1 Introduction

In the past years, a considerable amount of literature has been published on chaotic systems, for instance, Lorenz’s system [1], Rössler’s system [2], Sprott’s system [3], Chen and Ueta’s system [4], Chua’s circuit [5], Linz and Sprott’s system [6], Lü and Chen [7], Pehlivan and Uyaroglu [8], and so on. Since then, chaos theory has become a significant research issue in many chaos-based processes and information systems [9,10,11,12,13,14,15]. The complexity of chaotic systems has been applied in various engineering applications from image encryption [16,17,18], control and synchronization [19,20,21,22,23,24,25], weak signal detection [10] and forecasting water inrush in mines [26] to secure communication [27,28,29], cryptosystem design [30], audio encryption [31], permutation flow-shop scheduling problem [32], parallel distributed processing [33], and chaotic MIMO radar waveform design [34]. In the study of chaos, it is significant to form novel chaotic systems based on existing chaotic attractors. Chaotic systems have extremely complex nonlinear dynamics [35, 36]. The fundamental specification of these systems is high sensitivity to parametric uncertainties and initial states. A smooth autonomous chaotic system can exhibit attractors with different numbers of wings disregarding the number of equilibria [37, 38]. In fact, creation of complex multi-wing or multi-scroll chaotic attractors from three-dimensional autonomous systems has achieved rapid development [39, 40]. In terms of difficulty, formation of a three-dimensional autonomous system with a complex attractor is an important and stimulating task in theory and practical purposes [41]. Due to the above-mentioned properties of chaotic systems, chaos theory is employed in various scientific researches.

It is now well recognized from a variety of investigations that equilibrium points play important roles in theoretical design and dynamical analysis of chaotic systems [42, 43]. An equilibrium point of a dynamical system is the real solution of the differential equation \(\dot{x}=f(x)=0\). The conventional chaotic systems have a countable number of equilibrium points. In recent years, a few strange chaotic systems with uncountable equilibrium points have been introduced [44]. There are three families of chaotic systems with infinite number of equilibria: systems with line equilibria [45,46,47,48,49], systems with open-curve equilibria [50, 51], and systems with closed-curve equilibria [52,53,54]. The considerable properties of systems with infinite number of equilibria are rare and challenging to find. It is recognized that such systems with infinite number of equilibrium points exhibit hidden attractors, which have been studied as an exciting research subject in the recent years [55,56,57,58]. Nevertheless, there is still a necessity to discover new chaotic systems with different closed-curve equilibria [59, 60].

In this paper, a new chaotic system with infinite number of equilibria like a three-leaved clover is proposed. Using a systematic computer search, three dissipative chaotic systems with three-leaved-clover-shaped equilibria are found. To help understand the chaos generation of this system, some dynamical specifications containing phase portraits, bifurcation diagram, Lyapunov exponents, Kaplan–Yorke dimension and Poincaré map are discussed exhaustively. Furthermore, to investigate the applicability of the new chaotic system, an electronic circuit implementation and an RNG design are realized and the achieved random bit sequences are employed for image encryption. Finally, security analysis of image encryption processes has been executed.

This paper is organized as follows: In Sect. 2, the mathematical model of the new chaotic system is proposed. In Sect. 3, some discussion for the chaotic system containing dynamical specifications such as bifurcation diagrams, Lyapunov exponents, Kaplan–Yorke dimension, and Poincaré map is presented. An electronic circuit realization of the new chaotic system is performed in Sect. 4, while its engineering applications comprising the RNG design and image encryption are reported in Sect. 5. Finally, the last section presents conclusions of overall study.

2 Mathematical model of the chaotic system

In the search for chaotic flows with infinite number of equilibrium points, we consider the model of a novel three-dimensional chaotic system with equilibrium points like a three-leaved clover as

$$\begin{aligned} \dot{x}= & {} a_1 z \nonumber \\ \dot{y}= & {} -\,zf_1 (x,y,z) \nonumber \\ \dot{z}= & {} f_2 (x,y), \end{aligned}$$
(1)

where x, y, and z signify the state variables and \(f_1 (x,y,z)\) and \(f_2 (x,y)\) specify the nonlinear functions as:

$$\begin{aligned}&f_1 (x,y,z)=a_2 x+a_3 y+a_4 z+a_5 x^{2}+a_6 y^{2}\nonumber \\&\quad +\,a_7 z^{2}+a_8 xy+a_9 xz+a_{10} yz+a_{11}, \end{aligned}$$
(2)
$$\begin{aligned}&f_2 (x,y)=x^{4}+2x^{2}y^{2}+y^{4}-x^{3}+3xy^{2}, \end{aligned}$$
(3)

where \([a_1 , a_2 , \ldots , a_{11} ]\) are the constant parameters.

