Combination-Combination Synchronization of Four Nonlinear Complex Chaotic Systems

Abstract

This paper investigates the combination-combination synchronization of four nonlinear complex chaotic systems. Based on the Lyapunov stability theory, corresponding controllers to achieve combination-combination synchronization among four different nonlinear complex chaotic systems are derived. The special cases, such as combination synchronization and projective synchronization, are studied as well. Numerical simulations are given to illustrate the theoretical analysis.

1. Introduction

In 1982, Fowler et al. [1] generalized the real Lorenz model to a complex Lorenz model, which can be used to describe and simulate the physics of a detuned laser and the thermal convection of liquid flows [2, 3]. After that, many new chaotic and hyperchaotic complex systems have been reported and intensively studied, including the complex Van der Pol oscillators [4], the complex Chen and complex Lü systems [5], complex detuned laser system [6], complex hyperchaotic Lorenz system [7], complex modified hyperchaotic Lü system [8], and a novel hyperchaotic complex-variable system [9] which generates 2-, 3-, and 4-scroll attractors.

Since Pecora and Carroll [10] first proposed the drive-response concept for constructing synchronization of coupled chaotic systems, synchronization in chaotic systems has been extensively investigated due to their potential applications in the fields of secure communications; optical, chemical, physical, and biological systems; neural networks; and so forth [11–13]. When applying the complex systems in communications, the complex variables will double the number of variables and can increase the content and security of the transmitted information. Based on the Lyapunov stability theory, linear feedback controller was derived to achieve hybrid projective synchronization in a chaotic complex nonlinear system [14]. The authors [15] achieved adaptive antisynchronization of a class of chaotic complex nonlinear systems described by a united mathematical expression with fully uncertain parameters. In [16], the author investigated the modified projective phase synchronization of chaotic complex nonlinear systems. Based on the passive theory, the authors studied the projective synchronization of hyperchaotic complex nonlinear systems and its application in secure communications [17]. In [18], the authors achieved fast synchronization of a novel hyperchaotic complex system based on finite-time stability theory.

However, most of the existing synchronization schemes are based on the usual drive-response synchronization mode, which has one drive system and one response system. In [19], Luo et al. proposed the combination synchronization scheme, which has two drive systems and one response system. Zhou et al. investigated combination synchronization of three nonlinear complex hyperchaotic systems in [20]. Sun et al. [21] extended the combination synchronization scheme to the combination-combination synchronization scheme, where synchronization is achieved between two drive systems and two response systems. This synchronization scheme has advantages over the other synchronization schemes, such that it can provide greater security in secure communication.

For the nonlinear complex chaotic or hyperchaotic systems, there are no work on combination-combination synchronization for them. This paper aims to study the combination-combination synchronization of four nonlinear complex chaotic systems. The rest of this paper is organized as follows. Section 2 introduces the scheme of combination-combination synchronization. In Section 3, we investigate combination-combination synchronization of four complex nonlinear chaotic systems. Numerical simulations are conducted in Section 4. Finally, conclusions are given in Section 5.

2. The Scheme of Combination-Combination Synchronization

In the scheme of combination-combination synchronization, there are four nonlinear dynamical systems, two drive systems, and two response system.

The two drive systems are, respectively, given by 𝑥̇₁=𝑓₁𝑥₁,𝑥̇₂=𝑓₂𝑥₂. (1)(2) The two response systems are, respectively, described by 𝑦̇₁=𝑔₁𝑦₁𝑦̇+𝜑,₂=𝑔₂𝑦₂+𝜑^∗, (3)(4) where 𝑥₁=(𝑥₁₁,𝑥₁₂,…,𝑥_1𝑛)^𝑇, 𝑥₂=(𝑥₂₁,𝑥₂₂,…,𝑥_2𝑛)^𝑇, 𝑦₁=(𝑦₁₁,𝑦₁₂,…,𝑦_1𝑛)^𝑇, and 𝑦₂=(𝑦₂₁,𝑦₂₂,…,𝑦_2𝑛)^𝑇 are the state vectors of the systems (1), (2), (3), and (4), respectively; 𝑓₁(⋅),𝑓₂(⋅),𝑔₁(⋅),𝑔₂(⋅)∶𝑅^𝑛→𝑅^𝑛 are four continuous vector functions and 𝜑,𝜑^∗∶𝑅^𝑛×𝑅^𝑛×𝑅^𝑛×𝑅^𝑛→𝑅^𝑛 are two controller vectors which will be designed.

Definition 1 (see [21]). If there exist four constant matrices 𝐴, 𝐵, 𝐶, and 𝐷∈𝑅^𝑛 and 𝐶≠0 or 𝐷≠0 such that lim_𝑡→+∞𝐴𝑥₁+𝐵𝑥₂−𝐶𝑦₁−𝐷𝑦₂=0, (5) the drive systems (1) and (2) are realized combination-combination synchronization with the response systems (3) and (4), where ‖⋅‖ represents the matrix norm.

Remark 2. The combination-combination synchronization can be reduced to combination synchronization, projective synchronization, and even control problem, if we choose specific values of 𝐴, 𝐵, 𝐶, and 𝐷.

3. Combination-Combination Synchronization of Four Nonlinear Complex Chaotic Systems

In this section, we investigate the combination-combination synchronization of four nonlinear complex chaotic systems.

The first drive system [22] is given by 𝑥̇₁₁=𝛼₁𝑥₁₂−𝑥₁₁+𝑥₁₂𝑥₁₃,𝑥̇₁₂=𝛾₁𝑥₁₁−𝑥₁₂−𝑥₁₁𝑥₁₃,𝑥̇₁₃=−𝛽₁𝑥₁₃+12𝑥₁₁𝑥₁₂+𝑥₁₁𝑥₁₂, (6) and the second drive system [23] is described as follows: 𝑥̇₂₁=𝑎₁𝑥₂₁+𝑏₁𝑥₂₂𝑥₂₃,𝑥̇₂₂=𝑎₂𝑥₂₂+𝑏₂𝑥₂₁𝑥₂₃,𝑥̇₂₃=𝑎₃𝑥₂₃+𝑏₃2𝑥₂₁𝑥₂₂+𝑥₂₁𝑥₂₂. (7) The first response system [6] takes the following form: 𝑦̇₁₁=𝜎₃𝑦₁₂−𝜎₃1−𝑖𝛿₃𝑦₁₁+𝜑₁+𝑖𝜑₂,𝑦̇₁₂=𝛼₃−𝑦₁₃𝑦₁₁−1+𝑖𝛿₃𝑦₁₂+𝜑₃+𝑖𝜑₄,𝑦̇₁₃=−𝛽₃𝑦₁₃+12𝑦₁₁𝑦₁₂+𝑦₁₁𝑦₁₂+𝜑₅, (8) and the second response [9] is given by 𝑦̇₂₁=𝑦₂₂−𝛼₄𝑦₂₁+𝛽₄𝑦₂₂𝑦₂₃+𝜑^∗₁+𝑖𝜑^∗₂,𝑦̇₂₂=𝛾₄𝑦₂₂−𝑦₂₁𝑦₂₃+𝑦₂₃+𝜑^∗₃+𝑖𝜑^∗₄,𝑦̇₂₃=𝛿₄2𝑦₂₁𝑦₂₂+𝑦₂₁𝑦₂₂−𝜎₄𝑦₂₃+𝜑^∗₅, (9) where 𝛼₁, 𝛽₁, 𝛾₁, 𝑎₁, 𝑎₂, 𝑎₃, 𝑏₁, 𝑏₂, 𝑏₃, 𝛼₃, 𝛽₃, 𝛿₃, 𝜎₃, 𝛼₄, 𝛽₄, 𝛾₄, 𝜎₄, and 𝛿₄ are system parameters; 𝑥₁₁=𝑢₁+𝑖𝑢₂, 𝑥₁₂=𝑢₃+𝑖𝑢₄, 𝑥₂₁=𝑣₁+𝑖𝑣₂, 𝑥₂₂=𝑣₃+𝑖𝑣₄, 𝑦₁₁=𝑤₁+𝑖𝑤₂, 𝑦₁₂=𝑤₃+𝑖𝑤₄, 𝑦₂₁=𝜇₁+𝑖𝜇₂, and 𝑦₂₂=𝜇₃+𝑖𝜇₄ are complex variables; 𝑖=√−1; and 𝑢_𝑖, 𝑣_𝑖, 𝑤_𝑖, 𝜇_𝑖 (𝑖=1,2,3,4), 𝑥₁₃=𝑢₅, 𝑥₂₃=𝑣₅, 𝑦₁₃=𝑤₅, and 𝑦₂₃=𝜇₅ are real variables. The overbar represents complex conjugate function. 𝜑_𝑖 and 𝜑^∗_𝑖 (𝑖=1,2,3,4,5) are real controllers to be determined. Their chaotic attractors are illustrated in Figures 1, 2, 3, and 4, respectively.

