Combination-Combination Hyperchaos Synchronization of Complex Memristor Oscillator System

Abstract

The combination-combination synchronization scheme is based on combination of multidrive systems and combination of multiresponse systems. In this paper, we investigate combination-combination synchronization of hyperchaotic complex memristor oscillator system. Several sufficient conditions are provided to ascertain the combination of two drive hyperchaotic complex memristor oscillator systems to synchronize the combination of two response hyperchaotic complex memristor oscillator systems. These new conditions improve and extend the existing synchronization results for memristive systems. A numerical example is given to show the feasibility of theoretical results.

1. Introduction

Since the characteristics of passive two-terminal memristor are reviewed, lots of emulators and macromodels have been proposed. One remarkable use for memristor is in complex memristor oscillator systems, where nonlinear oscillators from Chua’s oscillators are replaced by using memristors [1–10]. The complex behaviors for such systems were analyzed in [1–5, 8–10]. When nonlinear memristor is applied to electronic device, the behavior of the new-type device becomes complicated and difficult to predict due to the unique nonlinear mechanism of memristor [11–14]. However, to apply to the practical engineering, its electrical characteristics must be able to be fully understood. So it is vital to understand the characteristics and behaviors of memristive devices. In [1], the transient chaos was studied for a smooth flux-controlled complex memristor oscillator system. In [2], the local and global behavior of the basic oscillator on complex memristor oscillator system was investigated. Itoh and Chua [3] discussed some interesting oscillation properties and rich nonlinear dynamics of several complex memristor oscillator systems from Chua’s oscillators. Talukdar et al. [7] reported the nonlinear dynamics of three memristor-based phase shift oscillators. As we all know, the electrical characteristics of complex memristor oscillator system can vary depending upon its circuit structure. The internal behavior of high order (fifth-order or more) complex memristor oscillator system still is not being analyzed [9, 10]. In high order complex memristor oscillator system, there are still many unknown fields waiting for us to exploit them. Specifically, the current studies about complex memristor oscillator system have mostly remained at the basic chaotic dynamic theory. Compared with common chaos with only one positive Lyapunov exponent, hyperchaos with more than one positive Lyapunov exponents can exhibit multidirectional expansion and extremely complex behaviors. Can we design high order complex memristor oscillator system to achieve the hyperchaotic effect? If this is really so, many different applications might be proposed, such as ultradense nonvolatile memory and high-performance secure communication [9, 10].

The topic of chaos synchronization of oscillator systems is extensively investigated because of possible relevance to information encryption [8–11, 15–28]. Various kinds of synchronization laws have been obtained, for example, complete synchronization [8, 11, 15–18], compound synchronization [10], antisynchronization [19–21], phase synchronization [22, 23], lag synchronization [24–26], projective synchronization [10, 27], and combination synchronization [9, 28]. However, in the existing literature, most of synchronization laws are based on one drive system and one response system. In this way, when chaos synchronization is applied to secure communication, the information signal is transmitted by only one chaotic system, which is not suitable for very high-performance secure communication. Can the information signal be transmitted by the combination of multiple chaotic systems? If we can successfully provide some ways to do this, then the transmitted signals will be more complex and unpredictable, and thereby the security of communication can be effectively enhanced. Thus, a theoretical question is whether we can design a synchronization scheme on the combination of multidrive systems and combination of multiresponse systems. Such a combination design may help to improve the secrecy in communications [9, 28].

Motivated by the above discussions, in this paper, firstly, a new hyperchaotic complex memristor oscillator system is introduced and studied. Secondly, we bring in the scheme of combination-combination synchronization and establish some synchronization criteria. Roughly stated, the main advantages of this paper include the following three points.(1)Hyperchaotic complex memristor oscillator system is systematically investigated. Hyperchaos is often considered better than common chaos in many engineering fields. Theoretically, hyperchaotic complex memristor oscillator system might improve the security of chaotic communication system.(2)The synchronization control between combination of two drive hyperchaotic complex memristor oscillator systems and combination of two response hyperchaotic complex memristor oscillator systems in drive-response coupled system is investigated. The generalization of synchronization scheme will provide a wider scope for the designs and applications of memristive system.(3)The proposed scheme of combination-combination synchronization in this paper can be applied to the general nonlinear systems.

The rest of this paper is arranged as follows. Section 2 describes some preliminaries and problem formulation. Section 3 derives some sufficient conditions on combination-combination synchronization for hyperchaotic complex memristor oscillator system and then presents a numerical example to demonstrate the validity of the theoretical results. Section 4 concludes the paper with some remarks.

2. Preliminaries

Consider a fifth-order complex memristor oscillator system with its dynamics described by the following ordinary differential equations: 𝑣̇₁(𝑡)=1𝐶₁ℓ₁(𝑡)−1𝐶₁𝑊₁𝜑(𝑡)𝑣₁(𝑡),𝑣̇₂(𝑡)1=−𝐶₂ℓ₁(𝑡)+1𝐶₂ℓ₂(𝑡),ℓ₁̇(𝑡)=1𝐿₁𝑣₂(𝑡)−1𝐿₁𝑣₁(𝑡)−𝑅𝐿₁ℓ₁(𝑡),ℓ₂̇(𝑡)=−1𝐿₂𝑣₂(𝑡),𝜑̇(𝑡)=𝑣₁(𝑡), (1) where 𝑣₁(𝑡) and 𝑣₂(𝑡) denote voltages, ℓ₁(𝑡) and ℓ₂(𝑡) represent currents, 𝐶₁ and 𝐶₂ denote capacitors, 𝐿₁ and 𝐿₂ represent inductors, 𝑊(𝜑(𝑡)) is memductance function, and 𝜑(𝑡) and 𝑅 are magnetic flux and resistor, respectively.

Similarly as in [1, 9, 10], a cubic memristor is chosen; then 𝑊𝜑(𝑡)=𝑎+3𝑏𝜑(𝑡)², (2) where 𝑎 and 𝑏 are parameters.

From (1) and (2), 𝑣̇₁(𝑡)=1𝐶₁ℓ₁(𝑡)−𝑎𝐶₁𝑣₁(𝑡)−3𝑏𝐶₁𝑣₁(𝑡)𝜑(𝑡)²,𝑣̇₂(𝑡)1=−𝐶₂ℓ₁(𝑡)+1𝐶₂ℓ₂(𝑡),ℓ₁̇(𝑡)=1𝐿₁𝑣₂(𝑡)−1𝐿₁𝑣₁(𝑡)−𝑅𝐿₁ℓ₁(𝑡),ℓ₂̇(𝑡)=−1𝐿₂𝑣₂(𝑡),𝜑̇(𝑡)=𝑣₁(𝑡). (3)

Let 𝑥₁₁(𝑡)=𝑣₁(𝑡), 𝑥₁₂(𝑡)=𝑣₂(𝑡), 𝑥₁₃(𝑡)=ℓ₁(𝑡), 𝑥₁₄(𝑡)=ℓ₂(𝑡), 𝑥₁₅(𝑡)=𝜑(𝑡), 𝛼₁=1/𝐶₁, 𝛼₂=𝑎/𝐶₁, 𝛼₃=3𝑏/𝐶₁, 𝛼₄=1/𝐶₂, 𝛼₅=1/𝐿₁, 𝛼₆=𝑅/𝐿₁, 𝛼₇=1/𝐿₂; then (3) can be rewritten as 𝑥̇₁₁=𝛼₁𝑥₁₃−𝛼₂𝑥₁₁−𝛼₃𝑥₁₁𝑥²₁₅,𝑥̇₁₂=−𝛼₄𝑥₁₃+𝛼₄𝑥₁₄,𝑥̇₁₃=𝛼₅𝑥₁₂−𝛼₅𝑥₁₁−𝛼₆𝑥₁₃,𝑥̇₁₄=−𝛼₇𝑥₁₂,𝑥̇₁₅=𝑥₁₁. (4)