The equilibrium of the general model (1) can be found as:

$$\begin{aligned}&z=0, \nonumber \\&z\,f_1 (x,y,z)=0, \nonumber \\&f_2 (x,y)=0, \end{aligned}$$
(4)

where it is concluded that in the plane \(z=0\), the equilibrium points of the general model (1) lay on the surface \(f_2 (x,y)=0\). Then, it is confirmed that the system (1) has infinite number of equilibrium points \(E(x^{*},y^{*},0)\) located on

$$\begin{aligned} x^{*4}+2x^{*2}y^{*2}+y^{*4}-x^{*3}+3x^{*} y^{*2}=0. \end{aligned}$$
(5)

Interestingly, Eq. (5) defines a three-leaved-clover-shaped curve of equilibrium points, as shown in Fig. 1.

Fig. 1
figure 1

Presentation of the three-leaved-clover–shaped equilibrium

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Table 1 Three chaotic flows with equilibrium points like a three-leaved clover
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An exhaustive computer search is performed considering millions of combinations of the parameters \(a_1, \ldots , a_{11}\) and initial conditions, seeking for the chaotic behavior with the largest Lyapunov exponent greater than 0.001. Cases CL1–CL3 in Table 1 are three chaotic systems obtained in this way. In addition to the cases CL1–CL3, dozens of chaotic systems with more complicated structures and inessential terms were found. Figure 2 demonstrates the strange chaotic attractors of the cases CL1–CL3 with infinite number of equilibria like a three-leaved clover.

Since the sum of the Lyapunov exponents of chaotic systems CL1–CL3 is negative, one can conclude that the new chaotic systems CL1–CL3 are dissipative. The equilibrium points, Lyapunov exponents, and Kaplan–Yorke dimensions are reported in Table 1 along with the initial states which are close to the chaotic attractors. As is common for strange chaotic attractors of three-dimensional autonomous systems, the dimensions of the attractors for the cases CL1–CL3 are only slightly greater than 2. Among dissipative cases CL1–CL3, the largest Kaplan–Yorke dimension is 2.1727 for CL1, although no effort is done to tune the parameters for high complexity.

Fig. 2
figure 2

Strange chaotic attractors of cases CL1–CL3

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3 Discussions

In this section, we study further as a simple example the system CL1 due to the fact that it has the biggest value of Kaplan–Yorke dimension (\(D_{\mathrm{KY}}\)). By investigating the effect of CL1’s parameters (b, c and d) of nonlinear terms on system’s behavior, our simulations indicated that CL1 system displays striking dynamics. Firstly, the value of the parameter b is varying in the range from 1.3 to 1.7 in order to show the dynamics of system CL1 by obtaining the bifurcation diagram of the variable z when the trajectories cut the plane \(x = 0\) with \({\text {d}x}/{\text {d}t} < 0\), as the control parameter b is decreased. For this reason, the proposed system CL1 is integrated numerically using the classical fourth-order Runge–Kutta integration algorithm. For each set of parameters used in this work, the time step is always \(\Delta t = 0.002\) and the calculations are performed using variables and parameters in extended precision mode. For each parameter settings, the system is integrated for a sufficiently long time and the transient is discarded.

The bifurcation diagram and the three Lyapunov exponents of the system CL1 are presented in Figs. 3 and 4, respectively. Here, Lyapunov exponents are calculated with the Wolf’s algorithm [61]. As can be seen from Figs. 3 and 4, there is the presence of a classical period-doubling route to chaos when decreasing the value of the parameter b. For \(b \ge 1.476\), system CL1 generates periodical oscillations. For instance, period-1, period-2, period-4, and period-8 oscillations of system CL1 are illustrated in Fig. 5, respectively. For \(b < 1.476\), system CL1 displays more complex behaviors, for example chaos (see the strange attractors of system CL1 in Fig. 2). The Poincaré map of system CL1 in y–z plane, when \(x = 0\) with \({\text {d}x}/{\text {d}t} < 0\) (see Fig. 6), also indicates the properties of chaos. Finally, it is noted that the system is unbounded for \(b < 1.216\).