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Figure 1 

Chaotic attractor for system (6). (a)–(c) Projections in 3D space; (d)–(f) projections in 2D plane.

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Figure 2 

Chaotic attractor for system (7). (a)–(c) Projections in 3D space; (d)–(f) projections in 2D plane.

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Figure 3 

Chaotic attractor for system (8). (a)–(c) Projections in 3D space; (d)–(f) projections in 2D plane.

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Figure 4 

Chaotic attractor for system (9). (a)–(c) Projections in 3D space; (d)–(f) projections in 2D plane.

For the convenience of our discussions, we assume 𝐴=diag(𝑘₁,𝑘₂,𝑘₃), 𝐵=diag(𝑙₁,𝑙₂,𝑙₃), 𝐶=diag(𝑚₁,𝑚₂,𝑚₃), and 𝐷=diag(𝑛₁,𝑛₂,𝑛₃) in our synchronization scheme.

We define error states between the drive systems (6) and (7) and the response systems (8) and (9) as 𝑒₁+𝑖𝑒₂=𝑘₁𝑥₁₁+𝑙₁𝑥₂₁−𝑚₁𝑦₁₁−𝑛₁𝑦₂₁,𝑒₃+𝑖𝑒₄=𝑘₂𝑥₁₂+𝑙₂𝑥₂₂−𝑚₂𝑦₁₂−𝑛₂𝑦₂₂,𝑒₅=𝑘₃𝑥₁₃+𝑙₃𝑥₂₃−𝑚₃𝑦₁₃−𝑛₃𝑦₂₃, (10) such that lim_𝑡→∞𝑘₁𝑥₁₁+𝑙₁𝑥₂₁−𝑚₁𝑦₁₁−𝑛₁𝑦₂₁=0,lim_𝑡→∞𝑘₂𝑥₁₂+𝑙₂𝑥₂₂−𝑚₂𝑦₁₂−𝑛₂𝑦₂₂=0,lim_𝑡→∞𝑘₃𝑥₁₃+𝑙₃𝑥₂₃−𝑚₃𝑦₁₃−𝑛₃𝑦₂₃=0. (11)

Thus, we have the following error dynamical system: 𝑒₁̇+𝑖𝑒₂̇=𝑘₁𝑥̇₁₁+𝑙₁𝑥̇₂₁−𝑚₁𝑦̇₁₁−𝑛₁𝑦̇₂₁,𝑒₃̇+𝑖𝑒₄̇=𝑘₂𝑥̇₁₂+𝑙₂𝑥̇₂₂−𝑚₂𝑦̇₁₂−𝑛₂𝑦̇₂₂,𝑒₅̇=𝑘₃𝑥̇₁₃+𝑙₃𝑥̇₂₃−𝑚₃𝑦̇₁₃−𝑛₃𝑦̇₂₃. (12)

Substituting (6)–(9) into (12) and separating the real and imaginary parts yields 𝑒₁̇=𝑘₁𝛼₁𝑢₃−𝑢₁+𝑢₃𝑢₅+𝑙₁𝑎₁𝑣₁+𝑏₁𝑣₃𝑣₅−𝑚₁𝜎₃𝑤₃−𝑤₁−𝛿₃𝑤₂+𝜑₁−𝑛₁𝜇₃−𝛼₄𝜇₁+𝛽₄𝜇₃𝜇₅+𝜑^∗₁,𝑒₂̇=𝑘₁𝛼₁𝑢₄−𝑢₂+𝑢₄𝑢₅+𝑙₁𝑎₁𝑣₂+𝑏₁𝑣₄𝑣₅−𝑚₁𝜎₃𝑤₄−𝑤₂+𝛿₃𝑤₁+𝜑₂−𝑛₁𝜇₄−𝛼₄𝜇₂+𝛽₄𝜇₄𝜇₅+𝜑^∗₂,𝐞̇_𝟑=𝐤_𝟐𝜸_𝟏𝐮_𝟏−𝐮_𝟓𝐮_𝟏−𝐮_𝟑+𝐥_𝟐𝐚_𝟐𝐯_𝟑+𝐛_𝟐𝐯_𝟏𝐯_𝟓−𝐦_𝟐𝜶_𝟑−𝐰_𝟓𝐰_𝟏−𝐰_𝟑−𝜹_𝟑𝐰_𝟒+𝝋_𝟑−𝐧_𝟐𝜸_𝟒𝝁_𝟑−𝝁_𝟏𝝁_𝟓+𝝁_𝟓+𝝋^∗_𝟑,𝑒₄̇=𝑘₂𝛾₁𝑢₂−𝑢₅𝑢₂−𝑢₄+𝑙₂𝑎₂𝑣₄+𝑏₂𝑣₂𝑣₅−𝑚₂𝛼₃−𝑤₅𝑤₂−𝑤₄−𝛿₃𝑤₃+𝜑₄−𝑛₂𝛾₄𝜇₄−𝜇₂𝜇₅+𝜑^∗₄,𝑒₅̇=𝑘₃−𝛽₁𝑢₅+𝑢₁𝑢₃+𝑢₂𝑢₄+𝑙₃𝑎₃𝑣₅+𝑏₃𝑣₁𝑣₃+𝑣₂𝑣₄−𝑚₃−𝛽₃𝑤₅+𝑤₁𝑤₃+𝑤₂𝑤₄+𝜑₅−𝑛₃𝛿₄𝜇₁𝜇₃+𝜇₂𝜇₄−𝜎₄𝜇₅+𝜑^∗₅. (13)

Denote 𝑈_𝑖=𝑚_𝑖𝜑_𝑖+𝑛_𝑖𝜑^∗_𝑖 (𝑖=1,2,3,4,5); then we obtain the following results.

