(*Example B*)

Let *<sup>f</sup>*(*u*, *<sup>v</sup>*, *<sup>w</sup>*) = *<sup>α</sup><sup>v</sup>* <sup>+</sup> *<sup>β</sup>v*<sup>2</sup> <sup>+</sup> *uw* and *<sup>g</sup>*(*u*, *<sup>v</sup>*) = *<sup>u</sup>*<sup>2</sup> <sup>−</sup> <sup>|</sup>*uv*<sup>|</sup> <sup>+</sup> *<sup>v</sup>*<sup>2</sup> <sup>−</sup> 1. Then, model (1) deduces the following system:

$$\begin{aligned} \dot{u} &= w, \\ \dot{\psi} &= -w(\alpha v + \beta v^2 + \mu w), \\ \dot{w} &= u^2 - |\mu v| + v^2 - 1, \end{aligned} \tag{4}$$

which was established by Wang, Pham, and Volos [19] in 2017. The chaotic system (4) has cloud-shaped curve equilibrium, as shown in Figure 2.

**Figure 2.** The cloud-shape of equilibrium points of system (4) in the plane *w* = 0.

(*Example C*)

Very recently, Zhu and Du [13] discovered and studied a new family of systems with different equilibrium (as shown in Figure 3) described by

$$\begin{aligned} \dot{u} &= w, \\ \dot{\upsilon} &= -w(av + \beta v^2 + \mu w), \\ \dot{\upsilon} &= |u|^k + |v|^k - 1, \end{aligned} \tag{5}$$

where *<sup>k</sup>* <sup>∈</sup>N. In fact, the chaotic system (5) can be obtained by putting *<sup>f</sup>*(*u*, *<sup>v</sup>*, *<sup>w</sup>*) = *<sup>α</sup><sup>v</sup>* <sup>+</sup> *<sup>β</sup>v*<sup>2</sup> <sup>+</sup> *uw* and *g*(*u*, *v*) = |*u*| *<sup>k</sup>* <sup>+</sup> <sup>|</sup>*v*<sup>|</sup> *<sup>k</sup>* <sup>−</sup> 1 into model (1). In [13], Zhu and Du analyzed the dynamical properties of their proposed systems using the methods of equilibrium points, eigenvalues, phase portraits, maximal Lyapunov exponents, and Kaplan–Yorke dimension; see [13] for more details.

**Figure 3.** Different shapes of equilibrium points of system (5), *k* = 1, 2, 3, 4, 5, from the interior to the outside, respectively, in the plane *w* = 0.

The results established in [11,13,19] are very important for indicating the existence of chaotic systems with different shapes of equilibrium points (see Table 1). Note that the first two equations of the systems proposed in [13,19] are the same as in [11]. The difference is the third equation. When we choose a different third equation, we can get different systems to display new features, such as different shapes of equilibrium point and other dynamic properties.


**Table 1.** Chaotic systems with infinitely many equilibrium points.

To the best of our knowledge, there is no paper devoted to the study of chaotic dynamical systems with eye-shaped curve equilibrium. Therefore, this study is an important ongoing research topic. In this paper, motivated and inspired by this, two functions, *f*(*u*, *v*, *w*) and *g*(*u*, *v*), are chosen in the following forms

$$\begin{aligned} f(u,v,w) &= av + \beta v^2 + \mu w, \\ g(u,v) &= u^2 - |u| + |v| + v^2. \end{aligned} \tag{6}$$

where *α* and *β* are two positive parameters. Substituting (6) into system (1), our new system is described as

$$\begin{aligned} \dot{u} &= w, \\ \dot{v} &= -w(av + \beta v^2 + \mu w), \\ \dot{w} &= u^2 - |u| + |v| + v^2. \end{aligned} \tag{7}$$

It is verified that system (7) has infinitely many equilibrium points (*u*∗, *v*∗). These equilibrium points are located on the curve in the coordinate plane described by

$$|\left(\mu^\*\right)^2 - \left|\mu^\*\right| + \left|v^\*\right| + \left(v^\*\right)^2 = 0. \tag{8}$$

It means that the new system (7) has eye-shaped curve equilibrium as shown in Figure 4. Note that the eye-shaped curve is different from some other shapes reported, such as line, square, circle, or cloud-shaped [11,19], and is symmetric about the *u*-axis, *v*-axis, and origin. Furthermore, system (7) has hidden attractors [17]. Above all, investigating system (7) will strengthen our understanding of hidden attractors.

