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This paper investigates hybrid synchronization of the uncertain generalized Lorenz system. Several useful criteria are given for synchronization of two generalized Lorenz systems, and the adaptive control law and the parameter update law are used. In comparison with those of existing synchronization methods, hybrid synchronization includes full-order synchronization, reduced-order synchronization, and modified projective synchronization. What is more, control of the stability point, complete synchronization, and antisynchronization can coexist in the same system. Numerical simulations show the effectiveness of this method in a class of chaotic systems.

In 1990, Pecora and Carroll made chaos synchronization come true [

In the above discussed literature, the given systems usually were typical benchmark chaotic systems, such as the Lorenz system, Chen system, and Lü system. In this paper, we consider the generalized Lorenz system. Based on the stability theory of systems, several useful criteria are given for discussing synchronization of two generalized Lorenz systems, and the adaptive control law and the parameter update law are also given. In comparison with those of existing synchronization methods, hybrid synchronization includes full-order synchronization, reduced-order synchronization, and the modified projective synchronization. What is more, control of the stability point, complete synchronization, and antisynchronization can coexist in the same system. The rest of this paper is organized as follows: Section

Let us consider a class of chaotic systems described by

A number of well-known chaotic systems have the form of (

For the drive and response chaotic systems

If

If

Our objective is to design the controller

For the drive system (

From (

Choose a candidate Lyapunov function as follows:

Then the differentiation of

In this section, we give some examples to show the effectiveness of this method.

In 1963, Lorenz found the first classical chaotic attractor [

If

The proof of Corollary

In the simulations, suppose that the “unknown” parameters of the drive and response Lü chaotic systems are chosen as

Synchronization errors between two Lü chaotic systems; the full-order hybrid synchronization is realized.

Estimated parameters of the Lü system finally evolve to the true values

If

From

Then the differentiation of

Obviously, the controllers

In the simulations, suppose that the “unknown” parameters of the drive and response Chen chaotic systems are chosen as

Complete synchronization and antisynchronization are realized for the Chen system.

The third state vector

Estimated parameters of the Chen system finally evolve to the true values

This paper has discussed hybrid synchronization of the uncertain generalized Lorenz system. Based on the stability theory systems, several useful criteria have been given for synchronization of two generalized Lorenz system systems, and the adaptive control law and the parameter update law were also given. In comparison with those of existing synchronization methods, hybrid synchronization includes full-order synchronization, reduced-order synchronization, and modified projective synchronization. Simulation results were presented to demonstrate the application of theoretical results. Our future work is to study hybrid synchronization of Markovian jump complex networks with time-varying delay.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work is supported by the National Natural Science Foundation of China (61673221), the Youth Fund Project of the Humanities and Social Science Research for the Ministry of Education of China (14YJCZH173), and the Science and Technology Research Key Program for the Education Department of Hubei Province of China (D20156001).