Theorem 3. If the controllers are chosen as follows: 𝑈₁=𝑘₁𝑢₁+𝑙₁𝑣₁−𝑚₁𝑤₁−𝑛₁𝜇₁+𝑎₁𝑘₁𝑢₂+𝑙₁𝑣₂−𝑚₁𝑤₂−𝑛₁𝜇₂+𝑘₁𝛼₁𝑢₃−𝑢₁+𝑢₃𝑢₅+𝑙₁𝑎₁𝑣₁+𝑏₁𝑣₃𝑣₅−𝑚₁𝜎₃𝑤₃−𝑤₁−𝛿₃𝑤₂−𝑛₁𝜇₃−𝛼₄𝜇₁+𝛽₄𝜇₃𝜇₅,𝐔_𝟐=𝐤_𝟏𝐮_𝟐+𝐥_𝟏𝐯_𝟐−𝐦_𝟏𝐰_𝟐−𝐧_𝟏𝝁_𝟐−𝐚_𝟏𝐤_𝟏𝐮_𝟏+𝐥_𝟏𝐯_𝟏−𝐦_𝟏𝐰_𝟏−𝐧_𝟏𝝁_𝟏−𝐚_𝟐𝐤_𝟐𝐮_𝟑+𝐥_𝟐𝐯_𝟑−𝐦_𝟐𝐰_𝟑−𝐧_𝟐𝝁_𝟑+𝐤_𝟏𝜶_𝟏𝐮_𝟒−𝐮_𝟐+𝐮_𝟒𝐮_𝟓+𝐥_𝟏𝐚_𝟏𝐯_𝟐+𝐛_𝟏𝐯_𝟒𝐯_𝟓−𝐦_𝟏𝝈_𝟑𝐰_𝟒−𝐰_𝟐+𝜹_𝟑𝐰_𝟏−𝐧_𝟏𝝁_𝟒−𝜶_𝟒𝝁_𝟐+𝜷_𝟒𝝁_𝟒𝝁_𝟓,𝑈₃=𝑘₂𝑢₃+𝑙₂𝑣₃−𝑚₂𝑤₃−𝑛₂𝜇₃+𝑎₂𝑘₁𝑢₂+𝑙₁𝑣₂−𝑚₁𝑤₂−𝑛₁𝜇₂+𝑎₃𝑘₂𝑢₄+𝑙₂𝑣₄−𝑚₂𝑤₄−𝑛₂𝜇₄+𝑘₂𝛾₁𝑢₁−𝑢₅𝑢₁−𝑢₃+𝑙₂𝑎₂𝑣₃+𝑏₂𝑣₁𝑣₅−𝑚₂𝛼₃−𝑤₅𝑤₁−𝑤₃−𝛿₃𝑤₄−𝑛₂𝛾₄𝜇₃−𝜇₁𝜇₅+𝜇₅,𝐔_𝟒=𝐤_𝟐𝐮_𝟒+𝐥_𝟐𝐯_𝟒−𝐦_𝟐𝐰_𝟒−𝐧_𝟐𝝁_𝟒−𝐚_𝟑𝐤_𝟐𝐮_𝟑+𝐥_𝟐𝐯_𝟑−𝐦_𝟐𝐰_𝟑−𝐧_𝟐𝝁_𝟑−𝐛_𝟏𝐤_𝟑𝐮_𝟓+𝐥_𝟑𝐯_𝟓−𝐦_𝟑𝐰_𝟓−𝐧_𝟑𝝁_𝟓+𝐤_𝟐𝜸_𝟏𝐮_𝟐−𝐮_𝟓𝐮_𝟐−𝐮_𝟒+𝐥_𝟐𝐚_𝟐𝐯_𝟒+𝐛_𝟐𝐯_𝟐𝐯_𝟓−𝐦_𝟐𝜶_𝟑−𝐰_𝟓𝐰_𝟐−𝐰_𝟒−𝜹_𝟑𝐰_𝟑−𝐧_𝟐𝜸_𝟒𝝁_𝟒−𝝁_𝟐𝝁_𝟓,𝑈₅=𝑘₃𝑢₅+𝑙₃𝑣₅−𝑚₃𝑤₅−𝑛₃𝜇₅+𝑏₁𝑘₂𝑢₄+𝑙₂𝑣₄−𝑚₂𝑤₄−𝑛₂𝜇₄+𝑘₃−𝛽₁𝑢₅+𝑢₁𝑢₃+𝑢₂𝑢₄+𝑙₃𝑎₃𝑣₅+𝑏₃𝑣₁𝑣₃+𝑣₂𝑣₄−𝑚₃−𝛽₃𝑤₅+𝑤₁𝑤₃+𝑤₂𝑤₄−𝑛₃𝛿₄𝜇₁𝜇₃+𝜇₂𝜇₄−𝜎₄𝜇₅, (14) then the drive systems (6) and (7) will achieve combination-combination synchronization with the response systems (8) and (9).