Similarly, we have changed the value of the parameter c and d to discover the dynamics of system CL1. The bifurcation diagrams of system CL1 for \(c\in [1.7, 3.1]\) and \(d\in [2.5, 3.2]\) are reported in Fig. 7. As shown in Fig. 7, the presence of a classical period-doubling route to chaos is observed when decreasing the value of the parameter c and increasing the value of the parameter d. Furthermore, the corresponding spectrums of the three Lyapunov exponents by varying the parameter c and the parameter d are shown in Fig. 8. It can be seen that the bifurcation diagrams well coincide with the spectrum of the Lyapunov exponents.

Fig. 3
figure 3

Bifurcation diagram of z versus b of the system CL1, for \(a = 0.29\), \(c = 2\) and \(d = 3\)

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Fig. 4
figure 4

Lyapunov exponents of the system CL by varying the parameter b, for \(a = 0.29\), \(c = 2\), and \(d = 3\)

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Fig. 5
figure 5

Four views of periodic behaviors of the system CL1: a period-1 oscillation (\(b = 1.65\)), b period-2 oscillation (\(b = 1.55\)), c period-4 oscillation (\(b = 1.49\)), d period-8 oscillation (\(b = 1.48\)), for \(a = 0.29\), \(c = 2\) and \(d = 3\)

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4 Circuit realization

The classical approach for the verification of the feasibility of theoretical chaotic models is the physical realization through electronic circuits [62,63,64,65,66]. Furthermore, the circuital realization of chaotic systems has been applied in numerous engineering applications, for example in secure communications [67, 68], liquid mixing [69], robotics [70], image encryption process [71], audio encryption scheme [31], target detection [72], or random signal generation [73, 74]. For this reason, analog and digital approaches have been applied to realize chaotic oscillators by using different kinds of electronic devices such as common off-the-shelf electronic components [75, 76], integrated circuit technology [77, 78], microcontroller [79], or field-programmable gate array (FPGA) [80,81,82].

Therefore, in this section, we will confirm the feasibility of one of the proposed systems CL1–CL3 by discussing its circuital realization by using the general operational amplifier-based approach. In more details, the system which has been chosen in this work is the system CL1. The three state variables (x, y, z) of the system CL1 have been rescaled as \(X = 10x, Y = 5y\), and \(Z = 10z\), in order to avoid the limitations of the components of electronic circuit. Therefore, the system CL1 is transformed into the following equivalent system:

$$\begin{aligned} \dot{{X}}= & {} 0.29{Z} \nonumber \\ \dot{Y}= & {} -\,{Z}\left( {\frac{1.4}{10}Y +\frac{{Z}^{2}}{200}+\frac{{ YZ}}{100}} \right) \nonumber \\ \dot{{Z}}= & {} \frac{{X}^{4}}{1000}+\frac{2{X}^{2}Y^{2}}{250} +\frac{Y^{4}}{62.5}-\frac{{X}^{3}}{100}+\frac{3{ XY}^{2}}{25} \end{aligned}$$
(6)
Fig. 6
figure 6

Poincaré map of system CL1 in the y–z plane, for \(a = 0.29\), \(b = 1.4\), \(c = 2\) and \(d = 3\)

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Figure 9 shows the schematic of the circuit for realizing the system (6). As shown in this figure, the circuit includes nine resistors, three capacitors, four operational amplifiers (TL081), and eleven analog multipliers (AD633). In this point, we should mention that the most common nonlinearities in chaotic oscillators are polynomials or the products of two state variables, as it happened in system CL1. For this reason, nonlinear circuits designed from a mathematical chaotic system usually require analog multipliers, such as AD633. In fact, despite the accuracy of analog devices, the nonideal effects are always present. The nonidealities of the analog multipliers may lead to further nonlinear terms, which will be considered as parasitic effects [83]. However, in this work, the parasitic effects of the analog multipliers have been neglected because according to our study, they do not affect significantly circuit’s behavior in regard to the expected behavior from the numerical simulation.

