, *w*) and *g*(*u*, *v*) are two nonlinear functions. The equilibrium points in model (1) can be obtained by calculating

$$\begin{aligned} w &= 0, \\ -wf(u, v, w) &= 0, \\ g(u, v) &= 0. \end{aligned} \tag{2}$$

It is obtained that the equilibrium points locate on a curve described by *g*(*u*, *v*) = 0 in the plane *w* = 0. In fact, by selecting appropriate functions *f* and *g*, some known systems, both chaotic and with different closed curve equilibrium, can be constructed.

(*Example A*)

Take *<sup>f</sup>*(*u*, *<sup>v</sup>*, *<sup>w</sup>*) = *<sup>α</sup><sup>v</sup>* + *<sup>β</sup>v*<sup>2</sup> + *uw* and *<sup>g</sup>*(*u*, *<sup>v</sup>*) = *<sup>u</sup>*<sup>2</sup> + *<sup>v</sup>*<sup>2</sup> − 1, then model (1) will deduce the following system

$$\begin{aligned} \dot{u} &= w, \\ \dot{v} &= -w(av + \beta v^2 + \mu w), \\ \dot{w} &= u^2 + v^2 - 1, \end{aligned} \tag{3}$$

which was introduced and studied by Gotthans, Sprott, and Petrzela [11] in 2016. The chaotic systems (3) has circle equilibrium (see Figure 1).

**Figure 1.** The circle-shape of equilibrium points of system (3) in the plane *w* = 0.