Proof. Construct the following Lyapunov function: 1𝑉=2𝑒²₁+𝑒²₂+𝑒²₃+𝑒²₄+𝑒²₅. (15) Taking the time derivative of 𝑉 along the trajectory of the error dynamical system (13) yields 𝑉̇=𝑒₁𝑒₁̇+𝑒₂𝑒₂̇+𝑒₃𝑒₃̇+𝑒₄𝑒₄̇+𝑒₅𝑒₅̇=𝑒₁𝑘₁𝛼₁𝑢₃−𝑢₁+𝑢₃𝑢₅+𝑙₁𝑎₁𝑣₁+𝑏₁𝑣₃𝑣₅−𝑚₁𝜎₃𝑤₃−𝑤₁−𝛿₃𝑤₂+𝜑₁−𝑛₁𝜇₃−𝛼₄𝜇₁+𝛽₄𝜇₃𝜇₅+𝜑^∗₁+𝑒₂𝑘₁𝛼₁𝑢₄−𝑢₂+𝑢₄𝑢₅+𝑙₁𝑎₁𝑣₂+𝑏₁𝑣₄𝑣₅−𝑚₁𝜎₃𝑤₄−𝑤₂+𝛿₃𝑤₁+𝜑₂−𝑛₁𝜇₄−𝛼₄𝜇₂+𝛽₄𝜇₄𝜇₅+𝜑^∗₂+𝑒₃𝑘₂𝛾₁𝑢₁−𝑢₅𝑢₁−𝑢₃+𝑙₂𝑎₂𝑣₃+𝑏₂𝑣₁𝑣₅−𝑚₂𝛼₃−𝑤₅𝑤₁−𝑤₃−𝛿₃𝑤₄+𝜑₃−𝑛₂𝛾₄𝜇₃−𝜇₁𝜇₅+𝜇₅+𝜑^∗₃+𝑒₄𝑘₂𝛾₁𝑢₂−𝑢₅𝑢₂−𝑢₄+𝑙₂𝑎₂𝑣₄+𝑏₂𝑣₂𝑣₅−𝑚₂𝛼₃−𝑤₅𝑤₂−𝑤₄−𝛿₃𝑤₃+𝜑₄−𝑛₂𝛾₄𝜇₄−𝜇₂𝜇₅+𝜑^∗₄+𝑒₅𝑘₃−𝛽₁𝑢₅+𝑢₁𝑢₃+𝑢₂𝑢₄+𝑙₃𝑎₃𝑣₅+𝑏₃𝑣₁𝑣₃+𝑣₂𝑣₄−𝑚₃−𝛽₃𝑤₅+𝑤₁𝑤₃+𝑤₂𝑤₄+𝜑₅−𝑛₃𝛿₄𝜇₁𝜇₃+𝜇₂𝜇₄−𝜎₄𝜇₅+𝜑^∗₅=𝑒₁𝑘₁𝛼₁𝑢₃−𝑢₁+𝑢₃𝑢₅+𝑙₁𝑎₁𝑣₁+𝑏₁𝑣₃𝑣₅−𝑚₁𝜎₃𝑤₃−𝑤₁−𝛿₃𝑤₂−𝑛₁𝜇₃−𝛼₄𝜇₁+𝛽₄𝜇₃𝜇₅−𝑚₁𝜑₁+𝑛₁𝜑^∗₁+𝑒₂𝑘₁𝛼₁𝑢₄−𝑢₂+𝑢₄𝑢₅+𝑙₁𝑎₁𝑣₂+𝑏₁𝑣₄𝑣₅−𝑚₁𝜎₃𝑤₄−𝑤₂+𝛿₃𝑤₁−𝑛₁𝜇₄−𝛼₄𝜇₂+𝛽₄𝜇₄𝜇₅−𝑚₁𝜑₂+𝑛₁𝜑^∗₂+𝑒₃𝑘₂𝛾₁𝑢₁−𝑢₅𝑢₁−𝑢₃+𝑙₂𝑎₂𝑣₃+𝑏₂𝑣₁𝑣₅−𝑚₂𝛼₃−𝑤₅𝑤₁−𝑤₃−𝛿₃𝑤₄−𝑛₂𝛾₄𝜇₃−𝜇₁𝜇₅+𝜇₅−𝑚₂𝜑₃+𝑛₂𝜑^∗₃+𝑒₄𝑘₂𝛾₁𝑢₂−𝑢₅𝑢₂−𝑢₄+𝑙₂𝑎₂𝑣₄+𝑏₂𝑣₂𝑣₅−𝑚₂𝛼₃−𝑤₅𝑤₂−𝑤₄−𝛿₃𝑤₃−𝑛₂𝛾₄𝜇₄−𝜇₂𝜇₅−𝑚₂𝜑₄+𝑛₂𝜑^∗₄+𝑒₅𝑘₃−𝛽₁𝑢₅+𝑢₁𝑢₃+𝑢₂𝑢₄+𝑙₃𝑎₃𝑣₅+𝑏₃𝑣₁𝑣₃+𝑣₂𝑣₄−𝑚₃−𝛽₃𝑤₅+𝑤₁𝑤₃+𝑤₂𝑤₄−𝑛₃𝛿₄𝜇₁𝜇₃+𝜇₂𝜇₄−𝜎₄𝜇₅−𝑚₃𝜑₅+𝑛₃𝜑^∗₅. (16)
Substituting (14) into (16) leads to 