Let parameters 𝛼₁=9, 𝛼₂=−10.8, 𝛼₃=10.8, 𝛼₄=1, 𝛼₅=30, 𝛼₆=30, 𝛼₇=15, the initial state 𝑥₁₁(0)=0.01, 𝑥₁₂(0)=0.01, 𝑥₁₃(0)=0.01, 𝑥₁₄(0)=0.01, 𝑥₁₅(0)=0.01, by mean of computer program with MATLAB, the corresponding Lyapunov exponents of system (4) are 0.279403, 0.023752, −0.038779, −7.391447, −16.291929. The numerical result is shown in Figure 1, where the first two Lyapunov exponents are positive. Clearly, it implies that memristor oscillator system (4) is hyperchaotic. Figure 2 describes the hyperchaotic attractors.

Figure 1 

Dynamics of Lyapunov exponents from memristor oscillator system (4).

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2 

3D projections of the hyperchaotic attractor from memristor oscillator system (4).

Many efforts have been paid in recent years to develop efficient circuit implementation for the generation of hyperchaotic memristor oscillator system. However, to the best of our knowledge, up to now, the hyperchaotic memristor oscillator system is still very unusual. Thus, hyperchaotic memristor oscillator system (4) is important for our understanding of hyperchaotic memristive devices.

In the following, we end this section with the scheme of combination-combination synchronization, which will be used in subsequent section.

In engineering field, generally, combination-combination synchronization is based on multiple drive systems and multiple response systems. In many engineering applications and hardware implementations, combination-combination synchronization constituting of two drive systems and two response systems is usually considered. Figure 3 describes a schematic diagram of combination-combination synchronization scheme. In fact, combination-combination synchronization has its unique physical interpretation. For example, in Figure 3, the combination of two drive systems generates the resultant signal; then the combination of two response systems tracks the resultant signal.

Figure 3 

The physical realization of combination-combination synchronization scheme.

Next, we give specific mathematical descriptions of combination-combination synchronization scheme.

Consider the drive system: 𝜒̇₁=𝑓₁𝜒₁. (5)

The other drive system is given by 𝜒̇₂=𝑓₂𝜒₂, (6) and two response systems are described by 𝜒̇₃=𝑓₃𝜒₃𝜒̇+𝑢,₄=𝑓₄𝜒₄+𝑢^∗, (7) where state vectors 𝜒₁=(𝜒₁₁,𝜒₁₂,…,𝜒_1𝑛)^𝑇, 𝜒₂=(𝜒₂₁,𝜒₂₂,…,𝜒_2𝑛)^𝑇, 𝜒₃=(𝜒₃₁,𝜒₃₂,…,𝜒_3𝑛)^𝑇, and 𝜒₄=(𝜒₄₁,𝜒₄₂,…,𝜒_4𝑛)^𝑇 and vector functions 𝑓₁(⋅),𝑓₂(⋅),𝑓₃(⋅),𝑓₄(⋅)∶ℜ^𝑛→ℜ^𝑛, 𝑢=(𝑢₁,𝑢₂,…,𝑢_𝑛)^𝑇, 𝑢^∗=(𝑢^∗₁,𝑢^∗₂,…,𝑢^∗_𝑛)^𝑇∶ℜ^𝑛×ℜ^𝑛×⋯×ℜ^𝑛→ℜ^𝑛 are the appropriate control inputs that will be designed in order to obtain a certain control objective.

Definition 1. The drive systems (5) and (6) are realized combination-combination synchronization with the response systems (7) if there exist 𝑛-dimensional constant diagonal matrices 𝐴₁,𝐴₂,𝐴₃≠0, and 𝐴₄≠0 such that lim_𝑡→+∞‖𝑒‖=lim_𝑡→+∞𝐴₁𝑋₁+𝐴₂𝑋₂−𝐴₃𝑋₃−𝐴₄𝑋₄=0, (8) where ‖⋅‖ is vector norm, 𝑒=(𝑒₁,𝑒₂,…,𝑒_𝑛)^𝑇 is the synchronization error vector, 𝑋₁=diag(𝜒₁₁,𝜒₁₂,…,𝜒_1𝑛), 𝑋₂=diag(𝜒₂₁,𝜒₂₂,…,𝜒_2𝑛), 𝑋₃=diag(𝜒₃₁,𝜒₃₂,…,𝜒_3𝑛), and 𝑋₄=diag(𝜒₄₁,𝜒₄₂,…,𝜒_4𝑛).

Remark 2. As stated earlier, according to Definition 1, the physical implication of combination-combination synchronization is rather intuitionistic. The resultant signal of the combination of two drive systems (5) and (6) is tracked by the synthetic signal of the combination of two response systems (7).

Remark 3. In [28], Sun et al. apply combination-combination design to study the synchronization control of some classical chaotic systems. Generally speaking, this synchronization method for chaotic system is quite novel.

Remark 4. In Definition 1, matrices 𝐴₁, 𝐴₂, 𝐴₃, and 𝐴₄ are often called the scaling matrices. It is not hard to find that the scheme of combination-combination synchronization in Definition 1 contains the combination of two drive systems and the combination of two response systems. Moreover, one advantage of combination-combination synchronization is that the drive systems or the response systems can be completely identical or different.

Remark 5. The scheme of combination-combination synchronization improves and generalizes some existing synchronization schemes. When scaling matrices 𝐴₃=0 or 𝐴₄=0, the combination-combination synchronization will degrade into combination synchronization. When scaling matrices 𝐴₁=𝐴₃=0 or 𝐴₁=𝐴₄=0 or 𝐴₂=𝐴₃=0 or 𝐴₂=𝐴₄=0, the combination-combination synchronization will be reduced to complete synchronization. When scaling matrices 𝐴₁=𝐴₂=𝐴₃=0 or 𝐴₁=𝐴₂=𝐴₄=0, the combination-combination synchronization will change into chaos control.

3. Synchronization Criteria

In this section, we will devote to investigate combination-combination synchronization of hyperchaotic complex memristor oscillator system.

