Elsevier

Physics Letters A

Volume 288, Issue 1, 10 September 2001, Pages 23-28
Physics Letters A

Chaos synchronization by using intermittent parametric adaptive control method

https://doi.org/10.1016/S0375-9601(01)00521-7Get rights and content

Abstract

The method using intermittent parametric adaptive control algorithm is proposed for the synchronization of two chaotic logistic maps with different parameters. A sufficient condition is drawn for the synchronization. This method is also robust when two chaotic systems are subjected to common noises. Based on this method, a scheme for feedback intensity is designed. With this scheme the synchronization is unconditionally achieved. Finally, numerical simulation results are shown to demonstrate the effectiveness and efficiency of the proposed method.

Introduction

Since Pecora and Carroll proposed a method to synchronize chaotic systems and realized it in electronic circuit [1], [2], chaos synchronization has attracted considerable interests especially for its potential application in secure communications [3], [4]. There are some methods that could be used for the synchronization, such as linear feedback and nonlinear feedback [5], [6]. In practice, the chaotic systems would be inevitably exposed to a noisy environment. It is difficult to find two exactly identical chaotic systems. Hence, the synchronization of non-identical systems [6], [7], [8], [9] (generalized synchronization) plays a significant role in practical applications.

In Refs. [9], [10], the generalized synchronization by using parametric adaptive control method is discussed, and satisfactory results are obtained. It is noted that, in Refs. [9], [10], feedback control exists continuously and the higher control price must be paid for the synchronization. In practical applications, however, continuous control is not necessary, and the control price can also be reduced. Recently, discontinuous control method has been increasingly developed for chaos synchronization [11], [12], [13], [14]. Compared with continuous control method, discontinuous control method can achieve target orbit more efficiently even under the circumstances that the control price is greatly reduced. In this Letter, we present the intermittent parametric adaptive control method for the synchronization of two logistic maps with different parameters, i.e., for the generalized synchronization of logistic maps. A sufficient condition is drawn for the synchronization, and the robustness of the synchronization with respect to common noise is studied. We also design a scheme by which the synchronization can be readily achieved. Finally, numerical simulation is conducted. And the results show that the generalized synchronization of logistic maps can be rapidly achieved with randomly chosen initial values. Moreover, the form of this scheme is simple and can be easily realized in experimental applications.

This Letter has been organized as follows. In Section 2, intermittent control method is applied to the generalized synchronization of two logistic maps. In Section 3, simulation results are presented to illustrate the effectiveness of the proposed method. Finally, some conclusions are given in Section 4.

Section snippets

The method using intermittent parametric adaptive control algorithm

We consider the systems represented by the following equations in discrete time: xn+1cxn(1−xn),yn+10yn(1−yn), where n=0,1,… is the discrete time, the parameters μc,μ0∈[3.57,4.0] and the initial values x0,y0∈(0,1) so that the systems exhibit chaotic behaviors [8]. Here , represent the drive system and the response system, respectively. The aim of synchronization is to make |ynxn|=0 as n→∞. Moreover, it is preferable with practical purpose that we accomplish the synchronization with simple

Results from numerical simulation

In this section we simulate the generalized synchronization of two chaotic logistic maps. The initial values are randomly chosen as x0=0.79 and y0=0.8, respectively, for the drive system and the response system. Their parameter values are μc=3.7 and μ0=3.9. Feedback control is not imposed on the response system until iteration number is 1000. Here, we only display the time evolution plots from the moment when control is imposed on.

Fig. 1 shows that the synchronization at different control

Conclusion

In conclusion, the generalized synchronization of two chaotic logistic maps is achieved by using intermittent parametric adaptive control method in this Letter. A sufficient condition is drawn for the synchronization and a control scheme for the weight of the feedback control is also designed. The scheme is correlated with the output of the response system and simple in form, whereas it can ensure that the synchronization is realized asymptotically and unconditionally. In addition, the method

Acknowledgements

We thank Yun Zhang and Prof. Xi-Ya Ding for their help with this manuscript.

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