Chaos synchronization and control theory have a great interest due to their potential applications in several fields during the recent decades.^[1–5] Different synchronization types have been introduced and many theoretical results have been achieved, such as complete synchronization,^[6–8] anti-synchronization,^[9–11] lag synchronization,^[12,13] projective synchronization,^[13,14] function projective synchronization,^[15] etc. Most of the pervious types of synchronization concentrate on synchronization between one drive and one response systems. There exist many techniques of control to achieve synchronization such as sliding mode control, random phase control, adaptive passive control, and others.^[16–24]

Synchronization of systems with more than one drive and response systems is very interesting and important, because the transmitted signals may own stronger anti-translated and anti-attack ability. Complexity and the formation of the drive signals are also peppy factors to include communication security. Researchers studied complex schemes to transmit the information signals. Runzi et al. investigated combination synchronization of two drive and one response systems,^[25,26] while the synchronization of two drive and two response systems (combination–combination synchronization) is studied by Sun et al.^[27] Mahmoud et al. introduced the generalization of combination–combination synchronization of chaotic systems.^[28,29] A new kind of compound synchronization (synchronization between three drive systems and one response system) has been illustrated in Ref. [30]. Zhang and Deng studied the double compound synchronization, which is the synchronization of four drive and two response systems.^[31] The compound–combination synchronization (synchronization of three drive and two response systems) was presented in Ref. [32].

In this paper, we introduce a new type of synchronization which is a generalization of many pervious types of synchronization and more complexity. This type is called double compound combination synchronization (DCCS) among eight chaotic systems (four drive and other four response chaotic or hyperchaotic systems). In the last years, a new circuit element, namely memristor which is basically a fourth class of electrical circuit, joining the inductor, the capacitor, and the resistor. Memristor produced the circuit systems which have chaotic and hyperchaotic property. In Ref. [33], the authors replaced Chua’s diodes with memristors which proposed several nonlinear oscillators from Chua’s oscillators. Wen et al. regarded the fuzzy modeling and synchronization of memristor-based Lorenz circuits with memristor-based Chua’s circuits.^[34] Memristor was postulated as the missing fourth passive circuit element.^[35] A great breakthrough that a passive two terminal physical implementation was immediately related to memristor theory was obtained by Hewlett–Packard Labs.^[36] This new circuit element can be useful for low-power computation and storage to store information and data.^[37] Memristor, also, can be used to implement programmable analog circuits, leveraging memristor’s fine-resolution programmable resistance without causing perturbations due to parasitic components,^[38] and soon.

This paper is organized as follows: Section 2, contains the design of DCCS scheme among eight n-dimensional chaotic systems. The definition of this new type of synchronization is given. In Section 3, the DCCS for eight identical memristor-based Chua oscillators as an example, is investigated by using the Lyapunov stability theory and nonlinear feedback control. The analytical expressions of the control functions are derived. In Section 4, the numerical treatments of testing the analytical formula of the control functions to achieve the DCCS for our example are illustrated by using fourth order Runge–Kutta method. The simulation of the chaotic memristor-based Chua oscillator is contracted by using MATLAB/Simulink. We gain a good agreement between numerical treatments and simulation results for our proposed synchronization. The conclusion is finally stated in the last section.

We will present the technique which is used to achieve the DCCS of four drive and four response n-dimensional chaotic systems. The first scaling drive system is given as follows: 1 the second scaling drive system is written as 2 the first base drive system is described as 3 the second base drive system is drown as: 4 and the first response system is written as: 5 the second response system is defined as: 6 the third response system is given by: 7 the fourth response system is: 8 where are the state variables of systems (1)–(8), are eight diagonal matrices of continuous functions and are four control functions of the response systems (5)–(8). The topological structure diagram of the drive systems and the response systems is shown in Fig. 1. Now, we will state the new definition of the DCCS as

Fig. 1.