Corresponding to (4), the other drive system is given as 𝑥̇₂₁=𝛽₁𝑥₂₃−𝛽₂𝑥₂₁−𝛽₃𝑥₂₁𝑥²₂₅,𝑥̇₂₂=−𝛽₄𝑥₂₃+𝛽₄𝑥₂₄,𝑥̇₂₃=𝛽₅𝑥₂₂−𝛽₅𝑥₂₁−𝛽₆𝑥₂₃,𝑥̇₂₄=−𝛽₇𝑥₂₂,𝑥̇₂₅=𝑥₂₁, (9) and two response systems are described as follows: 𝑥̇₃₁=𝛾₁𝑥₃₃−𝛾₂𝑥₃₁−𝛾₃𝑥₃₁𝑥²₃₅+𝑢₁,𝑥̇₃₂=−𝛾₄𝑥₃₃+𝛾₄𝑥₃₄+𝑢₂,𝑥̇₃₃=𝛾₅𝑥₃₂−𝛾₅𝑥₃₁−𝛾₆𝑥₃₃+𝑢₃,𝑥̇₃₄=−𝛾₇𝑥₃₂+𝑢₄,𝑥̇₃₅=𝑥₃₁+𝑢₅,𝑥̇₄₁=𝜂₁𝑥₄₃−𝜂₂𝑥₄₁−𝜂₃𝑥₄₁𝑥²₄₅+𝑢^∗₁,𝑥̇₄₂=−𝜂₄𝑥₄₃+𝜂₄𝑥₄₄+𝑢^∗₂,𝑥̇₄₃=𝜂₅𝑥₄₂−𝜂₅𝑥₄₁−𝜂₆𝑥₄₃+𝑢^∗₃,𝑥̇₄₄=−𝜂₇𝑥₄₂+𝑢^∗₄,𝑥̇₄₅=𝑥₄₁+𝑢^∗₅, (10)(11) where 𝛽₁, 𝛽₂, 𝛽₃, 𝛽₄, 𝛽₅, 𝛽₆, 𝛽₇, 𝛾₁, 𝛾₂, 𝛾₃, 𝛾₄, 𝛾₅, 𝛾₆, 𝛾₇, 𝜂₁, 𝜂₂, 𝜂₃, 𝜂₄, 𝜂₅, 𝜂₆, and 𝜂₇ are parameters and 𝑢₁, 𝑢₂, 𝑢₃, 𝑢₄, 𝑢₅, 𝑢^∗₁, 𝑢^∗₂, 𝑢^∗₃, 𝑢^∗₄, and 𝑢^∗₅ are the control inputs to be designed.

In our synchronization scheme, let 𝐴₁=diag𝑎₁₁,𝑎₁₂,𝑎₁₃,𝑎₁₄,𝑎₁₅,𝐴₂=diag𝑎₂₁,𝑎₂₂,𝑎₂₃,𝑎₂₄,𝑎₂₅,𝐴₃=diag𝑎₃₁,𝑎₃₂,𝑎₃₃,𝑎₃₄,𝑎₃₅,𝐴₄=diag𝑎₄₁,𝑎₄₂,𝑎₄₃,𝑎₄₄,𝑎₄₅. (12) Thus, 𝑒₁=𝑎₁₁𝑥₁₁+𝑎₂₁𝑥₂₁−𝑎₃₁𝑥₃₁−𝑎₄₁𝑥₄₁,𝑒₂=𝑎₁₂𝑥₁₂+𝑎₂₂𝑥₂₂−𝑎₃₂𝑥₃₂−𝑎₄₂𝑥₄₂,𝑒₃=𝑎₁₃𝑥₁₃+𝑎₂₃𝑥₂₃−𝑎₃₃𝑥₃₃−𝑎₄₃𝑥₄₃,𝑒₄=𝑎₁₄𝑥₁₄+𝑎₂₄𝑥₂₄−𝑎₃₄𝑥₃₄−𝑎₄₄𝑥₄₄,𝑒₅=𝑎₁₅𝑥₁₅+𝑎₂₅𝑥₂₅−𝑎₃₅𝑥₃₅−𝑎₄₅𝑥₄₅. (13)

Obviously, we have 𝑒₁̇=𝑎₁₁𝑥̇₁₁+𝑎₂₁𝑥̇₂₁−𝑎₃₁𝑥̇₃₁−𝑎₄₁𝑥̇₄₁,𝑒₂̇=𝑎₁₂𝑥̇₁₂+𝑎₂₂𝑥̇₂₂−𝑎₃₂𝑥̇₃₂−𝑎₄₂𝑥̇₄₂,𝑒₃̇=𝑎₁₃𝑥̇₁₃+𝑎₂₃𝑥̇₂₃−𝑎₃₃𝑥̇₃₃−𝑎₄₃𝑥̇₄₃,𝑒₄̇=𝑎₁₄𝑥̇₁₄+𝑎₂₄𝑥̇₂₄−𝑎₃₄𝑥̇₃₄−𝑎₄₄𝑥̇₄₄,𝑒₅̇=𝑎₁₅𝑥̇₁₅+𝑎₂₅𝑥̇₂₅−𝑎₃₅𝑥̇₃₅−𝑎₄₅𝑥̇₄₅. (14)

Combining with (4) and (9)–(11), then the synchronization error system (14) can be transformed into the following form: 𝑒₁̇=𝑎₁₁𝛼₁𝑥₁₃−𝛼₂𝑥₁₁−𝛼₃𝑥₁₁𝑥²₁₅+𝑎₂₁𝛽₁𝑥₂₃−𝛽₂𝑥₂₁−𝛽₃𝑥₂₁𝑥²₂₅−𝑎₃₁𝛾₁𝑥₃₃−𝛾₂𝑥₃₁−𝛾₃𝑥₃₁𝑥²₃₅+𝑢₁−𝑎₄₁𝜂₁𝑥₄₃−𝜂₂𝑥₄₁−𝜂₃𝑥₄₁𝑥²₄₅+𝑢^∗₁,𝑒₂̇=𝑎₁₂−𝛼₄𝑥₁₃+𝛼₄𝑥₁₄+𝑎₂₂−𝛽₄𝑥₂₃+𝛽₄𝑥₂₄−𝑎₃₂−𝛾₄𝑥₃₃+𝛾₄𝑥₃₄+𝑢₂−𝑎₄₂−𝜂₄𝑥₄₃+𝜂₄𝑥₄₄+𝑢^∗₂,𝑒₃̇=𝑎₁₃𝛼₅𝑥₁₂−𝛼₅𝑥₁₁−𝛼₆𝑥₁₃+𝑎₂₃𝛽₅𝑥₂₂−𝛽₅𝑥₂₁−𝛽₆𝑥₂₃−𝑎₃₃𝛾₅𝑥₃₂−𝛾₅𝑥₃₁−𝛾₆𝑥₃₃+𝑢₃−𝑎₄₃𝜂₅𝑥₄₂−𝜂₅𝑥₄₁−𝜂₆𝑥₄₃+𝑢^∗₃,𝑒₄̇=𝑎₁₄−𝛼₇𝑥₁₂+𝑎₂₄−𝛽₇𝑥₂₂−𝑎₃₄−𝛾₇𝑥₃₂+𝑢₄−𝑎₄₄−𝜂₇𝑥₄₂+𝑢^∗₄,𝑒₅̇=𝑎₁₅𝑥₁₁+𝑎₂₅𝑥₂₁−𝑎₃₅𝑥₃₁+𝑢₅−𝑎₄₅𝑥₄₁+𝑢^∗₅. (15)

