A new type of finite-time synchronization with two drive systems and two response systems is presented. Based on the finite-time 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. 1. 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
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