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	Fig. 1. (color online) The topological structure diagram of the drive systems (1)–(4) and the response systems (5)–(8).

Let A_i = diag (a_i1, a_i2, …, a_in), B_i = diag (b_i1, b_i2, …, b_in), i = 1, 2, 3, 4, therefore, the error system can be described as: 10 From systems (10) and (1)–(4), the error dynamical system can be given as: 11 Denote 12 We will introduce a theorem for the analytical expression of the control functions U = diag (U₁, U₂, …, U_n).

In this section, we apply the synchronization scheme of Section 2 to study the DCCS for eight identical memristor-based Chua oscillators as an example. The first scaling drive memristor-based Chua oscillator is given by 21 where c_1i, (i = 1, 2, …, 5) are constant parameters. We calculated the Lyapunov exponents of system (21) and they are: λ₁ = 0.2522, λ₂ = − 0.0296, λ₃ = − 0.0419, λ₄ = − 7.5153, for c₁₁ = 15.6, c₁₂ = 3, c₁₃ = 21, c₁₄ = 15, c₁₅ = 0.5 and x₁₀ = diag (0.2, 0.2, 0.2, 0.2). It is clear that one of them is positive which means that its solution is chaotic as shown in Fig. 2.

Fig. 2.

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	Fig. 2. (color online) Chaotic attractor of system (21) when c11 = 15.6, c12 = 3, c13 = 21, c14 = 15, c15 = 0.5 and x10 = diag (0.2, 0.2, 0.2, 0.2): (a) x14, x11, x13 space, (b) x11, x12 space, (c) x13, x14 space, (d) x12, x14 space.

We designed circuit implementation for system (21) as shown in Fig. 3. The circuit equations in terms of the circuit parameters are: 22 The circuit consists of four channels to realize the integration, addition, and subtraction of the state variables x₁₁, x₁₂, x₁₃, and x₁₄, respectively. These variables are corresponding to the state voltages of the four channels, respectively. For the same parameters values of system (21) in Fig. 2, the corresponding values of circuit elements are R₁ = 16.67 kΩ, R₂ = 2.18 kΩ, R₃ = 1.59 kΩ, R₄ = 40 kΩ, R₅ = 3.33 kΩ, R₆ = 100 kΩ, R₇ = 100 kΩ, R = 10 kΩ, and the capacitors are C₁ = 30 μF, C₂ = 25 μF, C₃ = 20 μF, and C₄ = 10 μF. The circuit implementation of system (22) is drown in Fig. 3. The corresponding simulation observations of the chaotic memristor-based Chua oscillator (21) appear in Fig. 4 which agree with numerical simulation in Fig. 2.

Fig. 3.

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	Fig. 3. Circuit implementation of system (21).

Fig. 4.

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	Fig. 4. (color online) Simulation observations of system (21) for the same parameters in Fig. 2: (a) x14, x11, x13 space; (b) x11, x12 space; (c) x13, x14 space; (d) x12, x14 space.

The second scaling drive memristor-based Chua oscillator is given by 23 and the two base drive memristor-based Chua oscillators are shown below 24 25 where c_ij, (i = 2, 3, 4; j = 1, 2, …, 5) are constant parameters. The four response memristor-based Chua oscillators are given, respectively, as 26 27 28 29 where d_kj, (k = 1, 2, …, 4; j = 1, 2, …, 5) are constant parameters and u_kl, (k = 1, 2, …, 4; l = 1, 2, …, 4) are the control functions.

According to Theorem 1, we obtain the control functions as follows: 30

From Eq. (15) the Lyapunov function for this example is: , which is positive and its derivative is negative using Eq. (30).