Fig. 7
figure 7

Bifurcation diagrams of a z versus c and b z versus d, of the system CL1

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Fig. 8
figure 8

Lyapunov exponents of the system CL1, by varying a the parameter c and b the parameter d

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By applying Kirchhoff’s circuit laws into the designed circuit, we get the following circuital equation:

$$\begin{aligned} \dot{X}= & {} \frac{1}{RC}\left( {\frac{R}{R_1 }Z} \right) \nonumber \\ \dot{Y}= & {} \frac{1}{RC}\left( -\frac{R}{R_2 10V}{} { YZ}-\frac{R}{R_3 100V}Z^{3}\right. \nonumber \\&\left. -\,\frac{R}{R_4 100V^{2}}{} { YZ}^{2} \right) \nonumber \\ \dot{Z}= & {} \frac{1}{RC}\left( \frac{R}{R_5 1000V^{3}}X^{4}+\frac{R}{R_6 1000V^{3}}X^{2}Y^{2}\right. \nonumber \\&\left. +\,\frac{R}{R_7 1000V^{3}}Y^{4}-\frac{R}{R_8 100V^{2}}X^{3}\right. \nonumber \\&\left. +\,\frac{R}{R_9 100V^{2}}{} { XY}^{2} \right) \end{aligned}$$
(7)

In system (7), X, Y, and Z correspond to the voltages on the integrators (U2–U4), respectively, while the power supply is \(\pm \,15\hbox {V}_{\mathrm{DC}}\). System (7) is normalized by using \(\tau = t/RC\). It can thus be suggested that system (7) is equivalent to system (6), with \(\frac{R}{R_1}=0.29\), \(\frac{R}{R_2}=1.4\), \(\frac{R}{R_3}=\frac{1}{2}\), \(\frac{R}{R_4}=1\), \(\frac{R}{R_5}=1\), \(\frac{R}{R_6}=8\), \(\frac{R}{R_7}=16\), \(\frac{R}{R_8}=1\), and \(\frac{R}{R_9}=12\). So, the values of circuit components are: \(R = 10\) k\(\Omega \), \(R_{1} = 66.666\) k\(\Omega \), \(R_{2} = 7.143\) k\(\Omega \), \(R_{3}=R_{4}=R_{5}=R_{8} = 1\) k\(\Omega \), \(R_{6} = 12.5\) k\(\Omega \), \(R_{7} = 0.625\) k\(\Omega \), \(R_{9} = 0.833\,\hbox { k}\Omega \), and \(C_{1}=C_{2}=C_{3}=C = 1\,\hbox { nF}\). The designed circuit has been implemented in Multisim, and PSpice results are reported in Fig. 10. It is easy to see the agreement between the circuit’s simulation results (Fig. 10) and numerical results (Fig. 2).

Fig. 9
figure 9

Schematic of the circuit including nine resistors, three capacitors, four operational amplifiers, and eleven analog multipliers. The power supplies of all operational amplifiers and analog multipliers are \(\pm \,15\hbox {V}_{\mathrm{DC}}\)

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Fig. 10
figure 10

PSpice chaotic attractors of the designed circuit in a \(X\hbox {-}Y\) plane, b \(X\hbox {-}Z\) plane, and c \(Y\hbox {-}Z\) plane

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5 Engineering application (RNG design and image encryption)

In this section, an RNG algorithm is designed using the developed chaotic systems, and the random numbers obtained from the RNG algorithm are applied to NIST-800-22 [84] randomness tests. Then, an image encryption algorithm with random bit sequences generated by the RNG algorithm is presented, and image encryption application is performed. Tests have been conducted to demonstrate the level of security of the image encryption process performed.

Table 2 NIST-800-22 test results of RNGs based on CL systems
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5.1 Design of RNG algorithm and NIST tests results

The random bit sequences to be used in the encryption process are very important for the security of the encryption. Chaotic systems are widely used in random number generation because of their rich dynamics. The design steps of the RNG algorithm are given below.