𝑉̇=𝑒₁𝑘₁𝛼₁𝑢₃−𝑢₁+𝑢₃𝑢₅+𝑙₁𝑎₁𝑣₁+𝑏₁𝑣₃𝑣₅−𝑚₁𝜎₃𝑤₃−𝑤₁−𝛿₃𝑤₂−𝑛₁𝜇₃−𝛼₄𝜇₁+𝛽₄𝜇₃𝜇₅−𝑘₁𝑢₁+𝑙₁𝑣₁−𝑚₁𝑤₁−𝑛₁𝜇₁+𝑎₁𝑘₁𝑢₂+𝑙₁𝑣₂−𝑚₁𝑤₂−𝑛₁𝜇₂+𝑘₁𝛼₁𝑢₃−𝑢₁+𝑢₃𝑢₅+𝑙₁𝑎₁𝑣₁+𝑏₁𝑣₃𝑣₅−𝑚₁𝜎₃𝑤₃−𝑤₁−𝛿₃𝑤₂−𝑛₁𝜇₃−𝛼₄𝜇₁+𝛽₄𝜇₃𝜇₅+𝑒₂𝑘₁𝛼₁𝑢₄−𝑢₂+𝑢₄𝑢₅+𝑙₁𝑎₁𝑣₂+𝑏₁𝑣₄𝑣₅−𝑚₁𝜎₃𝑤₄−𝑤₂+𝛿₃𝑤₁−𝑛₁𝜇₄−𝛼₄𝜇₂+𝛽₄𝜇₄𝜇₅−𝑘₁𝑢₂+𝑙₁𝑣₂−𝑚₁𝑤₂−𝑛₁𝜇₂−𝑎₁𝑘₁𝑢₁+𝑙₁𝑣₁−𝑚₁𝑤₁−𝑛₁𝜇₁−𝑎₂𝑘₂𝑢₃+𝑙₂𝑣₃−𝑚₂𝑤₃−𝑛₂𝜇₃+𝑘₁𝛼₁𝑢₄−𝑢₂+𝑢₄𝑢₅+𝑙₁𝑎₁𝑣₂+𝑏₁𝑣₄𝑣₅−𝑚₁𝜎₃𝑤₄−𝑤₂+𝛿₃𝑤₁−𝑛₁𝜇₄−𝛼₄𝜇₂+𝛽₄𝜇₄𝜇₅+𝑒₃𝑘₂𝛾₁𝑢₁−𝑢₅𝑢₁−𝑢₃+𝑙₂𝑎₂𝑣₃+𝑏₂𝑣₁𝑣₅−𝑚₂𝛼₃−𝑤₅𝑤₁−𝑤₃−𝛿₃𝑤₄−𝑛₂𝛾₄𝜇₃−𝜇₁𝜇₅+𝜇₅−𝑘₂𝑢₃+𝑙₂𝑣₃−𝑚₂𝑤₃−𝑛₂𝜇₃+𝑎₂𝑘₁𝑢₂+𝑙₁𝑣₂−𝑚₁𝑤₂−𝑛₁𝜇₂+𝑎₃𝑘₂𝑢₄+𝑙₂𝑣₄−𝑚₂𝑤₄−𝑛₂𝜇₄+𝑘₂𝛾₁𝑢₁−𝑢₅𝑢₁−𝑢₃+𝑙₂𝑎₂𝑣₃+𝑏₂𝑣₁𝑣₅−𝑚₂𝛼₃−𝑤₅𝑤₁−𝑤₃−𝛿₃𝑤₄−𝑛₂𝛾₄𝜇₃−𝜇₁𝜇₅+𝜇₅+𝑒₄𝑘₂𝛾₁𝑢₂−𝑢₅𝑢₂−𝑢₄+𝑙₂𝑎₂𝑣₄+𝑏₂𝑣₂𝑣₅−𝑚₂𝛼₃−𝑤₅𝑤₂−𝑤₄−𝛿₃𝑤₃−𝑛₂𝛾₄𝜇₄−𝜇₂𝜇₅−𝑘₂𝑢₄+𝑙₂𝑣₄−𝑚₂𝑤₄−𝑛₂𝜇₄−𝑎₃𝑘₂𝑢₃+𝑙₂𝑣₃−𝑚₂𝑤₃−𝑛₂𝜇₃−𝑏₁𝑘₃𝑢₅+𝑙₃𝑣₅−𝑚₃𝑤₅−𝑛₃𝜇₅+𝑘₂𝛾₁𝑢₂−𝑢₅𝑢₂−𝑢₄+𝑙₂𝑎₂𝑣₄+𝑏₂𝑣₂𝑣₅−𝑚₂𝛼₃−𝑤₅𝑤₂−𝑤₄−𝛿₃𝑤₃−𝑛₂𝛾₄𝜇₄−𝜇₂𝜇₅+𝑒₅𝑘₃−𝛽₁𝑢₅+𝑢₁𝑢₃+𝑢₂𝑢₄+𝑙₃𝑎₃𝑣₅+𝑏₃𝑣₁𝑣₃+𝑣₂𝑣₄−𝑚₃−𝛽₃𝑤₅+𝑤₁𝑤₃+𝑤₂𝑤₄−𝑛₃𝛿₄𝜇₁𝜇₃+𝜇₂𝜇₄−𝜎₄𝜇₅−𝑘₃𝑢₅+𝑙₃𝑣₅−𝑚₃𝑤₅−𝑛₃𝜇₅+𝑏₁𝑘₂𝑢₄+𝑙₂𝑣₄−𝑚₂𝑤₄−𝑛₂𝜇₄+𝑘₃−𝛽₁𝑢₅+𝑢₁𝑢₃+𝑢₂𝑢₄+𝑙₃𝑎₃𝑣₅+𝑏₃𝑣₁𝑣₃+𝑣₂𝑣₄−𝑚₃−𝛽₃𝑤₅+𝑤₁𝑤₃+𝑤₂𝑤₄−𝑛₃𝛿₄𝜇₁𝜇₃+𝜇₂𝜇₄−𝜎₄𝜇₅, (17) and we get 𝑉̇=𝑒₁−𝑒₁−𝑎₁𝑒₂+𝑒₂−𝑒₂+𝑎₁𝑒₁+𝑎₂𝑒₃+𝑒₃−𝑒₃−𝑎₂𝑒₂−𝑎₃𝑒₄+𝑒₄−𝑒₄+𝑎₃𝑒₃+𝑏₁𝑒₅+𝑒₅−𝑒₅−𝑏₁𝑒₄=−𝑒²₁−𝑒²₂−𝑒²₃−𝑒²₄−𝑒²₅. (18) Since 𝑉̇≤0 as 𝑡→∞, according to the Lyapunov stability theory, we know 𝑒_𝑖→0 (𝑖=1,2,3,4,5); that is, lim_𝑡→∞‖𝑒‖=0. Therefore, the drive systems (6) and (7) will achieve combination-combination synchronization with the response systems (8) and (9).
This completes the proof.