Denote 𝑈₁=𝑎₃₁𝑢₁+𝑎₄₁𝑢^∗₁,𝑈₂=𝑎₃₂𝑢₂+𝑎₄₂𝑢^∗₂,𝑈₃=𝑎₃₃𝑢₃+𝑎₄₃𝑢^∗₃,𝑈₄=𝑎₃₄𝑢₄+𝑎₄₄𝑢^∗₄,𝑈₅=𝑎₃₅𝑢₅+𝑎₄₅𝑢^∗₅. (16) Then (15) can be rewritten as 𝑒₁̇=𝑎₁₁𝛼₁𝑥₁₃−𝛼₂𝑥₁₁−𝛼₃𝑥₁₁𝑥²₁₅+𝑎₂₁𝛽₁𝑥₂₃−𝛽₂𝑥₂₁−𝛽₃𝑥₂₁𝑥²₂₅−𝑎₃₁𝛾₁𝑥₃₃−𝛾₂𝑥₃₁−𝛾₃𝑥₃₁𝑥²₃₅−𝑎₄₁𝜂₁𝑥₄₃−𝜂₂𝑥₄₁−𝜂₃𝑥₄₁𝑥²₄₅−𝑈₁,𝑒₂̇=𝑎₁₂−𝛼₄𝑥₁₃+𝛼₄𝑥₁₄+𝑎₂₂−𝛽₄𝑥₂₃+𝛽₄𝑥₂₄−𝑎₃₂−𝛾₄𝑥₃₃+𝛾₄𝑥₃₄−𝑎₄₂−𝜂₄𝑥₄₃+𝜂₄𝑥₄₄−𝑈₂,𝑒₃̇=𝑎₁₃𝛼₅𝑥₁₂−𝛼₅𝑥₁₁−𝛼₆𝑥₁₃+𝑎₂₃𝛽₅𝑥₂₂−𝛽₅𝑥₂₁−𝛽₆𝑥₂₃−𝑎₃₃𝛾₅𝑥₃₂−𝛾₅𝑥₃₁−𝛾₆𝑥₃₃−𝑎₄₃𝜂₅𝑥₄₂−𝜂₅𝑥₄₁−𝜂₆𝑥₄₃−𝑈₃,𝑒₄̇=−𝑎₁₄𝛼₇𝑥₁₂−𝑎₂₄𝛽₇𝑥₂₂+𝑎₃₄𝛾₇𝑥₃₂+𝑎₄₄𝜂₇𝑥₄₂−𝑈₄,𝑒₅̇=𝑎₁₅𝑥₁₁+𝑎₂₅𝑥₂₁−𝑎₃₅𝑥₃₁−𝑎₄₅𝑥₄₁−𝑈₅. (17)

In fact, (16) is called the controller to be designed.

Theorem 6. If the control law is chosen as follows: 𝑈₁=𝑎₁₁𝛼₁𝑥₁₃+1−𝛼₂𝑥₁₁−𝛼₃𝑥₁₁𝑥²₁₅+𝑎₂₁𝛽₁𝑥₂₃+1−𝛽₂𝑥₂₁−𝛽₃𝑥₂₁𝑥²₂₅−𝑎₃₁𝛾₁𝑥₃₃+1−𝛾₂𝑥₃₁−𝛾₃𝑥₃₁𝑥²₃₅−𝑎₄₁𝜂₁𝑥₄₃+1−𝜂₂𝑥₄₁−𝜂₃𝑥₄₁𝑥²₄₅+𝑎₁₂𝑥₁₂+𝑎₂₂𝑥₂₂−𝑎₃₂𝑥₃₂−𝑎₄₂𝑥₄₂,𝑈₂=𝑎₁₂𝑥₁₂−𝛼₄𝑥₁₃+𝛼₄𝑥₁₄+𝑎₂₂𝑥₂₂−𝛽₄𝑥₂₃+𝛽₄𝑥₂₄−𝑎₃₂𝑥₃₂−𝛾₄𝑥₃₃+𝛾₄𝑥₃₄−𝑎₄₂𝑥₄₂−𝜂₄𝑥₄₃+𝜂₄𝑥₄₄−𝑎₁₁𝑥₁₁+𝑎₂₁𝑥₂₁−𝑎₃₁𝑥₃₁−𝑎₄₁𝑥₄₁+𝑎₁₃𝑥₁₃+𝑎₂₃𝑥₂₃−𝑎₃₃𝑥₃₃−𝑎₄₃𝑥₄₃,𝑈₃=𝑎₁₃𝛼₅𝑥₁₂−𝛼₅𝑥₁₁+1−𝛼₆𝑥₁₃+𝑎₂₃𝛽₅𝑥₂₂−𝛽₅𝑥₂₁+1−𝛽₆𝑥₂₃−𝑎₃₃𝛾₅𝑥₃₂−𝛾₅𝑥₃₁+1−𝛾₆𝑥₃₃−𝑎₄₃𝜂₅𝑥₄₂−𝜂₅𝑥₄₁+1−𝜂₆𝑥₄₃−𝑎₁₂𝑥₁₂+𝑎₂₂𝑥₂₂−𝑎₃₂𝑥₃₂−𝑎₄₂𝑥₄₂+𝑎₁₄𝑥₁₄+𝑎₂₄𝑥₂₄−𝑎₃₄𝑥₃₄−𝑎₄₄𝑥₄₄,𝑈₄=−𝑎₁₃𝑥₁₃+𝑎₂₃𝑥₂₃−𝑎₃₃𝑥₃₃−𝑎₄₃𝑥₄₃+𝑎₁₄𝑥₁₄−𝛼₇𝑥₁₂+𝑎₂₄𝑥₂₄−𝛽₇𝑥₂₂−𝑎₃₄𝑥₃₄−𝛾₇𝑥₃₂−𝑎₄₄𝑥₄₄−𝜂₇𝑥₄₂+𝑎₁₅𝑥₁₅+𝑎₂₅𝑥₂₅−𝑎₃₅𝑥₃₅−𝑎₄₅𝑥₄₅,𝑈₅=𝑎₁₅𝑥₁₁+𝑥₁₅+𝑎₂₅𝑥₂₁+𝑥₂₅−𝑎₃₅𝑥₃₁+𝑥₃₅−𝑎₄₅𝑥₄₁+𝑥₄₅−𝑎₁₄𝑥₁₄+𝑎₂₄𝑥₂₄−𝑎₃₄𝑥₃₄−𝑎₄₄𝑥₄₄, (18) then the drive systems (4) and (9) will achieve combination-combination synchronization with the response systems (10) and (11).

