Lag synchronization of Rossler system and Chua circuit via a scalar signal
Introduction
Recently, synchronization in chaotic or hyperchaotic systems has attracted great attention and has been extensively studied [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17]. Different methods have been developed in order to synchronize chaotic systems. For instance, the well-known scheme proposed in [1] consisted in taking a chaotic system, duplicating some subsystem and driving the duplicate and the original subsystem with signals from the unduplicated part. When all the Lyapunov exponents of the driven subsystem (response system) are less than zero, the response system synchronizes with the drive system, assuming that both systems start in the same basin of attraction. Until now, only some attempts have been made to give an answer to this question. In [2], a linear combination of the original state variables is used to synchronize hyperchaos in Rossler's systems. However, this technique is short of a systematic methodology for synchronization, since the arbitrary selection for the coefficients of the linear combination are somewhat difficult. An interesting result has been given in [3], where a parameter control method is proposed for hyperchaos synchronization. In [4], [5], [6], a coupled of Lorenz or Rossler or Chen systems was synchronized by using active control. Several adaptive observers had been designed to synchronize certain chaotic systems [7], [8], [9]. A nonlinear observer as the synchronizing system was reported in [16], [17]. Based on the approach is given by Grassi and Mascolo, Rossler system and Matsumoto–Chua–Kobayashi circuit had been synchronized.
There are also different regimes of synchronization in interacting chaotic systems: complete synchronization, which implies coincidence of states of interacting systems, i.e., [1], [2], [3], [4], [5], [6], [7], [8], [9]. Generalized synchronization, which is defined as the presence of some functional relation between the states of response and driver for drive-response systems, i.e., there exists a function F such that [10], [11]. Phase synchronization, which means entrainment of phases of chaotic oscillators, (n and m are integers), whereas their amplitudes remain chaotic and uncorrelated [10], [13]. Lag synchronization appears as a coincidence of shifted-in-time states of two systems, with positive τ, and has been studied in between symmetrically coupled non-identical oscillators [10], [12] and in time-delayed systems [14]. Lastly, it was recently discovered [15] that dissipative chaotic systems with a time-delayed feedback can drive identical systems in such a way that the driven systems anticipate the drivers by synchronizing with their future states, .
In this Letter, we will focus on the lag synchronization of chaotic systems. A modified nonlinear observer, namely, lag-in-time observer, to synchronize the given chaotic system is proposed, and to verify the technique, lag synchronization in two well-known chaotic systems is also demonstrated.
The organization of the remaining part is as follows. In Section 2, chaotic lag synchronization is restated as a nonlinear lag observer design issue. Hence, a linear and time-invariant error system is derived. Furthermore, a necessary and sufficient condition of asymptotical stability for error system is given. By applying the method proposed in the previous section, Rossler system and a modified Chua circuit are synchronized in Sections 3 and 4, respectively. Finally, some conclusions are drawn in Section 5.
Section snippets
Lag observer design for synchronizing chaotic systems
Consider the following two chaotic systems, the dynamics of which are described by where , and is a nonlinear vector field, systems (1) and (2) are said to be lag synchronized if where e represents the lag synchronization error and is called the synchronization lag.
Definition 1 A nonlinear and lag-in-time observer is a dynamic system that designed to be driven by the output of another dynamic system (plant) and having the property that the
Lag synchronization in Rossler systems with chaos
Consider 3D Rossler system [4] where , , are state variables of system (12), are constant scalars. To insure the existence of chaotic behavior of system (12), the parameters, α, β, γ, should be taken in a range. Throughout this section, we select , , , with which phase plot of system (12) is shown in Fig. 2.
In light of the method mentioned above, Rossler system (12) can be rewritten as where
Lag synchronization in Chua circuit with chaos
In this section, we will achieve the lag synchronization of chaos in a modified Chua circuit, which is described by the following equations [4]: where , , are state variables of system (17), p, q are constant scalars. To insure the existence of chaotic behavior of system (17), the parameters, p, q, should be taken in a range. Throughout this section, we select , , with which phase plot of system (17) is shown in Fig. 5.
Similarly, we
Conclusions
In this Letter a simple and rigorous method for lag synchronizing of a wide class of chaotic systems via a scalar transmitted signal has been studied. The proposed methodology does not require either the computation of the Lyapunov exponents or initial conditions belonging to the same basin of attraction. Simulation results for Rossler's system and Chua circuit have shown the usefulness of the suggested method.
Acknowledgments
The authors would like to thank the reviewer for his valuable comments and suggestions. The work described in this Letter was partially supported by the National Natural Science Foundation of China (Grant No. 60271019), the Doctorate Foundation of the Ministry of Education of China (Grant No. 20020611007), the Basic and Applying Basic Research Foundation of Chongqing University (Grant No. 713411003).
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