Finite-Time Combination-Combination Synchronization for Hyperchaotic Systems

A new type of finite-time synchronization with two drive systems and two response systems is presented. Based on the finitetime stability theory, step-by-step control and nonlinear control method, a suitable controller is designed to achieve finite-time combination-combination synchronization among four hyperchaotic systems. Numerical simulations are shown to verify the feasibility and effectiveness of the proposed control technique.


Introduction
As a new subject in 1980s, chaos almost covers all the fields of science.It is known that chaos is an interesting nonlinear phenomenon which may lead to irregularity and unpredictability in the dynamic system, and it has been intensively studied in the last three decades.Since Pecora and Carroll proposed the PC method to synchronize two chaotic systems in 1990 [1,2], the study of synchronization of chaotic systems has been widely investigated due to their potential applications in various fields, for instance, in chemical reactions, biological systems, and secure communication.Over the past decades, a variety of control approaches such as adaptive control [3], linear feedback control [4], active control [5], and backstepping control [6] have been proposed for various types of synchronization, which include complete synchronization [7], projective synchronization [8,9], general synchronization [10], lag synchronization [11], and novel compound synchronization [12].
Most of the aforementioned works are based on the synchronization scheme which consists of one drive system and one response system and can be seen as one-to-one system.However, we found it not secure and flexible enough in many real world applications, for instance, in secure communication.Recently, Runzi et al. presented a new type of synchronization with two drive systems and one response system [13].Then, Sun et al. extended multi-to-one system to multi-to-two systems and reported a new type of synchronization, namely, combination-combination synchronization, where synchronization is achieved between two drive systems and two response systems [14].The type of synchronization can improve the security of communication; for instance, we can split the transmitted signals into several parts, then load each part in different drive systems, and then restore it to the original signals by combining the received signals of different response systems correctly.
Notice that the mentioned literatures mainly investigated the asymptotic synchronization of chaotic systems.However, in the view of practical application, optimizing the synchronization time is more important than achieving synchronization asymptotically [15][16][17][18][19]. Recently, based on the stepby-step control method, Wang et al. realized the finite-time synchronization of two chaotic systems by designing a proper controller [15].The method has the ability to achieve global stability in finite time.In addition, the step-by-step technique has the advantage of reducing controller complexity.
Motivated by the previous discussion, this paper aims to study the finite-time synchronization between a combination of two drive systems and a combination of two response systems in drive-response synchronization scheme.We have applied the finite-time stability theory to our analysis to achieve finite-time combination-combination synchronization.The step-by-step control method and nonlinear control technique are adopted to synchronize four different

The Control Scheme
Thus, we can get the error system as follows: Our aim is to design a suitable controller, such that the drive systems ( 8) and ( 9) are realized as combinationcombination synchronization with the response systems (10) and (11) in finite time.Then, the problem is changed to design a suitable controller, such that the error system (12) achieves the finite-time stability at the origin.
The design plan and its steps are as follows. Step where is a proper rational number, and  is a positive odd number,  > .
Substituting  4 into the fourth equation of ( 12), we get Choose a candidate Lyapunov function Thus, the derivative of  4 along the solution of error equation ( 15) is According to Lemma 2, the system ( 15) is finite-time stability, which implies that there exists  1 > 0, such that  4 ≡ 0 for  ≥  1 .
Step 2. Choose where For  >  1 , substituting  1 into the first equation of ( 12), we get Choose a candidate Lyapunov function Thus, the derivative of  1 along the solution of error equation ( 20) is According to Lemma 2, the system ( 20) is finite-time stability, which implies that there exists  2 > 0, such that  1 ≡ 0 for  ≥  2 .
Step 3. Choose where For  >  2 , substituting  2 into the second equation of ( 12), we get Choose a candidate Lyapunov function Thus, the derivative of  2 along the solution of error equation (25) is According to Lemma 2, the system (25) is finite-time stability, which implies that there exists  3 > 0, such that  2 ≡ 0 for  ≥  3 .

Journal of Chaos
Choose a candidate Lyapunov function Thus, the derivative of  3 along the solution of error equation ( 30) is According to Lemma 2, the system (30) is finite-time stability, which implies that there exists  4 > 0, such that  3 ≡ 0 for  ≥  4 .
The controller is designed as follows: According to what we discussed previously, we can obtain this conclusion that the error system (12) achieves finite-time stability under the control of the controller (33).Furthermore, the drive systems ( 8) and ( 9) are realized as combinationcombination synchronization with the response systems (10) and (11) in finite time  ≤   , where   =  1 +  2 +  3 +  4 .

Numerical Simulation
To verify the effectiveness of the proposed finite-time synchronization method, we consider the hyperchaotic Chen system with the parameters  = 35,  = 3,  = 12,  = 7, and  = 0.5.The hyperchaotic attractor of the system is shown in Figure 1.

Conclusion
In this paper, the problem of finite-time combinationcombination synchronization with two drive systems and two response systems was investigated.Based on the finitetime stability theory, the step-by-step control and nonlinear control approach, a suitable controller was introduced.The simulation results demonstrated that the proposed controller works well for synchronizing four hyperchaotic systems in finite time.