In this section, numerical treatments are illustrated to state the effectiveness of the suggested scheme of DCCS of eight identical memristor-based Chua oscillators. If we take, c_i1 = d_i1 = 15.6, c_i2 = d_i2 = 3, c_i3 = d_i3 = 21, c_i4 = d_i4 = 15, c_i5 = d_i5 = 0.5, (i = 1, 2, 3, 4), and the initial conditions for the four drive systems (21)–(25) and the four response systems (26)–(29), are respectively, The corresponding numerical results are shown in Figs. 5 and 6. Figure 5 presents the state variables of the four drive systems and the four response systems. The synchronization errors converge to zero as shown in Fig. 6.

Fig. 5.

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Fig. 5. (color online) State variables synchronization for the four drive systems (21)–(25) (solid curves) and the four response system (26)–(29) (dashed curves): (a) (x11 + 2x21)(x31 + 5x41) and (2y11 + 4y21 + y31 + 2y41) versus t; (b) (−x12 + 2x22)(x32 + x42) and (3y12 + y22 − y32 + 3y42) versus t; (c) (2x13 + 2x23) (−x33 − x43) and (2y13 + 3y23 + 2y33 − y43) versus t; (d) (x14 + 2x24)(2x34 + x44) and (2y14 − 2y24 + 3y34 + y44) versus t.

Fig. 6.

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	Fig. 6. (color online) Synchronization errors for the four drive systems (21)–(25) and the four response systems (26)–(29): (a) (t, e1) diagram; (b) (t, e2) diagram; (c) (t, e3) diagram; (d) (t, e4) diagram.

Synchronization results also have been calculated from MATLAB/Simulink and these results are shown in Figs. 7 and 8. Figure 7 shows the same results of Fig. 5, while figure 8 explains the relationship between the drive and response systems which clearly appears as the form of a line y = x, this means that there is an agreement between the results of Figs. 8 and 6. The synchronization setup time^[39] with different values of k₁, k₂, k₃, and k₄ are calculated as shown in Fig. 9. According to Fig. 9, the synchronization setup time (e) approaches to zero fast for reasonable large values of k_i.

Fig. 7.

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Fig. 7. (color online) Synchronization results from simulation observation of four drive (21)–(25) and four response (26)–(29) memristor-based Chua chaotic systems: (a) (x11 + 2x21)(x31 + 5x41) and (2y11 + 4y21 + y31 + 2y41) versus t; (b) (−x12 + 2x22)(x32 + x42) and (3y12 + y22 − y32 + 3y42) versus t; (c) (2x13 + 2x23) (−x33 − x43) and (2y13 + 3y23 + 2y33 − y43) versus t; (d) (x14 + 2x24)(2x34 + x44) and (2y14 − 2y24 + 3y34 + y44) versus t.

Fig. 8.

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Fig. 8. (color online) Synchronization results from simulation observation of drive (21)–(25) and response (26)–(29) memristor-based Chua chaotic systems: (a) (x11 + 2x21)(x31 + 5x41), (2y11 + 4y21 + y31 + 2y41) space; (b) (−x12 + 2x22)(x32 + x42), (3y12 + y22 − y32 + 3y42) space; (c) (2x13 + 2x23) (− x33 − x43), (2y13 + 3y23 + 2y33 − y43) space; (d) (x14 + 2x24)(2x34 + x44), (2y14 − 2y24 + 3y34 + y44) space.

Fig. 9.

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	Fig. 9. (color online) Synchronization setup time with different parameter: (a) k1, (b) k2, (c) k3, (d) k4.

The object of this paper is to propose a new type of synchronization among eight n-dimensional chaotic systems. The proposed scheme is used to achieve the DCCS between four drive and four response n-dimensional chaotic systems. Using the Lyapunov stability theory and nonlinear feedback control, analytical expressions are obtained for achieving DCCS. As an example, we consider the eight identical memristor-based Chua chaotic systems. Numerical results have been calculated to illustrate the validity of our scheme. Based on complexity of DCCS, our results make more secure and interesting to transmit and receive signals in applications of communications. The numerical results of DCCS are shown in Figs. 5 and 6, and agree with the MATLAB/Simulink results as shown in Figs. 7 and 8.