Proof. Consider the following Lyapunov function: 𝑉(𝑒(𝑡))=𝑉𝑒₁,𝑒₂,𝑒₃,𝑒₄,𝑒₅=12𝑒²₁+𝑒²₂+𝑒²₃+𝑒²₄+𝑒²₅. (19)
Calculating the upper right Dini derivative of 𝑉 along the trajectory of (17), then 𝐷⁺𝑉=𝑒₁𝑒₁̇+𝑒₂𝑒₂̇+𝑒₃𝑒₃̇+𝑒₄𝑒₄̇+𝑒₅𝑒₅̇=𝑒₁𝑎₁₁𝛼₁𝑥₁₃−𝛼₂𝑥₁₁−𝛼₃𝑥₁₁𝑥²₁₅+𝑎₂₁𝛽₁𝑥₂₃−𝛽₂𝑥₂₁−𝛽₃𝑥₂₁𝑥²₂₅−𝑎₃₁𝛾₁𝑥₃₃−𝛾₂𝑥₃₁−𝛾₃𝑥₃₁𝑥²₃₅−𝑎₄₁𝜂₁𝑥₄₃−𝜂₂𝑥₄₁−𝜂₃𝑥₄₁𝑥²₄₅−𝑈₁+𝑒₂𝑎₁₂−𝛼₄𝑥₁₃+𝛼₄𝑥₁₄+𝑎₂₂−𝛽₄𝑥₂₃+𝛽₄𝑥₂₄−𝑎₃₂−𝛾₄𝑥₃₃+𝛾₄𝑥₃₄−𝑎₄₂−𝜂₄𝑥₄₃+𝜂₄𝑥₄₄−𝑈₂+𝑒₃𝑎₁₃𝛼₅𝑥₁₂−𝛼₅𝑥₁₁−𝛼₆𝑥₁₃+𝑎₂₃𝛽₅𝑥₂₂−𝛽₅𝑥₂₁−𝛽₆𝑥₂₃−𝑎₃₃𝛾₅𝑥₃₂−𝛾₅𝑥₃₁−𝛾₆𝑥₃₃−𝑎₄₃𝜂₅𝑥₄₂−𝜂₅𝑥₄₁−𝜂₆𝑥₄₃−𝑈₃+𝑒₄−𝑎₁₄𝛼₇𝑥₁₂−𝑎₂₄𝛽₇𝑥₂₂+𝑎₃₄𝛾₇𝑥₃₂+𝑎₄₄𝜂₇𝑥₄₂−𝑈₄+𝑒₅𝑎₁₅𝑥₁₁+𝑎₂₅𝑥₂₁−𝑎₃₅𝑥₃₁−𝑎₄₅𝑥₄₁−𝑈₅=𝑒₁Δ₁−𝑈₁+𝑒₂Δ₂−𝑈₂+𝑒₃Δ₃−𝑈₃+𝑒₄Δ₄−𝑈₄+𝑒₅Δ₅−𝑈₅, (20) where Δ₁=𝑎₁₁𝛼₁𝑥₁₃−𝛼₂𝑥₁₁−𝛼₃𝑥₁₁𝑥²₁₅+𝑎₂₁𝛽₁𝑥₂₃−𝛽₂𝑥₂₁−𝛽₃𝑥₂₁𝑥²₂₅−𝑎₃₁𝛾₁𝑥₃₃−𝛾₂𝑥₃₁−𝛾₃𝑥₃₁𝑥²₃₅−𝑎₄₁𝜂₁𝑥₄₃−𝜂₂𝑥₄₁−𝜂₃𝑥₄₁𝑥²₄₅,Δ₂=𝑎₁₂−𝛼₄𝑥₁₃+𝛼₄𝑥₁₄+𝑎₂₂−𝛽₄𝑥₂₃+𝛽₄𝑥₂₄−𝑎₃₂−𝛾₄𝑥₃₃+𝛾₄𝑥₃₄−𝑎₄₂−𝜂₄𝑥₄₃+𝜂₄𝑥₄₄,Δ₃=𝑎₁₃𝛼₅𝑥₁₂−𝛼₅𝑥₁₁−𝛼₆𝑥₁₃+𝑎₂₃𝛽₅𝑥₂₂−𝛽₅𝑥₂₁−𝛽₆𝑥₂₃−𝑎₃₃𝛾₅𝑥₃₂−𝛾₅𝑥₃₁−𝛾₆𝑥₃₃−𝑎₄₃𝜂₅𝑥₄₂−𝜂₅𝑥₄₁−𝜂₆𝑥₄₃,Δ₄=−𝑎₁₄𝛼₇𝑥₁₂−𝑎₂₄𝛽₇𝑥₂₂+𝑎₃₄𝛾₇𝑥₃₂+𝑎₄₄𝜂₇𝑥₄₂,Δ₅=𝑎₁₅𝑥₁₁+𝑎₂₅𝑥₂₁−𝑎₃₅𝑥₃₁−𝑎₄₅𝑥₄₁. (21)
On the basis of (18), we have Δ₁−𝑈₁=−𝑎₁₁𝑥₁₁+𝑎₂₁𝑥₂₁−𝑎₃₁𝑥₃₁−𝑎₄₁𝑥₄₁−𝑎₁₂𝑥₁₂+𝑎₂₂𝑥₂₂−𝑎₃₂𝑥₃₂−𝑎₄₂𝑥₄₂,Δ₂−𝑈₂=𝑎₁₁𝑥₁₁+𝑎₂₁𝑥₂₁−𝑎₃₁𝑥₃₁−𝑎₄₁𝑥₄₁−𝑎₁₂𝑥₁₂+𝑎₂₂𝑥₂₂−𝑎₃₂𝑥₃₂−𝑎₄₂𝑥₄₂−𝑎₁₃𝑥₁₃+𝑎₂₃𝑥₂₃−𝑎₃₃𝑥₃₃−𝑎₄₃𝑥₄₃,Δ₃−𝑈₃=𝑎₁₂𝑥₁₂+𝑎₂₂𝑥₂₂−𝑎₃₂𝑥₃₂−𝑎₄₂𝑥₄₂−𝑎₁₃𝑥₁₃+𝑎₂₃𝑥₂₃−𝑎₃₃𝑥₃₃−𝑎₄₃𝑥₄₃−𝑎₁₄𝑥₁₄+𝑎₂₄𝑥₂₄−𝑎₃₄𝑥₃₄−𝑎₄₄𝑥₄₄,Δ₄−𝑈₄=𝑎₁₃𝑥₁₃+𝑎₂₃𝑥₂₃−𝑎₃₃𝑥₃₃−𝑎₄₃𝑥₄₃−𝑎₁₄𝑥₁₄+𝑎₂₄𝑥₂₄−𝑎₃₄𝑥₃₄−𝑎₄₄𝑥₄₄−𝑎₁₅𝑥₁₅+𝑎₂₅𝑥₂₅−𝑎₃₅𝑥₃₅−𝑎₄₅𝑥₄₅,Δ₅−𝑈₅=𝑎₁₄𝑥₁₄+𝑎₂₄𝑥₂₄−𝑎₃₄𝑥₃₄−𝑎₄₄𝑥₄₄−𝑎₁₅𝑥₁₅+𝑎₂₅𝑥₂₅−𝑎₃₅𝑥₃₅−𝑎₄₅𝑥₄₅. (22) That is, Δ₁−𝑈₁=−𝑒₁−𝑒₂,Δ₂−𝑈₂=𝑒₁−𝑒₂−𝑒₃,Δ₃−𝑈₃=𝑒₂−𝑒₃−𝑒₄,Δ₄−𝑈₄=𝑒₃−𝑒₄−𝑒₅,Δ₅−𝑈₅=𝑒₄−𝑒₅. (23)
Together with (20) and (23), then 𝐷⁺𝑉=𝑒₁−𝑒₁−𝑒₂+𝑒₂𝑒₁−𝑒₂−𝑒₃+𝑒₃𝑒₂−𝑒₃−𝑒₄+𝑒₄𝑒₃−𝑒₄−𝑒₅+𝑒₅𝑒₄−𝑒₅=−𝑒²₁−𝑒²₂−𝑒²₃−𝑒²₄−𝑒²₅=−𝑒^𝑇𝑒, (24) where 𝑒=(𝑒₁,𝑒₂,𝑒₃,𝑒₄,𝑒₅)^𝑇.
Let 𝑡>0 be arbitrarily given; integrating the above Equation (24) from 0 to 𝑡, then ^𝑡₀‖𝑒(𝑠)‖²d𝑠=^𝑡₀−𝑉̇d𝑠=𝑉(𝑒(0))−𝑉(𝑒(𝑡))≤𝑉(𝑒(0)), (25) where ‖⋅‖ is the Euclidean vector norm.
According to Barbalat’s lemma, we have ‖𝑒(𝑡)‖²→0 as 𝑡→+∞. Hence, (𝑒₁,𝑒₂,𝑒₃,𝑒₄,𝑒₅)→(0,0,0,0,0) as 𝑡→+∞. It implies that the drive systems (4) and (9) can achieve combination-combination synchronization with the response systems (10) and (11). The proof is completed.