  • Step 1 Enter initial conditions and system parameters for each chaotic system

    CL1 \(\Rightarrow \) (− 0.54, − 0.69, 0.37), CL2 \(\Rightarrow \) (− 0.29, 0.37, 0.2), CL3 \(\Rightarrow \) (0.19, 0.29, − 0.34);

  • Step 2 Determination of sampling interval (\(\Delta h = 0.5\));

  • Step 3 Analysis of chaotic system using RK4 algorithm;

  • Step 4 Using the sampling step interval, obtain the float value from the analyzed system;

  • Step 5 Convert 32-bit binary value to float value;

  • Step 6 Selecting the LSB-10 bit from the 32-bit number array and adding it to the random number sequence;

  • Step 7 For each phase until a 1 million-bit number sequence is obtained, repeat range step 3–6;

  • Step 8 Bitwise XOR operation of 1 M. bit sequences generated for each phase;

  • Step 9 Applying NIST tests to the obtained 1 M. random bit sequences;

The RNG algorithm described above is applied separately for each chaotic system, and NIST tests are applied by obtaining different random bit sequences from each chaotic system. NIST 800-22 tests are used to determine the randomness degrees of random bit sequences. In NIST 800-22 tests, random number sequences are subjected to 16 different tests. To have sufficient randomness of the sequence of bits, it must pass all tests. Table 2 shows the NIST 800-22 test results of the bit sequences obtained from the XOR operation of the random bit sequences obtained from the x, y, and z phases of three different chaotic systems. According to the test results, the bit sequences obtained from each chaotic system have passed all tests.

Fig. 11
figure 11

a Original image b CL1 encrypted image c CL2 encrypted image d CL3 encrypted image e decrypted image

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5.2 Image encryption algorithm and its applications

After the RNG algorithm design, the encryption process is performed using the obtained random bit sequences. The image pixel values to be encrypted in the encryption algorithm are converted into a binary bit sequence. With these values, bit sequences obtained from the developed RNG algorithm are subjected to XOR processing and encryption is performed. Three different encryption processes have been performed with random bit sequences obtained from each chaotic system. In Fig. 11a, 256 * 256 size pepper.jpg original image is shown. Fig. 11b–d shows the results of cryptography performed with random bit sequences from CL1, CL2, and CL3 chaotic systems, respectively. Figure 11e shows the decrypted image resulting from decryption. Comparing the results in Fig. 11, it can be said that the encryption and decryption processes have been successfully performed for all the processes.

5.3 Security analysis results

In this section, the security analysis results of the encryption processes are presented. Security analysis reveals the quality of the encryption process. Histogram, correlation, entropy, and linear-differential attack (NPCR–UACI) analyses of cryptographic operations are performed in security analysis. Firstly, histogram distributions of cryptographic processes are examined. Figure 12 shows histogram distribution graphs of original, encrypted, and decrypted pictures. Following the encryption process, the number of pixel values in the image is almost equal, indicating that a good encryption is being performed. When the results in Fig. 12 are examined, it is seen that the encryption results of four different systems have a good histogram distribution.

Fig. 12
figure 12

The results of histogram analysis, a original image, b CL1 encrypted image, c CL2 encrypted image, d CL3 encrypted image, e decrypted image

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Correlation coefficient analysis [85] examines the relationship between random variables in the encryption process. This relationship should not be linear. The correlation coefficient is calculated using the following equation. In the equation, x and y represent the values of two adjacent pixels in the image, and N is the number of selected pixel pairs.

$$\begin{aligned}&E(x)=\frac{1}{N}{\mathop {\sum }\limits _{i=1}^N} {x_i } , \nonumber \\&D(x)=\frac{1}{N}{\mathop {\sum }\limits _{i=1}^N} {(x_i -E(x))^{2},} \nonumber \\&\hbox {cov}(x,y)=\frac{1}{N}\left( {{\mathop {\sum }\limits _{i=1}^N} {(x_i -E(x))(y_i -E(y))} } \right) , \nonumber \\&r_{xy} =\frac{\hbox {cov}(x,y)}{\sqrt{D(x)D(y)}}. \end{aligned}$$
(8)

The correlation graphs of the original and decrypted image are shown in Fig. 13a and f, respectively. These two graphs show that they are the same. Figure 13 shows the correlation distribution graphs of the encryption processes. Table 3 shows the correlation coefficient values of the encryption processes. When the graphs and the correlation coefficients are evaluated together, it is seen that the encryption processes have a good correlation distribution.