**Figure 4.** The eye-shape of equilibrium points of system (7) in the plane *w* = 0.

For *α* = 5, *β* = 30, and initial conditions (0.06, 0.01, 0.01), the new system (7) has chaotic attractors (see Figures 5 and 6). For the simulation, we used the Wolf et al. method to calculate the Lyapunov exponents [20], the time of computation was 1000, and we obtained the Lyapunov exponents (0.0424, 0, −0.2484). The method of Wolf et al. is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems to monitor the long-term growth rate of small volume elements in an attractor. In addition, the corresponding Kaplan–Yorke dimension of system (7) is 2.1707. Poincaré return maps are often used to transform complicated behavior of a dynamic system in phase space to discrete maps in a lower dimensional space to reveal the complicated behaviors. Poincaré return maps corresponding with phase portraits in Figure 6 are presented in Figure 7; there are some dense points in the Poincaré section, and it can be determined that the motion is a chaotic state. These results reveal that the system is chaotic.

**Figure 5.** 3D view of the chaotic attractor and eye-shape of equilibrium points located in the plane w = 0 of system (7) for *α* = 5, *β* = 30.

**Figure 6.** The projection of the trajectory of system (7) in (**a**) u-v plane, (**b**) u-w plane, (**c**) v-w plane for *α* = 5, *β* = 30.

**Figure 7.** The Poincaré section of system (7) for (**a**) *z* = 0.2, (**b**) *y* = 0.2, (**c**) *x* = −0.2 for *α* = 5, *β* = 30.

Gradually changing the value of the parameter *β* or *α*, the bifurcation plot of the system can be discovered in Figure 8. Figures 9 and 10 reveal the diagram of Maximal Lyapunov Exponents and the diagram of Kaplan–Yorke dimension of system (7) for *α* = 5, *β* ∈ [28, 48], respectively.

**Figure 8.** Bifurcation plot of system (7) for (**a**) *α* = 5, *β* ∈ [28, 48] and (**b**) *β* = 30, *α* ∈ [3, 5.5].

**Figure 9.** Maximal Lyapunov Exponents spectrum of system (7) for *α* = 5, *β* ∈ [28, 48].

**Figure 10.** Kaplan–Yorke dimension of system (7) for *α* = 5, *β* ∈ [28, 48].

The new system with eye-shaped equilibrium has periodic behavior in the range 36 ≤ *β* ≤ 48. For instance, the system can display period-1 behavior for *α* = 5, *β* = 45, period-2 behavior for *α* = 5, *β* = 38, and period-4 behavior for *α* = 5, *β* = 36 (see Figure 11a–c, respectively).

**Figure 11.** Periodic behavior of system (7) in the u-w plane: (**a**) period-1 (*β* = 45), (**b**) period-2 (*β* = 38), (**c**) period-4 (*β* = 36).

## **3. Anti-Synchronization of New Systems**

Synchronization of chaos is a phenomenon that may occur when two, or more, dissipative chaotic systems are coupled. Since the pioneering work of Pecora and Carroll related to synchronization in chaotic systems [21], some methods of chaotic synchronization have been presented related to complete, generalized, lag, and imperfect phase synchronization [22]. Many papers on applications of chaos synchronization for cryptographic [23], kinetics [24], physiology [25], neural networks [26], and economics [27] have appeared.

In the following, we consider the anti-synchronization of the systems with eye-shaped equilibrium related to the driver-response system. The driver system with eye-shaped closed curve equilibrium is as follows:

$$\begin{aligned} \dot{u} &= w\_\prime\\ \dot{v} &= -w(\alpha v + \beta v^2 + \mu w)\_\prime\\ \dot{w} &= u^2 - |u| + |v| + v^2 \end{aligned}$$

where *u*, *v*, and *w* are are three state variables, and the value of *α* = 5, *β* = 30.

The response system is described as

$$\begin{aligned} \dot{u}\_1 &= w\_1 + h\_{1\prime} \\ \dot{w}\_1 &= -w\_1(av\_1 + \beta v\_1^2 + u\_1 w\_1) + h\_{2\prime} \\ \dot{w}\_1 &= u\_1^2 - |u\_1| + |v\_1| + v\_1^2 + h\_{3\prime} \end{aligned} \tag{9}$$

where the control is **h** = [*h*1, *h*2, *h*3] *T*.