If we choose specific values of 𝑘_𝑖, 𝑙_𝑖, 𝑚_𝑖, and 𝑛_𝑖 (𝑖=1,2,3), then we can have the following corollaries.

Corollary 4. (i) Suppose that 𝑛₁=𝑛₂=𝑛₃=0, and if the controllers are chosen as follows: 𝜑₁=1𝑚₁𝑘₁𝑢₁+𝑙₁𝑣₁−𝑚₁𝑤₁+𝑎₁𝑘₁𝑢₂+𝑙₁𝑣₂−𝑚₁𝑤₂+𝑘₁𝛼₁𝑢₃−𝑢₁+𝑢₃𝑢₅+𝑙₁𝑎₁𝑣₁+𝑏₁𝑣₃𝑣₅−𝑚₁𝜎₃𝑤₃−𝑤₁−𝛿₃𝑤₂,𝜑₂=1𝑚₁𝑘₁𝑢₂+𝑙₁𝑣₂−𝑚₁𝑤₂−𝑎₁𝑘₁𝑢₁+𝑙₁𝑣₁−𝑚₁𝑤₁−𝑎₂𝑘₂𝑢₃+𝑙₂𝑣₃−𝑚₂𝑤₃+𝑘₁𝛼₁𝑢₄−𝑢₂+𝑢₄𝑢₅+𝑙₁𝑎₁𝑣₂+𝑏₁𝑣₄𝑣₅−𝑚₁𝜎₃𝑤₄−𝑤₂+𝛿₃𝑤₁,𝜑₃=1𝑚₂𝑘₂𝑢₃+𝑙₂𝑣₃−𝑚₂𝑤₃+𝑎₂𝑘₁𝑢₂+𝑙₁𝑣₂−𝑚₁𝑤₂+𝑎₃𝑘₂𝑢₄+𝑙₂𝑣₄−𝑚₂𝑤₄+𝑘₂𝛾₁𝑢₁−𝑢₅𝑢₁−𝑢₃+𝑙₂𝑎₂𝑣₃+𝑏₂𝑣₁𝑣₅−𝑚₂𝛼₃−𝑤₅𝑤₁−𝑤₃−𝛿₃𝑤₄,𝜑₄=1𝑚₂𝑘₂𝑢₄+𝑙₂𝑣₄−𝑚₂𝑤₄−𝑎₃𝑘₂𝑢₃+𝑙₂𝑣₃−𝑚₂𝑤₃−𝑏₁𝑘₃𝑢₅+𝑙₃𝑣₅−𝑚₃𝑤₅+𝑘₂𝛾₁𝑢₂−𝑢₅𝑢₂−𝑢₄+𝑙₂𝑎₂𝑣₄+𝑏₂𝑣₂𝑣₅−𝑚₂𝛼₃−𝑤₅𝑤₂−𝑤₄−𝛿₃𝑤₃,𝜑₅=1𝑚₃𝑘₃𝑢₅+𝑙₃𝑣₅−𝑚₃𝑤₅+𝑏₁𝑘₂𝑢₄+𝑙₂𝑣₄−𝑚₂𝑤₄+𝑘₃−𝛽₁𝑢₅+𝑢₁𝑢₃+𝑢₂𝑢₄+𝑙₃𝑎₃𝑣₅+𝑏₃𝑣₁𝑣₃+𝑣₂𝑣₄−𝑚₃−𝛽₃𝑤₅+𝑤₁𝑤₃+𝑤₂𝑤₄, (19) then the drive systems (6) and (7) will achieve combination synchronization with the response system (8).
(ii) Suppose that 𝑚₁=𝑚₂=𝑚₃=0, and if the controllers are chosen as follows: 𝝋^∗_𝟏=𝟏𝐧_𝟏𝐤_𝟏𝐮_𝟏+𝐥_𝟏𝐯_𝟏−𝐧_𝟏𝝁_𝟏+𝐚_𝟏𝐤_𝟏𝐮_𝟐+𝐥_𝟏𝐯_𝟐−𝐧_𝟏𝝁_𝟐+𝐤_𝟏𝜶_𝟏𝐮_𝟑−𝐮_𝟏+𝐮_𝟑𝐮_𝟓+𝐥_𝟏𝐚_𝟏𝐯_𝟏+𝐛_𝟏𝐯_𝟑𝐯_𝟓−𝐧_𝟏𝝁_𝟑−𝜶_𝟒𝝁_𝟏+𝜷_𝟒𝝁_𝟑𝝁_𝟓,𝝋^∗_𝟐=𝟏𝐧_𝟏𝐤_𝟏𝐮_𝟐+𝐥_𝟏𝐯_𝟐−𝐧_𝟏𝝁_𝟐−𝐚_𝟏𝐤_𝟏𝐮_𝟏+𝐥_𝟏𝐯_𝟏−𝐧_𝟏𝝁_𝟏−𝐚_𝟐𝐤_𝟐𝐮_𝟑+𝐥_𝟐𝐯_𝟑−𝐧_𝟐𝝁_𝟑+𝐤_𝟏𝜶_𝟏𝐮_𝟒−𝐮_𝟐+𝐮_𝟒𝐮_𝟓+𝐥_𝟏𝐚_𝟏𝐯_𝟐+𝐛_𝟏𝐯_𝟒𝐯_𝟓−𝐧_𝟏𝝁_𝟒−𝜶_𝟒𝝁_𝟐+𝜷_𝟒𝝁_𝟒𝝁_𝟓,𝝋^∗_𝟑=𝟏𝐧_𝟐𝐤_𝟐𝐮_𝟑+𝐥_𝟐𝐯_𝟑−𝐧_𝟐𝝁_𝟑+𝐚_𝟐𝐤_𝟏𝐮_𝟐+𝐥_𝟏𝐯_𝟐−𝐧_𝟏𝝁_𝟐+𝐚_𝟑𝐤_𝟐𝐮_𝟒+𝐥_𝟐𝐯_𝟒−𝐧_𝟐𝝁_𝟒+𝐤_𝟐𝜸_𝟏𝐮_𝟏−𝐮_𝟓𝐮_𝟏−𝐮_𝟑+𝐥_𝟐𝐚_𝟐𝐯_𝟑+𝐛_𝟐𝐯_𝟏𝐯_𝟓−𝐧_𝟐𝜸_𝟒𝝁_𝟑−𝝁_𝟏𝝁_𝟓+𝝁_𝟓,𝝋^∗_𝟒=𝟏𝐧_𝟐𝐤_𝟐𝐮_𝟒+𝐥_𝟐𝐯_𝟒−𝐧_𝟐𝝁_𝟒−𝐚_𝟑𝐤_𝟐𝐮_𝟑+𝐥_𝟐𝐯_𝟑−𝐧_𝟐𝝁_𝟑−𝐛_𝟏𝐤_𝟑𝐮_𝟓+𝐥_𝟑𝐯_𝟓−𝐧_𝟑𝝁_𝟓+𝐤_𝟐𝜸_𝟏𝐮_𝟐−𝐮_𝟓𝐮_𝟐−𝐮_𝟒+𝐥_𝟐𝐚_𝟐𝐯_𝟒+𝐛_𝟐𝐯_𝟐𝐯_𝟓−𝐧_𝟐𝜸_𝟒𝝁_𝟒−𝝁_𝟐𝝁_𝟓,𝝋^∗_𝟓=𝟏𝐧_𝟑𝐤_𝟑𝐮_𝟓+𝐥_𝟑𝐯_𝟓−𝐧_𝟑𝝁_𝟓+𝐛_𝟏𝐤_𝟐𝐮_𝟒+𝐥_𝟐𝐯_𝟒−𝐧_𝟐𝝁_𝟒+𝐤_𝟑−𝜷_𝟏𝐮_𝟓+𝐮_𝟏𝐮_𝟑+𝐮_𝟐𝐮_𝟒+𝐥_𝟑𝐚_𝟑𝐯_𝟓+𝐛_𝟑𝐯_𝟏𝐯_𝟑+𝐯_𝟐𝐯_𝟒−𝐧_𝟑𝜹_𝟒𝝁_𝟏𝝁_𝟑+𝝁_𝟐𝝁_𝟒−𝝈_𝟒𝝁_𝟓, (20) then the drive systems (6) and (7) will achieve combination synchronization with the response system (9).

