The antiphase and complete lag synchronization of hyperchaotic Lü systems with unknown parameters is investigated. Based on the Lyapunov stability theory, the sufficient conditions for achieving hybrid lag synchronization are derived. The optimized parameter observers are approached analytically via adaptive control approach. Numerical simulation results are presented to verify the effectiveness of the proposed scheme. 1. Introduction Chaos has been thoroughly studied over the past two decades for its “random” behavior and sensitive dependence on the initial conditions. Despite the complexity and unpredictability of chaotic behavior [1], it can be controlled and two chaotic systems can be synchronized [2]. Since Pecora and Carroll introduced a method to synchronize two identical chaotic dynamical systems [3], the synchronization of chaotic dynamical systems attracted much attention due to its theoretical challenge and potential application in secure communications, chemical reactions, biomedical science, social science, and many other fields [4–6]. Various synchronizations have been presented, such as complete synchronization (CS), phase synchronization (PS), lag synchronization (LS) or anticipated synchronization (AS), and generalized synchronization (GS) [7–16]. Subsequently, many effective synchronization methods have been proposed, such as linear or nonlinear feedback synchronization, adaptive synchronization, lag synchronization, Q-S synchronization, and anticipated synchronization [17–26]. It is an interesting problem that part of the states of the interactive chaotic system are synchronized in one type of synchronization while other states synchronized in another type of synchronization. This phenomenon is taken as mixed synchronization. Due to the potential applications of it, some types of mixed synchronization are introduced recently. In [27], some variables may converge into synchronization while other variables are in antisynchronization state in Chen-Lee chaotic systems. In [28], using a scalar coupling, some of the state variables may be in complete synchronization while others may be in antisynchronization state in two unidirectionally coupled chaotic oscillators. It is found that lag synchronization has important technological implications. Generally, lag synchronization can be trivially accomplished by coupling the response system to a past state of the drive system or by mismatching of the system parameters. Therefore, inspired by [29, 30], it is invited to investigate the coexistence lag synchronization of chaotic systems via linear
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