In order to reveal the difference between the driver system (7) and the response system (9), the state errors can be defined as

$$\begin{aligned} \varepsilon\_1 &= u + u\_{1\prime} \\ \varepsilon\_2 &= v + v\_{1\prime} \\ \varepsilon\_3 &= w + w\_{1\prime} \end{aligned} \tag{10}$$

and we obtain

$$\begin{aligned} \dot{\upsilon}\_1 &= \dot{\upsilon} + \dot{\upsilon}\_{1\prime} \\ \dot{\upsilon}\_2 &= \dot{\upsilon} + \dot{\upsilon}\_{1\prime} \\ \dot{\upsilon}\_3 &= \dot{\upsilon} + \dot{\upsilon}\_{1\prime} \end{aligned} \tag{11}$$

Combining (7), (9), (10), and (11), we get the state errors system

$$\begin{aligned} \dot{e}\_1 &= c\_3 + h\_1, \\ \dot{e}\_2 &= -w(av + \beta v^2 + \mu w) - w\_1(av\_1 + \beta v\_1^2 + u\_1 w\_1) + h\_2, \\ \dot{e}\_3 &= u^2 - |u| + |v| + v^2 + u\_1^2 - |u\_1| + |v\_1| + v\_1^2 + h\_3. \end{aligned} \tag{12}$$

We choose the control proposed by

$$\begin{aligned} h\_1 &= -e\_3 - k\_1 e\_1, \\ h\_2 &= w(av + \beta v^2 + uw) + w\_1(av\_1 + \beta v\_1^2 + u\_1 w\_1) - k\_2 e\_2, \\ h\_3 &= -u^2 + |u| - |v| - v^2 - u\_1^2 + |u\_1| - |v\_1| - v\_1^2 - k\_3 e\_3. \end{aligned} \tag{13}$$

where *ki* > 0 (*i* = 1, 2, 3) are the positive gain constants used to control the rate of anti-synchronization. By substituting (12) into (11), we get the state errors system

$$\begin{aligned} \dot{e}\_1 &= -k\_1 e\_1, \\ \dot{e}\_2 &= -k\_2 e\_2, \\ \dot{e}\_3 &= -k\_3 e\_3. \end{aligned} \tag{14}$$

Obviously, the eigenvalues (−*k*1, −*k*2, −*k*3) of the Jacobian matrix of the state errors system are negative. Then, the complete anti-synchronization between the driver system (7) and the response system (9) is proved.

In numerical simulation, we assume the initial values of the driver system (7) and the response system (9) to be

$$\begin{aligned} u(0) &= 0.06, \\ v(0) &= 0.01, \\ w(0) &= 0.01, \\ u\_1(0) &= -0.20, \\ v\_1(0) &= -0.09, \\ w\_1(0) &= 0.07. \end{aligned} \tag{15}$$

Then, the initial values of the state errors system (12) are

$$\begin{aligned} \varepsilon\_1(0) &= 0.40, \\ \varepsilon\_2(0) &= -0.08, \\ \varepsilon\_3(0) &= 0.08. \end{aligned} \tag{16}$$

The positive gain constants here are selected as *k*<sup>1</sup> = *k*<sup>2</sup> = *k*<sup>3</sup> = 3. It is obvious in Figure 12 that there exists anti-synchronization of the respective states of the new systems with two closed curve equilibrium (7) and (9). The time history of the synchronization errors *e*1,*e*2,*e*<sup>3</sup> is shown in Figure 13 which plots the anti-synchronization of the driver-response system.

**Figure 12.** Anti-synchronization of the driver-response system: (**a**) *u*, *u*1, (**b**) *v*, *v*1, (**c**) *w*, *w*1, the driver system (**dashed lines**), the response system (**solid lines**).

**Figure 13.** Time history of the anti-synchronization of the state errors system: (**a**) *e*<sup>1</sup> − *t*, (**b**) *e*<sup>2</sup> − *t*, (**c**) *e*<sup>3</sup> − *t*.

## **4. Conclusions**

In this work, we propose and study the following new system:

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

with eyed-shaped equilibrium points which are located on two closed curves passing the same point. In Section 2, some dynamical properties of the proposed system are presented, which were investigated using bifurcation diagram, phase portrait, maximal Lyapunov exponents, and Kaplan-Yorke dimension. Furthermore, the anti-synchronization of systems is obtained by using active control in Section 3. This study will broaden the current knowledge of chaotic systems with infinitely many equilibria.

**Author Contributions:** Both authors contributed equally to this work. Both authors read and approved the final manuscript.

**Funding:** The first author is funded by National Natural Science Foundation of China (Grant No.11672207). The second author is supported by Grant No. MOST 107-2115-M-017-004-MY2 of the Ministry of Science and Technology of the Republic of China.

**Acknowledgments:** The authors wish to express their hearty thanks to the anonymous referees for their valuable suggestions and comments.

**Conflicts of Interest:** The authors declare no conflict of interest.