Fig. 13
figure 13

The results of correlation analysis, a original image, b CL1 encrypted image, c CL2 encrypted image, d CL3 encrypted image, e decrypted image

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The complexity of the encrypted data is another criterion that gives information about the quality of the encryption. In information entropy analysis, the complexity of the encrypted data is determined. The formula used in information entropy analysis is given below. The optimal information entropy value is accepted as 8 [86]. The closer the information is to the entropy value of 8, the better the quality of the encryption. In Table 3, information entropy values of all encryption processes are given. It seems that all of these values are very close to 8.

$$\begin{aligned} \mathrm{Shan}En(x)=-\,{\mathop {\sum }\limits _{i=1}^N} {(p_i (x))^{2}(\log _2 p_i (x))^{2}} . \end{aligned}$$
(9)

Number of pixels change rate (NPCR) and unified average changing intensity (UACI) [87] are cryptanalysis methods that are used to detect the resilience of the encryption process to differential attack attacks. The relation between the original and encrypted image is determined by the NPCR method. The equations used for NPCR calculation are given below. The NPCR optimal value is determined to be \(\hbox {NPCR}_{\mathrm{opt}} = 99.61\%\) [88]. Table 3 shows that the NPCR values of the encryption processes performed in the study are very close to the optimum value.

$$\begin{aligned}&\mathrm{NPCR}(A,B)=\left( {\mathop {\sum }\limits _{i,j}} {D(i,j)/N} \right) \times 100\% \nonumber \\&D(i,j)=\left\{ {\begin{array}{lll} 1 &{}\quad \hbox {if} &{}\quad A(i,j)\ne B(i,j) \\ 0 &{}\quad \hbox {if} &{}\quad A(i,j)=B(i,j) \\ \end{array}} \right. \end{aligned}$$
(10)

The equation used to calculate the UACI value, which expresses the average intensity between two images, is given below. UACI optimal value is determined as \(\hbox {UACI}_{\mathrm{opt}} = 33.46\%\) [88]. When the UACI values of the encryption processes in Table 3 are examined, it is seen that these values are very close to the optimal values.

$$\begin{aligned}&\mathrm{UACI}(A,B)\nonumber \\&\quad =\frac{1}{N}\left( {{\mathop {\sum }\limits _{i,j}} {\left( {\left| {A(i,j)-B(i,j)} \right| } \right) /2^{L}-1} } \right) \times 100\%\nonumber \\ \end{aligned}$$
(11)

Remark 1

It is very important to show that the generated chaotic systems can be applied to engineering problems as well as the production, analysis, and examination of dynamic behaviors of chaotic systems. In particular, chaotic systems are widely used in the design of random number generators and data security. However, there are many studies that only refer such applications [16,17,18, 46, 73, 84, 85, 88]. In this study, RNG and image encryption applications were performed in this section to demonstrate the usefulness of different chaotic systems (CL1, CL2 and CL3) in engineering applications, and good enough results were obtained. Consequently, it was shown that the proposed chaotic systems can be employed in engineering applications.

Table 3 The security analysis results of encryption operations
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6 Conclusion

In this paper, a new class of three-dimensional autonomous chaotic systems with equilibrium points like a three-leaved clover is presented. Dynamical specifications of these chaotic systems are demonstrated by the bifurcation diagrams, Lyapunov exponents, Kaplan-Yorke dimension, and Poincaré map. These chaotic flows belong to the recently proposed class of chaotic systems with hidden attractors. We hope that this paper can stimulate for the further study on chaotic systems with infinite number of equilibrium points like a three-leaved clover. A new RNG algorithm is designed using chaotic systems developed in the study, and NIST 800-22 randomness tests are applied to random bit sequences generated by the RNG algorithm. It has been found that all random bit sequences generated have high randomness and all NIST tests pass. Image encryption with random bit sequences is performed, and security analysis of the encryption process is performed. When security analysis results are evaluated, it is concluded that secure encryption is performed with bit sequences with high randomness. As a future work, the real circuit of the proposed system, due to its rich dynamical behavior, will be built for using it in a real “chaotic” application (encryption scheme). Moreover, it should be noted that the powerful robust control approaches such as robust tracking and model following [89,90,91,92,93,94,95], robust PID feedback [96, 97], disturbance-observer-based robust control [98,99,100,101], robust linear matrix inequality (LMI) [102,103,104,105,106], robust \(H_{\infty }\) control [107,108,109], sliding mode control [110,111,112,113,114,115,116,117,118], adaptive robust fuzzy control [119,120,121,122,123,124], terminal sliding mode control [125,126,127,128,129,130,131], and robust backstepping control [132,133,134] can be employed for the control or synchronization of this class of chaotic systems in the future researches.