Corollary 5. (i) Suppose that 𝑘₁=𝑘₂=𝑘₃=0, 𝑛₁=𝑛₂=𝑛₃=0, and 𝑚₁=𝑚₂=𝑚₃=1, and if the controllers are chosen as follows: 𝜑₁=𝑙₁𝑣₁−𝑤₁+𝑎₁𝑙₁𝑣₂−𝑤₂+𝑙₁𝑎₁𝑣₁+𝑏₁𝑣₃𝑣₅−𝜎₃𝑤₃−𝑤₁−𝛿₃𝑤₂,𝜑₂=𝑙₁𝑣₂−𝑤₂−𝑎₁𝑙₁𝑣₁−𝑤₁−𝑎₂𝑙₂𝑣₃−𝑤₃+𝑙₁𝑎₁𝑣₂+𝑏₁𝑣₄𝑣₅−𝜎₃𝑤₄−𝑤₂+𝛿₃𝑤₁,𝜑₃=𝑙₂𝑣₃−𝑤₃+𝑎₂𝑙₁𝑣₂−𝑤₂+𝑎₃𝑙₂𝑣₄−𝑤₄+𝑙₂𝑎₂𝑣₃+𝑏₂𝑣₁𝑣₅−𝛼₃−𝑤₅𝑤₁−𝑤₃−𝛿₃𝑤₄,𝜑₄=𝑙₂𝑣₄−𝑤₄−𝑎₃𝑙₂𝑣₃−𝑤₃−𝑏₁𝑙₃𝑣₅−𝑤₅+𝑙₂𝑎₂𝑣₄+𝑏₂𝑣₂𝑣₅−𝛼₃−𝑤₅𝑤₂−𝑤₄−𝛿₃𝑤₃,𝜑₅=𝑙₃𝑣₅−𝑤₅+𝑏₁𝑙₂𝑣₄−𝑤₄+𝑙₃𝑎₃𝑣₅+𝑏₃𝑣₁𝑣₃+𝑣₂𝑣₄−−𝛽₃𝑤₅+𝑤₁𝑤₃+𝑤₂𝑤₄, (21) then the drive system (7) will achieve projective synchronization with the response system (8).
(ii) Suppose that 𝑙₁=𝑙₂=𝑙₃=0, 𝑛₁=𝑛₂=𝑛₃=0, and 𝑚₁=𝑚₂=𝑚₃=1, and if the controllers are chosen as follows: 𝜑₁=𝑘₁𝑢₁−𝑤₁+𝛼₃𝑘₁𝑢₂−𝑤₂+𝑘₁𝛼₁𝑢₃−𝑢₁+𝑢₃𝑢₅−𝜎₃𝑤₃−𝑤₁−𝛿₃𝑤₂,𝜑₂=𝑘₁𝑢₂−𝑤₂−𝛼₃𝑘₁𝑢₁−𝑤₁−𝛽₃𝑘₂𝑢₃−𝑤₃+𝑘₁𝛼₁𝑢₄−𝑢₂+𝑢₄𝑢₅−𝜎₃𝑤₄−𝑤₂+𝛿₃𝑤₁,𝜑₃=𝑘₂𝑢₃−𝑤₃+𝛽₃𝑘₁𝑢₂−𝑤₂+𝛿₃𝑘₂𝑢₄−𝑤₄+𝑘₂𝛾₁𝑢₁−𝑢₅𝑢₁−𝑢₃−𝛼₃−𝑤₅𝑤₁−𝑤₃−𝛿₃𝑤₄,𝜑₄=𝑘₂𝑢₄−𝑤₄−𝛿₃𝑘₂𝑢₃−𝑤₃−𝜎₃𝑘₃𝑢₅−𝑤₅+𝑘₂𝛾₁𝑢₂−𝑢₅𝑢₂−𝑢₄−𝛼₃−𝑤₅𝑤₂−𝑤₄−𝛿₃𝑤₃,𝜑₅=𝑘₃𝑢₅−𝑤₅+𝜎₃𝑘₂𝑢₄−𝑤₄+𝑘₃−𝛽₁𝑢₅+𝑢₁𝑢₃+𝑢₂𝑢₄−−𝛽₃𝑤₅+𝑤₁𝑤₃+𝑤₂𝑤₄, (22) then the drive system (6) will achieve projective synchronization with the response system (8).
(iii) Suppose that 𝑘₁=𝑘₂=𝑘₃=0, 𝑚₁=𝑚₂=𝑚₃=0, and 𝑛₁=𝑛₂=𝑛₃=1, and if the controllers are chosen as follows: 𝜑^∗₁=𝑙₁𝑣₁−𝜇₁+𝑎₁𝑙₁𝑣₂−𝜇₂+𝑙₁𝑎₁𝑣₁+𝑏₁𝑣₃𝑣₅−𝜇₃−𝛼₄𝜇₁+𝛽₄𝜇₃𝜇₅,𝜑^∗₂=𝑙₁𝑣₂−𝜇₂−𝑎₁𝑙₁𝑣₁−𝜇₁−𝑎₂𝑙₂𝑣₃−𝜇₃+𝑙₁𝑎₁𝑣₂+𝑏₁𝑣₄𝑣₅−𝜇₄−𝛼₄𝜇₂+𝛽₄𝜇₄𝜇₅,𝜑^∗₃=𝑙₂𝑣₃−𝜇₃+𝑎₂𝑙₁𝑣₂−𝜇₂+𝑎₃𝑙₂𝑣₄−𝜇₄+𝑙₂𝑎₂𝑣₃+𝑏₂𝑣₁𝑣₅−𝛾₄𝜇₃−𝜇₁𝜇₅+𝜇₅,𝜑^∗₄=𝑙₂𝑣₄−𝜇₄−𝑎₃𝑙₂𝑣₃−𝜇₃−𝑏₁𝑙₃𝑣₅−𝜇₅+𝑙₂𝑎₂𝑣₄+𝑏₂𝑣₂𝑣₅−𝛾₄𝜇₄−𝜇₂𝜇₅,𝜑^∗₅=𝑙₃𝑣₅−𝜇₅+𝑏₁𝑙₂𝑣₄−𝜇₄+𝑙₃𝑎₃𝑣₅+𝑏₃𝑣₁𝑣₃+𝑣₂𝑣₄−𝛿₄𝜇₁𝜇₃+𝜇₂𝜇₄−𝜎₄𝜇₅, (23) then the drive system (7) will achieve projective synchronization with the response system (9).
(iv) Suppose that 𝑙₁=𝑙₂=𝑙₃=0, 𝑚₁=𝑚₂=𝑚₃=0, and 𝑛₁=𝑛₂=𝑛₃=1, and if the controllers are chosen as follows: 𝜑^∗₁=𝑘₁𝑢₁−𝜇₁+𝛼₃𝑘₁𝑢₂−𝜇₂+𝑘₁𝛼₁𝑢₃−𝑢₁+𝑢₃𝑢₅−𝜇₃−𝛼₄𝜇₁+𝛽₄𝜇₃𝜇₅,𝜑^∗₂=𝑘₁𝑢₂−𝜇₂−𝛼₃𝑘₁𝑢₁−𝜇₁−𝛽₃𝑘₂𝑢₃−𝜇₃+𝑘₁𝛼₁𝑢₄−𝑢₂+𝑢₄𝑢₅−𝜇₄−𝛼₄𝜇₂+𝛽₄𝜇₄𝜇₅,𝜑^∗₃=𝑘₂𝑢₃−𝜇₃+𝛽₃𝑘₁𝑢₂−𝜇₂+𝛿₃𝑘₂𝑢₄−𝜇₄+𝑘₂𝛾₁𝑢₁−𝑢₅𝑢₁−𝑢₃−𝛾₄𝜇₃−𝜇₁𝜇₅+𝜇₅,𝜑^∗₄=𝑘₂𝑢₄−𝜇₄−𝛿₃𝑘₂𝑢₃−𝜇₃−𝜎₃𝑘₃𝑢₅−𝜇₅+𝑘₂𝛾₁𝑢₂−𝑢₅𝑢₂−𝑢₄−𝛾₄𝜇₄−𝜇₂𝜇₅,𝜑^∗₅=𝑘₃𝑢₅−𝜇₅+𝜎₃𝑘₂𝑢₄−𝜇₄+𝑘₃−𝛽₁𝑢₅+𝑢₁𝑢₃+𝑢₂𝑢₄−𝛿₄𝜇₁𝜇₃+𝜇₂𝜇₄−𝜎₄𝜇₅, (24) then the drive system (6) will achieve projective synchronization with the response system (9).

Corollary 6. (i) Suppose that 𝑘₁=𝑘₂=𝑘₃=0, 𝑙₁=𝑙₂=𝑙₃=0, 𝑛₁=𝑛₂=𝑛₃=0, and 𝑚₁=𝑚₂=𝑚₃=1, and if the controllers are chosen as follows: 𝜑₁=−𝑤₁−𝛼₃𝑤₂−𝜎₃𝑤₃−𝑤₁−𝛿₃𝑤₂,𝜑₂=−𝑤₂+𝛼₃𝑤₁+𝛽₃𝑤₃−𝜎₃𝑤₄−𝑤₂+𝛿₃𝑤₁,𝜑₃=−𝑤₃−𝛽₃𝑤₂−𝛿₃𝑤₄−𝛼₃−𝑤₅𝑤₁−𝑤₃−𝛿₃𝑤₄,𝜑₄=−𝑤₄+𝛿₃𝑤₃+𝜎₃𝑤₅−𝛼₃−𝑤₅𝑤₂−𝑤₄−𝛿₃𝑤₃,𝜑₅=−𝑤₅−𝜎₃𝑤₄−−𝛽₃𝑤₅+𝑤₁𝑤₃+𝑤₂𝑤₄, (25) then system (8) is stabilized to the equilibrium 𝑂(0,0,0,0,0).
(ii) Suppose that 𝑘₁=𝑘₂=𝑘₃=0, 𝑙₁=𝑙₂=𝑙₃=0, 𝑚₁=𝑚₂=𝑚₃=0, and 𝑛₁=𝑛₂=𝑛₃=1, and if the controllers are chosen as follows: 𝜑^∗₁=−𝜇₁−𝛼₄𝜇₂−𝜇₃−𝛼₄𝜇₁+𝛽₄𝜇₃𝜇₅,𝜑^∗₂=−𝜇₂+𝛼₄𝜇₁+𝛽₄𝜇₃−𝜇₄−𝛼₄𝜇₂+𝛽₄𝜇₄𝜇₅,𝜑^∗₃=−𝜇₃−𝛽₄𝜇₂−𝛾₄𝜇₄−𝛾₄𝜇₃−𝜇₁𝜇₅+𝜇₅,𝜑^∗₄=−𝜇₄+𝛾₄𝜇₃+𝜎₄𝜇₅−𝛾₄𝜇₄−𝜇₂𝜇₅,𝜑^∗₅=−𝜇₅−𝜎₄𝜇₄−𝛿₄𝜇₁𝜇₃+𝜇₂𝜇₄−𝜎₄𝜇₅, (26) then system (9) is stabilized to the equilibrium 𝑂(0,0,0,0,0).