Remark 7. The nonlinear degree of control input in (18) is somewhat high. In the sense of control theory, the control input in (18) may not be the best. However, in many applications, firstly one often needs to understand whether the control system designed has the desired properties such as the combination-combination synchronization behavior. In fact, as the works in [9, 28], because of the unique construction about combination design, the control input in synchronization scheme via combination design is usually a bit more complicated. How to design easy and efficient synchronization criteria will be the topic of future research.

Remark 8. For the controller design of combination-combination synchronization, once we get proper controller (16), then the 𝑢_𝑖 and 𝑢^∗_𝑖 (𝑖=1,2,3,4,5) in the response systems (10) and (11) can be obtained via 𝑎_3𝑖𝑢_𝑖+𝑎_4𝑖𝑢^∗_𝑖=𝑈_𝑖 (𝑖=1,2,3,4,5). Obviously, these 𝑢_𝑖 and 𝑢^∗_𝑖 (𝑖=1,2,3,4,5) have wide choices, which can provide the designer with the richness of flexibility. Of course, for the whole controller design of combination-combination synchronization, we merely concern the controller (16).

Next we extend Theorem 6 to other possible cases.

Corollary 9. If the control law is chosen as follows: 𝑈₁∶=𝑎₃₁𝑢₁=𝑎₁₁𝛼₁𝑥₁₃+1−𝛼₂𝑥₁₁−𝛼₃𝑥₁₁𝑥²₁₅+𝑎₂₁𝛽₁𝑥₂₃+1−𝛽₂𝑥₂₁−𝛽₃𝑥₂₁𝑥²₂₅−𝑎₃₁𝛾₁𝑥₃₃+1−𝛾₂𝑥₃₁−𝛾₃𝑥₃₁𝑥²₃₅+𝑎₁₂𝑥₁₂+𝑎₂₂𝑥₂₂−𝑎₃₂𝑥₃₂,𝑈₂∶=𝑎₃₂𝑢₂=𝑎₁₂𝑥₁₂−𝛼₄𝑥₁₃+𝛼₄𝑥₁₄+𝑎₂₂𝑥₂₂−𝛽₄𝑥₂₃+𝛽₄𝑥₂₄−𝑎₃₂𝑥₃₂−𝛾₄𝑥₃₃+𝛾₄𝑥₃₄−𝑎₁₁𝑥₁₁+𝑎₂₁𝑥₂₁−𝑎₃₁𝑥₃₁+𝑎₁₃𝑥₁₃+𝑎₂₃𝑥₂₃−𝑎₃₃𝑥₃₃,𝑈₃∶=𝑎₃₃𝑢₃=𝑎₁₃𝛼₅𝑥₁₂−𝛼₅𝑥₁₁+1−𝛼₆𝑥₁₃+𝑎₂₃𝛽₅𝑥₂₂−𝛽₅𝑥₂₁+1−𝛽₆𝑥₂₃−𝑎₃₃𝛾₅𝑥₃₂−𝛾₅𝑥₃₁+1−𝛾₆𝑥₃₃−𝑎₁₂𝑥₁₂+𝑎₂₂𝑥₂₂−𝑎₃₂𝑥₃₂+𝑎₁₄𝑥₁₄+𝑎₂₄𝑥₂₄−𝑎₃₄𝑥₃₄,𝑈₄∶=𝑎₃₄𝑢₄=−𝑎₁₃𝑥₁₃+𝑎₂₃𝑥₂₃−𝑎₃₃𝑥₃₃+𝑎₁₅𝑥₁₅+𝑎₂₅𝑥₂₅−𝑎₃₅𝑥₃₅+𝑎₁₄𝑥₁₄−𝛼₇𝑥₁₂+𝑎₂₄𝑥₂₄−𝛽₇𝑥₂₂−𝑎₃₄𝑥₃₄−𝛾₇𝑥₃₂,𝑈₅∶=𝑎₃₅𝑢₅=𝑎₁₅𝑥₁₁+𝑥₁₅+𝑎₂₅𝑥₂₁+𝑥₂₅−𝑎₃₅𝑥₃₁+𝑥₃₅−𝑎₁₄𝑥₁₄+𝑎₂₄𝑥₂₄−𝑎₃₄𝑥₃₄, (26) then the drive systems (4) and (9) will achieve combination synchronization with the response system (10).

Corollary 10. If the control law is chosen as follows: 𝑈₁∶=𝑎₄₁𝑢^∗₁=𝑎₁₁𝛼₁𝑥₁₃+1−𝛼₂𝑥₁₁−𝛼₃𝑥₁₁𝑥²₁₅+𝑎₂₁𝛽₁𝑥₂₃+1−𝛽₂𝑥₂₁−𝛽₃𝑥₂₁𝑥²₂₅−𝑎₄₁𝜂₁𝑥₄₃+1−𝜂₂𝑥₄₁−𝜂₃𝑥₄₁𝑥²₄₅+𝑎₁₂𝑥₁₂+𝑎₂₂𝑥₂₂−𝑎₄₂𝑥₄₂,𝑈₂∶=𝑎₄₂𝑢^∗₂=𝑎₁₂𝑥₁₂−𝛼₄𝑥₁₃+𝛼₄𝑥₁₄+𝑎₂₂𝑥₂₂−𝛽₄𝑥₂₃+𝛽₄𝑥₂₄−𝑎₄₂𝑥₄₂−𝜂₄𝑥₄₃+𝜂₄𝑥₄₄−𝑎₁₁𝑥₁₁+𝑎₂₁𝑥₂₁−𝑎₄₁𝑥₄₁+𝑎₁₃𝑥₁₃+𝑎₂₃𝑥₂₃−𝑎₄₃𝑥₄₃,𝑈₃∶=𝑎₄₃𝑢^∗₃=𝑎₁₃𝛼₅𝑥₁₂−𝛼₅𝑥₁₁+1−𝛼₆𝑥₁₃+𝑎₂₃𝛽₅𝑥₂₂−𝛽₅𝑥₂₁+1−𝛽₆𝑥₂₃−𝑎₄₃𝜂₅𝑥₄₂−𝜂₅𝑥₄₁+1−𝜂₆𝑥₄₃−𝑎₁₂𝑥₁₂+𝑎₂₂𝑥₂₂−𝑎₄₂𝑥₄₂+𝑎₁₄𝑥₁₄+𝑎₂₄𝑥₂₄−𝑎₄₄𝑥₄₄,𝑈₄∶=𝑎₄₄𝑢^∗₄=−𝑎₁₃𝑥₁₃+𝑎₂₃𝑥₂₃−𝑎₄₃𝑥₄₃+𝑎₁₅𝑥₁₅+𝑎₂₅𝑥₂₅−𝑎₄₅𝑥₄₅+𝑎₁₄𝑥₁₄−𝛼₇𝑥₁₂+𝑎₂₄𝑥₂₄−𝛽₇𝑥₂₂−𝑎₄₄𝑥₄₄−𝜂₇𝑥₄₂,𝑈₅∶=𝑎₄₅𝑢^∗₅=𝑎₁₅𝑥₁₁+𝑥₁₅+𝑎₂₅𝑥₂₁+𝑥₂₅−𝑎₄₅𝑥₄₁+𝑥₄₅−𝑎₁₄𝑥₁₄+𝑎₂₄𝑥₂₄−𝑎₄₄𝑥₄₄, (27) then the drive systems (4) and (9) will achieve combination synchronization with the response system (11).

