Elsevier

Chaos, Solitons & Fractals

Volume 18, Issue 1, September 2003, Pages 141-148
Chaos, Solitons & Fractals

An adaptive chaos synchronization scheme applied to secure communication

https://doi.org/10.1016/S0960-0779(02)00585-4Get rights and content

Abstract

This paper deals with the problem of synchronization of a class of continuous-time chaotic systems using the drive-response concept. An adaptive observer-based response system is designed to synchronize with a given chaotic drive system whose dynamical model is subjected to unknown parameters. Using the Lyapunov stability theory an adaptation law is derived to estimate the unknown parameters. We show that synchronization is achieved asymptotically. The approach is next applied to chaos-based secure communication. To demonstrate the efficiency of the proposed scheme numerical simulations are presented.

Introduction

In recent years, there has been increasing interest in the study of synchronizing chaotic systems [1], [2], [3], [4]. In their seminal paper, Pecora and Carroll [5] addressed the synchronization of chaotic systems using a drive-response conception. The idea is to use the output of the drive system to control the response system so that they oscillate in a synchronized manner. Since then, several other synchronization schemes have been developed, such as mutual coupling by Chua et al. [6] and inverse system approach by Hasler and coworkers [7], [8]. More recently, the synchronization has been regarded as a special case of observer design problem [9], [10], [11], [12]. In most of the research done on synchronizing chaotic system, perfect knowledge of these systems was assumed, yet such perfection is not realistic. Actually a few attempts to synchronize uncertain chaotic systems have been proposed. In [13] we have considered the presence of unknown disturbances and achieved synchronization using a reduced-order observer. In [14] a robust sliding observer was suggested to overcome the effect of parameter uncertainties. In [15], [16] adaptive observers were used to synchronize Lur’e type chaotic systems (i.e., where the nonlinearity is a function of the output).

In this work we suggest an adaptive observer for a larger class of chaotic systems. We use the Lyapunov approach to derive an updating law for the estimation of the unknown parameters. We show that under mild conditions, synchronization is asymptotically achieved and the parameters are correctly estimated. We also show that this method can be applied to secure message transmission using parameter modulation. The outline of this paper is as follows. In Section 2 we present the adaptive observer-based response system design and we prove its synchronization. In Section 3 we present some illustrative examples. In Section 4 we explain how can the proposed synchronization scheme be used for secure digital message transmission and we give some simulation results. Finally in Section 5 we include some concluding remarks.

Section snippets

Adaptive synchronization

Chaotic systems are generally described by a set of nonlinear differential equations. It is very common, however, to be able to separate the dynamics into linear and nonlinear parts. If we furthermore consider that the chaotic system is subjected to unknown parameters, the chaotic dynamics can therefore be described by the following equations:ẋ=Ax+f(x)+BΦ(x)θy=Cxwhere x∈Rn and y∈Rm are respectively the state vector and the output of the drive system. θ∈Rp represents a constant vector of

Illustrative examples

In this section, we consider two well-known chaotic systems to which we apply the chaotic synchronization scheme proposed in the foregoing section.

Secure communication using parameter modulation

Secure communication has been an interesting field of application of chaotic synchronization since the last decade [21], [22], [23]. Due to their unpredictability and broad band spectrum, chaotic signals have been used to encode information by simple masking (addition) or using modulation. As a matter of fact, since the synchronization scheme proposed in the previous section can correctly estimate the unknown constant uncertainty of the drive system parameter, one can expect that it can also

Conclusion

In this paper we showed that given a single driving signal of a drive chaotic system, we can concurrently obtain synchronization and estimation of a constant unknown parameter at the response system side. The result is obtained using an adaptive observer. We demonstrated that information about the parameters of a chaotic system is embedded in the time series data of a state variable and can be extracted under mild conditions. Consequently, a parameter of the drive system can be stirred to vary

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