Remark 7. The above corollaries can be easily obtained from Theorem 3, and their proofs are similar to that of Theorem 3, so we omit the proofs here.

4. Numerical Simulations

In this section, three numerical examples are presented to illustrate the theoretical analysis. In the following numerical simulations, the fourth-order Runge-kutta method is employed with time step size 0.001. The system parameters are selected as 𝛼₁=30, 𝛾₁=90, 𝛽₁=11, 𝑎₁=9.5, 𝑎₂=−19, 𝑎₃=−3, 𝑏₁=−1, 𝑏₂=1, 𝑏₃=1, 𝛼₃=50.0625, 𝛽₃=0.75, 𝜎₃=5, 𝛿₃=0.25, 𝛼₄=3.5, 𝛽₄=0.599, 𝛾₄=3, 𝛿₄=2, and 𝜎₄=9, so that the four nonlinear complex chaotic systems exhibit chaotic behaviors, respectively.

Firstly, consider the combination-combination synchronization of the two drive systems (6) and (7) and the response systems (8) and (9) with the controllers (14). We assume 𝑘₁=𝑘₂=𝑘₃=1, 𝑙₁=𝑙₂=𝑙₃=1, 𝑚₁=𝑚₂=𝑚₃=1, and 𝑛₁=𝑛₂=𝑛₃=1, and the initial states for the drive systems and response systems are arbitrarily given by (𝑥₁₁(0),𝑥₁₂(0),𝑥₁₃(0))=(2+4𝑖,1+3𝑖,2), (𝑥₂₁(0),𝑥₂₂(0),𝑥₂₃(0))=(−2−𝑖,5−3𝑖,4), (𝑦₁₁(0),𝑦₁₂(0),𝑦₁₃(0))=(2+𝑖,5+3𝑖,4), and (𝑦₂₁(0),𝑦₂₂(0),𝑦₂₃(0))=(5+2𝑖,−1+𝑖,−4); that is, (𝑢₁(0),𝑢₂(0),𝑢₃(0),𝑢₄(0),𝑢₅(0))=(2,4,1,3,2), (𝑣₁(0),𝑣₂(0),𝑣₃(0),𝑣₄(0),𝑣₅(0))=(−2,−1,5,−3,4), (𝑤₁(0),𝑤₂(0),𝑤₃(0),𝑤₄(0),𝑤₅(0))=(2,1,5,3,4), and (𝜇₁(0),𝜇₂(0),𝜇₃(0),𝜇₄(0),𝜇₅(0))=(5,2,−1,1,−4), respectively. The corresponding numerical results are shown in Figure 5. Figure 5 displays time response of the combination-combination synchronization errors 𝑒₁, 𝑒₂, 𝑒₃, 𝑒₄, and 𝑒₅, where 𝑒_𝑖=𝑢_𝑖+𝑣_𝑖−𝑤_𝑖−𝜇_𝑖 (𝑖=1,2,3,4,5). The errors converge to zero which implies that the drive systems (6) and (7) and the response systems (8) and (9) have achieved combination-combination synchronization.

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Figure 5 

Combination-combination synchronization errors 𝑒1, 𝑒2, 𝑒3, 𝑒4, and 𝑒5 between the drive systems (6) and (7) and the response systems (8) and (9), where 𝑒𝑖=𝑢𝑖+𝑣𝑖−𝑤𝑖−𝜇𝑖 (𝑖=1,2,3,4,5).

Secondly, consider the combination synchronization of the two drive systems (6) and (7) and the response system (8) with the controllers (19). We assume 𝑘₁=𝑘₂=𝑘₃=1, 𝑙₁=𝑙₂=𝑙₃=1, 𝑚₁=𝑚₂=𝑚₃=1, and 𝑛₁=𝑛₂=𝑛₃=0. The corresponding numerical results are shown in Figure 6. Figure 6 displays time response of the combination synchronization errors 𝑒₁, 𝑒₂, 𝑒₃, 𝑒₄, and 𝑒₅, where 𝑒_𝑖=𝑢_𝑖+𝑣_𝑖−𝑤_𝑖 (𝑖=1,2,3,4,5). The errors converge to zero which implies that the drive systems (6) and (7) and the response system (8) have achieved combination synchronization.

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Figure 6 

Combination synchronization errors 𝑒1, 𝑒2, 𝑒3, 𝑒4, and 𝑒5 between the drive systems (6) and (7) and the response system (8), where 𝑒𝑖=𝑢𝑖+𝑣𝑖−𝑤𝑖 (𝑖=1,2,3,4,5).

Finally, consider another special case, that is, when 𝑘₁=𝑘₂=𝑘₃=0, 𝑙₁=𝑙₂=𝑙₃=0, 𝑚₁=𝑚₂=𝑚₃=1, and 𝑛₁=𝑛₂=𝑛₃=0, system (8) will be stabilized to its equilibrium 𝑂(0,0,0,0,0). Figure 7 shows the time evolution of the states 𝑤₁, 𝑤₂, 𝑤₃, 𝑤₄, and 𝑤₅ of system (8) with controller (25), which illustrates that system (8) is stabilized to the equilibrium 𝑂(0,0,0,0,0).

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Figure 7 

Time evolution of the states of system (8).

5. Conclusions

In this paper, we investigate the combination-combination synchronization of four nonlinear complex chaotic systems. Based on the Lyapunov stability theory, corresponding controllers to achieve combination-combination synchronization among four different nonlinear complex chaotic systems are derived. The special cases, such as combination synchronization and projective synchronization, are studied as well. This synchronization scheme has advantages over the usual drive-response synchronization, such as being able to provide greater security in secure communication. In [24], the authors applied combination synchronization in secure communication; the signal was divided into two parts, and each part was transmitted by a different chaotic system (the drive system), which implied that the signal transmitted by this model may have stronger antiattack ability and antitranslated capability than that transmitted by the usual transmission model. When applying the nonlinear complex systems in communications, the complex variables will double the number of variables and can increase the content and security of the transmitted information. Thus combination-combination synchronization of complex nonlinear systems can find better applications in security communication, such as wireless communication [25].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the Youth Foundation of Yunnan University of Nationalities under Grant no. 11QN07, the Natural Science Foundation of Yunnan Province under Grants no. 2009CD019 and no. 2011FZ172, and the Natural Science Foundation of China under Grant no. 61263042.