Chaos synchronization of nonlinear Bloch equations
Introduction
Chaos is very interesting nonlinear phenomenon and has been intensively studied in the last three decades [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]. It is found to be useful or has great potential in many disciplines [1]. Especially, the subject of chaotic synchronization has received considerable attentions since 1990. In the literature, various synchronization schemes, such as variable structure control, OGY method, parameters adaptive control, observer-based control, active control, time-delay feedback approach, backstepping design technique, and so on, have been successfully applied to the chaos synchronization. Using these methods, numerous works for the synchronization problem of well-known chaotic systems such as Lorenz, Chen, Lü, and Rossler systems have been done by many scientist.
On the other hand, the dynamics of an ensemble of spins which do not exhibit mutual coupling, except for some interactions leading to relaxation, is well described by the simple Bloch equations. Recently, Abergel [12] examined the linear set of equations originally proposed by Bloch to describe the dynamics of an ensemble of spins with minimal coupling, and incorporated certain nonlinear effects that were caused by a radiation damping based feedback field. Ucar et al. [13] extend the calculation of Abergel [12] and demonstrate that is is possible to synchronize two of these nonlinear Bloch equations. However, these works are based on the exactly knowing of the system parameters. But in real situation, some or all of the parameters are unknown.
In this paper, the chaotic synchronization of nonlinear Bloch equations with uncertain parameters is investigated. A class of novel nonlinear control scheme for the synchronization is proposed, and the synchronization is achieved by the Lyapunov stability theory.
The organization of this paper is as follows. In Section 2, the problem statement and master–slave synchronization scheme are presented for the chaotic system. In Section 3, we provide an numerical example to demonstrate the effectiveness of the proposed method. Finally concluding remark is given.
Section snippets
Chaos synchronization
In dimensionless units, the dynamic model of nonlinear modified Bloch equations with feedback field [12] is given bywhere δ, λ and ψ are the system parameters and τ1 and τ2 are the longitudinal time and transverse relaxation times, respectively. In the work of Abergel [12], the dynamic behavior of the system has been extensively investigated for a fixed subset of the system parameters (δ, λ, τ1, τ2) and for a space area
Numerical example
In this section, to verify and demonstrate the effectiveness of the proposed method, we discuss the simulation result for Bloch equations. In the numerical simulations, the fourth-order Runge–Kutta method is used to solve the systems with time step size 0.001.
For this numerical simulation, we assume that the initial condition, (xm(0), ym(0), zm(0)) = (0.5, −0.5, 0), and (xs(0), ys(0), zs(0)) = (−0.5, 0.5, 0.3) is employed. Hence the error system has the initial values e1(0) = −1, e2(0) = 1 and e3(0) = 0.3. In
Concluding remark
In this paper, we investigate the synchronization of controlled nonlinear Bloch equations. We have proposed a novel nonlinear control scheme for asymptotic chaos synchronization using the Lyapunov method. Finally, a numerical simulation is provided to show the effectiveness of our method.
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