Corollary 11. If the control law is chosen as follows: 𝑈₁∶=𝑎₃₁𝑢₁=𝑎₁₁𝛼₁𝑥₁₃+1−𝛼₂𝑥₁₁−𝛼₃𝑥₁₁𝑥²₁₅−𝑎₃₁𝛾₁𝑥₃₃+1−𝛾₂𝑥₃₁−𝛾₃𝑥₃₁𝑥²₃₅+𝑎₁₂𝑥₁₂−𝑎₃₂𝑥₃₂,𝑈₂∶=𝑎₃₂𝑢₂=𝑎₁₂𝑥₁₂−𝛼₄𝑥₁₃+𝛼₄𝑥₁₄−𝑎₃₂𝑥₃₂−𝛾₄𝑥₃₃+𝛾₄𝑥₃₄−𝑎₁₁𝑥₁₁−𝑎₃₁𝑥₃₁+𝑎₁₃𝑥₁₃−𝑎₃₃𝑥₃₃,𝑈₃∶=𝑎₃₃𝑢₃=𝑎₁₃𝛼₅𝑥₁₂−𝛼₅𝑥₁₁+1−𝛼₆𝑥₁₃−𝑎₃₃𝛾₅𝑥₃₂−𝛾₅𝑥₃₁+1−𝛾₆𝑥₃₃−𝑎₁₂𝑥₁₂−𝑎₃₂𝑥₃₂+𝑎₁₄𝑥₁₄−𝑎₃₄𝑥₃₄,𝑈₄∶=𝑎₃₄𝑢₄=−𝑎₁₃𝑥₁₃−𝑎₃₃𝑥₃₃+𝑎₁₄𝑥₁₄−𝛼₇𝑥₁₂−𝑎₃₄𝑥₃₄−𝛾₇𝑥₃₂+𝑎₁₅𝑥₁₅−𝑎₃₅𝑥₃₅,𝑈₅∶=𝑎₃₅𝑢₅=𝑎₁₅𝑥₁₁+𝑥₁₅−𝑎₃₅𝑥₃₁+𝑥₃₅−𝑎₁₄𝑥₁₄−𝑎₃₄𝑥₃₄, (28) then the drive system (4) will achieve complete synchronization with the response system (10).

According to Corollary 11, on the complete synchronization criteria between (4) and (11), or (9) and (10), or (9) and (11), one can make some parallel promotions. So we will not repeat all of those corollaries here.

Corollary 12. If the control law is chosen as follows: 𝑈₁∶=𝑎₃₁𝑢₁=−𝑎₃₁𝛾₁𝑥₃₃+1−𝛾₂𝑥₃₁−𝛾₃𝑥₃₁𝑥²₃₅−𝑎₃₂𝑥₃₂,𝑈₂∶=𝑎₃₂𝑢₂=−𝑎₃₂𝑥₃₂−𝛾₄𝑥₃₃+𝛾₄𝑥₃₄+𝑎₃₁𝑥₃₁−𝑎₃₃𝑥₃₃,𝑈₃∶=𝑎₃₃𝑢₃=−𝑎₃₃𝛾₅𝑥₃₂−𝛾₅𝑥₃₁+1−𝛾₆𝑥₃₃+𝑎₃₂𝑥₃₂−𝑎₃₄𝑥₃₄,𝑈₄∶=𝑎₃₄𝑢₄=𝑎₃₃𝑥₃₃−𝑎₃₄𝑥₃₄−𝛾₇𝑥₃₂−𝑎₃₅𝑥₃₅,𝑈₅∶=𝑎₃₅𝑢₅=−𝑎₃₅𝑥₃₁+𝑥₃₅+𝑎₃₄𝑥₃₄, (29) then system (10) is asymptotically stabilizable.

Corollary 13. If the control law is chosen as follows: 𝑈₁∶=𝑎₄₁𝑢^∗₁=−𝑎₄₁𝜂₁𝑥₄₃+1−𝜂₂𝑥₄₁−𝜂₃𝑥₄₁𝑥²₄₅−𝑎₄₂𝑥₄₂,𝑈₂∶=𝑎₄₂𝑢^∗₂=−𝑎₄₂𝑥₄₂−𝜂₄𝑥₄₃+𝜂₄𝑥₄₄+𝑎₄₁𝑥₄₁−𝑎₄₃𝑥₄₃,𝑈₃∶=𝑎₄₃𝑢^∗₃=−𝑎₄₃𝜂₅𝑥₄₂−𝜂₅𝑥₄₁+1−𝜂₆𝑥₄₃+𝑎₄₂𝑥₄₂−𝑎₄₄𝑥₄₄,𝑈₄∶=𝑎₄₄𝑢^∗₄=𝑎₄₃𝑥₄₃−𝑎₄₄𝑥₄₄−𝜂₇𝑥₄₂−𝑎₄₅𝑥₄₅,𝑈₅∶=𝑎₄₅𝑢^∗₅=−𝑎₄₅𝑥₄₁+𝑥₄₅+𝑎₄₄𝑥₄₄, (30) then system (11) is asymptotically stabilizable.

Remark 14. Just like Remark 7, the nonlinear degree of control input in the above corollaries is somewhat high. How to design some less conservative criteria, this issue, will be the topic of future research.

Remark 15. In many existing works, the combination of multiple drive systems may induce the combination states to be asymptotically stable or emanative. And theoretically, a direct consequence of such combination of multiple drive systems is the information signal extraction which is extremely difficult or completely useless. In this paper, the combination of two drive systems can be always hyperchaotic; thus, the dynamic behavior is more abundant and complex (see the subsequent numerical example). In theory, when it is used to chaotic communication, this would greatly enhance the security and reliability for high-performance communication.

Remark 16. Theoretically, in the past chaotic communication via drive-response model, the transmitted signal only depends on one drive system. However, in theory, when designing secure communication via combination-combination synchronization, the transmitted signal and received signal can both be split into multiple different chaotic systems. Compared to some conventional design methods, this design approach maybe have stronger antiattack ability.

In the following, we give a numerical example to illustrate the superiority of theoretical results via computer simulations.

