Impulsive synchronization of chaotic systems subject to time delay

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Abstract

This paper studies synchronization of coupled chaotic systems with time delay by means of impulsive control. Some criteria on synchronization of such chaotic systems are established utilizing quadratic Lyapunov functions and differential inequalities in the spirit of the well-known Razumikhin technique. Those criteria may be applied to chaos-based secure communication systems with transmission delay, where a driving system and a response system are employed. Simulation results are given to demonstrate the analytical results.

Introduction

Chaos synchronization has been an active area of research due to its potential applications to secure communication [1], [2], [3], [4], [5], [6], [7], [8], [9], [10]. Several chaos-based secure communication schemes have been proposed. In these schemes, message signals are masked or modulated (encrypted) by using chaotic signals and the resulting encrypted signals are transmitted across a public channel. Perfect synchronization is usually expected to recover the information signals. In other words, the recovery of the information signals requires the receiver’s own copy of the chaotic signal which is synchronized with the transmitter’s one [8]. Different types of synchronization techniques (some of which are robust to parameter mismatch and channel noise) have been developed in the literature [11], [12], [13], [14], chief among them is the technique proposed by Pecora and Carroll in [8]. In that paper, it has been demonstrated that the dynamics of the drive and response systems become synchronized when the Lyapunov exponents of the response system are less than zero and when both systems start in the same basin of attraction. Low-dimensional chaotic systems have been employed in this particular type of synchronization and the resulting dynamics have been investigated by using the second Lyapunov method [2]. On the other hand, the synchronization of hyperchaotic systems possessing more than one positive Lyapunov exponent has been studied in [3], [4], [5], [9] by applying an observer-design method. Further analysis of this method and its applications to synchronization can be also found in [15], [16]. Some analytical results leading to a specific kind of generalized synchronization whose synchronizing manifold is linear has been presented in [10].

Most recently, another synchronization technique, based on impulsive control, has been reported and developed in [1], [38]. The technique allows the coupling and synchronization of two or more chaotic systems by using only small synchronizing impulses [7], [17], [18], [19], [20], [21]. These impulses are samples of the state variables (or functions of the state variables) of the drive system at discrete moments. They drive the response system discretely at these moments. When the attractivity in the large of the synchronization error between the drive and the response chaotic systems is achieved, the two coupled systems are said to be synchronized. An upper bound on the time intervals between the impulses is obtained in [17], [18], and a generalization of this type of synchronization to time-varying impulse intervals is investigated in [22], [23]. Since it is practically impossible to design two identical chaotic systems for synchronization, the robustness of synchronization toward parameter mismatch (with the absence of delay) has been also examined in [24]. Furthermore, this technique has been applied to a number of chaos-based secure communication schemes which exhibit good performance as far as synchronization and security are concerned [17], [18]. One particular advantage of this method is its ability to be combined with conventional cryptographic techniques. Detailed experiments and performance analysis of impulsive synchronization with the intent of testing its accurate recovery of message signals and its applicability to secure communication have appeared in the literature [25], [26], [27].

In general, transmission delays in communication systems are inevitable. Therefore it is very crucial to examine the robustness of synchronization in the presence of time delay when designing a chaos-based secure communication scheme. However, when time delay is taken into consideration, the resulting dynamical systems become infinite dimensional, and consequently, the mathematical analysis becomes more complex. There have been several investigations in the literature to study the existence, uniqueness, boundedness and stability of solutions of a particular class of delayed impulsive systems [28], [29]. In fact, the stability of linear continuous-time systems possessing delayed discrete-time controllers in networked control systems have been also analyzed [30], [31], [32]. Such studies were based on the notions of Lyapunov–Krasovskii functionals and Lyapunov–Razumikhin functions [33], [34], [35]. The asymptotic stability of singularly perturbed delayed impulsive systems with uncertainty from nonlinear perturbations has been also explored in [36].

In this paper, we shall investigate synchronization of coupled chaotic systems with time delay by means of impulsive control. By utilizing quadratic Lyapunov functions and differential inequalities in the spirit of the well-known Razumikhin technique, we shall establish some criteria on synchronization of such chaotic systems. These criteria may be applied to chaos-based secure communication systems with transmission delay, where a driving system and a response system are employed.

The rest of this paper is organized as follows. In Section 2, we formulate the problem and introduce a lemma which plays a key role in the proof of the main result to be given later. We then establish the main result on synchronization in Section 3. Next we demonstrate the analytical results in Section 4 by a numerical example, where Chua’s circuit is used in the simulations. Finally, conclusions are given in Section 5.

Section snippets

Problem formulation

Chaos-based communication system usually consists of two chaotic systems at the transmitter and the receiver ends, respectively. At the transmitter end, we have ẋ(t)=Ax(t)+Φ1(x(t)) and, at the receiver end, we have {u̇(t)=Au(t)+Φ2(u(t),x(tr),u(tr)),tτkΔu(t)=Bke(t),t=τk,k=1,2,, where the matrix ARn×n, and the matrix BkRn×n and the functions Φ1,Φ2 are continuous functions in their respective domain of definition. Δu(tk)=u(tk+)u(tk), u(tk+)=limttk+u(t), for k=1,2,. A typical form of

Synchronization criteria

In this section, we shall establish some synchronization criteria using impulsive control and Lemma 2.1.

Theorem 3.1

Assume that

  • (i)

    there exist a positive definite matrix P and constants αi,i=1,2 ,   with α2R+ such that[ATP+PA+PD1α1PPD2D2TPα2P]0;

  • (ii)

    there exists a real number β(0,1) such that for each   k=1,2, ,(I+BkT)P(I+Bk)βP0;

  • (iii)

    there exists a positive number τ such thatτkτk1τ,k=2,3,,andτβ(βα1+α2)+lnβ0.

Then the trivial solution of(2.4)is globally asymptotically stable, i.e, system(2.1)and system(2.2)are

Numerical example

In this section, we illustrate the synchronization criteria obtained in the previous section by an example.

Example 4.1

Consider the Chua circuit

{ẋ1=a1[x2h(x1)]ẋ2=x1x2+x3ẋ3=a2x2 where h(x1)=m1x1+12(m0m1)(|x1+a3||x1a3|). When a1=9, a2=14.286, a3=1, m0=1/7, m1=1.5/7, we obtain the double scroll attractor shown in Fig. 1. Chua’s circuit can be rewritten in the form of (2.1), i.e. ẋ(t)=Ax(t)+Φ1(x(t)) where A=[a1m1a101110a20],Φ1=[1/2a1(m0m1)(|x1+a3||x1a3|)00].

We then choose Φ2 in (2.2) the same

Conclusion

We have investigated the synchronization of coupled chaotic systems with time delay by means of impulsive control and established some criteria on synchronization of such chaotic systems by utilizing quadratic Lyapunov functions and differential inequalities in the spirit of the well-known Razumikhin technique. Those criteria may be applied to chaos-based secure communication systems with transmission delay, where a driving system and a response system are employed. Simulation results have been

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    This work was partially supported by the Natural Sciences and Engineering Research Council of Canada.

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