Assume 𝛼₁=𝛽₁=𝛾₁=𝜂₁=9, 𝛼₂=𝛽₂=𝛾₂=𝜂₂=−10.8, 𝛼₃=𝛽₃=𝛾₃=𝜂₃=10.8, 𝛼₄=𝛽₄=𝛾₄=𝜂₄=1, 𝛼₅=𝛽₅=𝛾₅=𝜂₅=30, 𝛼₆=𝛽₆=𝛾₆=𝜂₆=30, 𝛼₇=𝛽₇=𝛾₇=𝜂₇=15. Let 𝐴₁=diag(𝑎₁₁,𝑎₁₂,𝑎₁₃,𝑎₁₄,𝑎₁₅)=diag(1,1,1,1,1), 𝐴₂=diag(𝑎₂₁,𝑎₂₂,𝑎₂₃,𝑎₂₄,𝑎₂₅)=diag(1,1,1,1,1), 𝐴₃=diag(𝑎₃₁,𝑎₃₂,𝑎₃₃,𝑎₃₄,𝑎₃₅)=diag(1,1,1,1,1), 𝐴₄=diag(𝑎₄₁,𝑎₄₂,𝑎₄₃,𝑎₄₄,𝑎₄₅)=diag(1,1,1,1,1), according to Theorem 6; then the control law can be designed as 𝑈₁=9𝑥₁₃+11.8𝑥₁₁−10.8𝑥₁₁𝑥²₁₅+9𝑥₂₃+11.8𝑥₂₁−10.8𝑥₂₁𝑥²₂₅−9𝑥₃₃+11.8𝑥₃₁−10.8𝑥₃₁𝑥²₃₅−9𝑥₄₃+11.8𝑥₄₁−10.8𝑥₄₁𝑥²₄₅+𝑥₁₂+𝑥₂₂−𝑥₃₂−𝑥₄₂,𝑈₂=𝑥₁₂−𝑥₁₃+𝑥₁₄+𝑥₂₂−𝑥₂₃+𝑥₂₄−𝑥₃₂−𝑥₃₃+𝑥₃₄−𝑥₄₂−𝑥₄₃+𝑥₄₄−𝑥₁₁+𝑥₂₁−𝑥₃₁−𝑥₄₁+𝑥₁₃+𝑥₂₃−𝑥₃₃−𝑥₄₃,𝑈₃=30𝑥₁₂−30𝑥₁₁−29𝑥₁₃+30𝑥₂₂−30𝑥₂₁−29𝑥₂₃−30𝑥₃₂−30𝑥₃₁−29𝑥₃₃−30𝑥₄₂−30𝑥₄₁−29𝑥₄₃−𝑥₁₂+𝑥₂₂−𝑥₃₂−𝑥₄₂+𝑥₁₄+𝑥₂₄−𝑥₃₄−𝑥₄₄,𝑈₄=−𝑥₁₃+𝑥₂₃−𝑥₃₃−𝑥₄₃+𝑥₁₄−15𝑥₁₂+𝑥₂₄−15𝑥₂₂−𝑥₃₄−15𝑥₃₂−𝑥₄₄−15𝑥₄₂+𝑥₁₅+𝑥₂₅−𝑥₃₅−𝑥₄₅,𝑈₅=𝑥₁₁+𝑥₁₅+𝑥₂₁+𝑥₂₅−𝑥₃₁+𝑥₃₅−𝑥₄₁+𝑥₄₅−𝑥₁₄+𝑥₂₄−𝑥₃₄−𝑥₄₄. (31)

By (16), it follows that 𝑈₁=𝑢₁+𝑢^∗₁,𝑈₂=𝑢₂+𝑢^∗₂,𝑈₃=𝑢₃+𝑢^∗₃,𝑈₄=𝑢₄+𝑢^∗₄,𝑈₅=𝑢₅+𝑢^∗₅. (32) then 𝑢_𝑖 and 𝑢^∗_𝑖 (𝑖=1,2,3,4,5) can be chosen according to different requirements. For example, it allows 𝑢₁=9𝑥₁₃+11.8𝑥₁₁−10.8𝑥₁₁𝑥²₁₅+9𝑥₂₃+11.8𝑥₂₁−10.8𝑥₂₁𝑥²₂₅−9𝑥₃₃+11.8𝑥₃₁−10.8𝑥₃₁𝑥²₃₅−9𝑥₄₃+11.8𝑥₄₁−10.8𝑥₄₁𝑥²₄₅,𝑢^∗₁=𝑥₁₂+𝑥₂₂−𝑥₃₂−𝑥₄₂,𝑢₂=𝑥₁₂−𝑥₁₃+𝑥₁₄+𝑥₂₂−𝑥₂₃+𝑥₂₄−𝑥₃₂−𝑥₃₃+𝑥₃₄−𝑥₄₂−𝑥₄₃+𝑥₄₄,𝑢^∗₂=−𝑥₁₁+𝑥₂₁−𝑥₃₁−𝑥₄₁+𝑥₁₃+𝑥₂₃−𝑥₃₃−𝑥₄₃,𝑢₃=30𝑥₁₂−30𝑥₁₁−29𝑥₁₃+30𝑥₂₂−30𝑥₂₁−29𝑥₂₃−30𝑥₃₂−30𝑥₃₁−29𝑥₃₃−30𝑥₄₂−30𝑥₄₁−29𝑥₄₃,𝑢^∗₃=−𝑥₁₂+𝑥₂₂−𝑥₃₂−𝑥₄₂+𝑥₁₄+𝑥₂₄−𝑥₃₄−𝑥₄₄,𝑢₄=−𝑥₁₃+𝑥₂₃−𝑥₃₃−𝑥₄₃+𝑥₁₄−15𝑥₁₂+𝑥₂₄−15𝑥₂₂−𝑥₃₄−15𝑥₃₂−𝑥₄₄−15𝑥₄₂,𝑢^∗₄=𝑥₁₅+𝑥₂₅−𝑥₃₅−𝑥₄₅,𝑢₅=𝑥₁₁+𝑥₁₅+𝑥₂₁+𝑥₂₅−𝑥₃₁+𝑥₃₅−𝑥₄₁+𝑥₄₅,𝑢^∗₅=−𝑥₁₄+𝑥₂₄−𝑥₃₄−𝑥₄₄. (33)

Of course, the 𝑢_𝑖 and 𝑢^∗_𝑖  (𝑖=1,2,3,4,5) in (33) are just one case rather than the norm. The designers can choose what they need via the combination-combination controller (31).

Simulation result of the combination of two drive systems (4) and (9) is depicted in Figure 4. The computer simulation suggests that the combination of two drive systems (4) and (9) has a hyperchaotic attractor, as shown in Figure 4, which has verified that the combination of drive systems (4) and (9) remains hyperchaotic. Meanwhile, according to Theorem 6, the combination of two drive systems (4) and (9) can achieve synchronization with the combination of two response systems (10) and (11). Figure 5 depicts the time response of the synchronization error 𝑒(𝑡)=(𝑒₁(𝑡),𝑒₂(𝑡),𝑒₃(𝑡),𝑒₄(𝑡),𝑒₅(𝑡))^𝑇.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 4 

3D projections of the hyperchaotic attractor from combination of two drive systems.

(a)

(b)

(c)

(d)

(e)

Figure 5 

Time response curve for synchronization error 𝑒(𝑡)=(𝑒1(𝑡),𝑒2(𝑡),𝑒3(𝑡),𝑒4(𝑡),𝑒5(𝑡))𝑇.

4. Concluding Remarks

A fifth-order complex memristor oscillator system with hyperchaotic effect is presented in this paper. Some sufficient conditions are derived to guarantee the hyperchaotic complex memristor oscillator system for realizing combination-combination synchronization. The hyperchaotic complex memristor oscillator system is useful in secure communication systems with enhanced security features that protect against deciphering. Also, the scheme of combination-combination synchronization might overcome some design issues associated with previous schemes for using chaos in communications. Application of new intelligent control methods to complex memristor oscillator model yields a deep insight into the behaviors under investigation.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The work is supported by the Research Project of Hubei Provincial Department of Education of China under Grant T201412.