Robust Synchronization of Hyperchaotic Systems with Uncertainties and External Disturbances

Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 523572, 8 pages http://dx.doi.org/10.1155/2014/523572 Research Article Robust Synchronization of Hyperchaotic Systems with Uncertainties and External Disturbances Qing Wang, Yongguang Yu, and Hu Wang Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China Correspondence should be addressed to Yongguang Yu; Received 4 December 2013; Revised 21 March 2014; Accepted 21 March 2014; Published 7 April 2014 Academic Editor: Zlatko Jovanoski Copyright © 2014 Qing Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The robust synchronization of hyperchaotic systems with uncertainties and external disturbances is investigated. Based on the Lyapunov stability theory, the appropriate adaptive controllers and parameter update laws are designed to achieve the synchronization of uncertain hyperchaotic systems. The robust synchronization of two hyperchaotic Chen systems is taken as an example to verify the feasibility of the presented schemes. The size of the subcontroller gain’s influences on the convergence speed is discussed. Finally, numerical simulations are given to demonstrate the effectiveness of the proposed synchronization schemes. 1. Introduction Since the method of synchronization between identical chaotic systems with different initial conditions was presented by Pecora and Carroll [1], chaos synchronization has attracted considerable attention because of its wide range of applications in many important research fields, such as secure communication, chemical reactions, artificial neural networks, and biological systems [2, 3]. The idea of synchronization is to use the output of the drive system to control the response system so that the output of the response system follows the one of the drive system asymptotically. Up to now, many types of synchronization phenomena have been reported, such as generalized synchronization [4, 5], adaptive synchronization [6–9], projective synchronization [10–12], impulsive synchronization [13], lag synchronization [14, 15], and function projective synchronization [16, 17]. And a wide variety of control approaches, such as backstepping design technique [18], fuzzy sliding mode control [19, 20], adaptive control [21], optimal control [22], and 𝐻∞ control [23], have been proposed to synchronize chaotic systems. However, most of the reported schemes are mainly concerned with the synchronization of chaotic systems without uncertainties and external disturbances. This behaviour results in that the complexity of chaotic dynamics is limited while, in real world applications, there exist a mass of phenomena of the uncertainties and external disturbances in chaotic systems. In this regard, a number of researchers have paid their attention to the synchronization of chaotic systems with uncertainties and external disturbances [24–30]. The corresponding works have solved the problem of synchronization of chaotic systems with uncertainties and external disturbances. However, these studies have not considered the synchronization of the hyperchaotic systems. The chaos-hyperchaos transition occurs when the second Lyapunov exponent becomes positive [31]. Compared with chaotic systems with one positive Lyapunov exponent, hyperchaotic systems are characterized by at least two positive Lyapunov exponents which indicates that they have more complex dynamics and much wider application. Moreover, the chaotic systems with higher-dimensional attractors have much more randomness and higher unpredictability of the corresponding system [32]. So the hyperchaos may be more useful in some fields such as communication and encryption. Motivated by this, robust synchronization of uncertain hyperchaotic systems is investigated. Aghababa [33] proposed finite-time chaos control and synchronization of fractionalorder nonautonomous chaotic (hyperchaotic) systems. In [34, 35], Fu considered robust adaptive modified function projective synchronization and robust adaptive antisynchronization of different hyperchaotic systems subject to external disturbances. Jawaada et al. [36] proposed robust active sliding mode antisynchronization of hyperchaotic systems with uncertainties and external disturbances. Li et al. [37] 2 Journal of Applied Mathematics studied generalized function projective synchronization of two different hyperchaotic systems with unknown parameters. However, these studies have not considered the effects of both uncertainties and the different kinds of characteristics of external disturbances in the dynamics of the hyperchaotic systems. In this paper, depending on the characteristics of external disturbance signals, appropriate adaptive controllers and parameter update laws are proposed for the robust synchronization of hyperchaotic systems with uncertainties and external disturbances. The uncertainties are bounded by a nonlinear state-dependent function, instead of a real constant. In the first adaptive controller, assuming that external disturbances are square integrable signals, the adaptivebased controllers and parameter update laws are designed to remove the effects of uncertainties. In the second proposed adaptive controller, the uncertainties and the external disturbances are not square integrable signals. The uncertainties and external disturbances are combined into an uncertain time-varying function with unknown bound. The robustness properties with respect to uncertainties and external disturbances are provided by the proposed controllers. The organization of this paper is as follows. In Section 2, the robust synchronization of uncertain hyperchaotic Chen systems is formulated. According to the Lyapunov stability theory, the robust adaptive synchronization techniques and the error dynamical system’s stability analysis are proposed in Section 3. Then, the simulation results in Section 4 are given to illustrate the effectiveness of the advised methods. Finally, the concluding remarks are given in Section 5. Throughout the paper, for a vector 𝑉 ∈ 𝑅𝑛 , ‖𝑉‖ denotes the Euclidean vector norm and ‖𝑉‖2𝑄 := 𝑉𝑇 𝑄𝑉, with the weighting matrix 𝑄. Make ‖𝑉‖2 := ‖𝑉‖2𝑄=𝐼 = 𝑉𝑇 𝑉. Furthermore, 𝑇 𝑉 ∈ 𝐿 2 [0, 𝑇], if ∫0 ‖𝑉(𝑡)‖2 𝑑𝑡 < ∞, 𝑇 ∈ [0, ∞). Consider the nonlinear systems in the form of 𝑋 (0) = 𝑋0 ∈ 𝑅𝑛 , 𝑌̇ = 𝐺 (𝑌) + Δ𝐺 + ℎ + 𝑢, 𝑌 (0) = 𝑌0 ∈ 𝑅𝑛 , 𝑇 where 𝐸 = [𝑒1 , 𝑒2 , . . . , 𝑒𝑛 ]𝑇 . 𝑦̇ = 𝑑𝑥 − 𝑥𝑧 + 𝑐𝑦, 𝑧̇ = 𝑥𝑦 − 𝑏𝑧, (4) 𝑤̇ = 𝑦𝑧 + 𝑟𝑤, where 𝑥, 𝑦, 𝑧, and 𝑤 are state variables and 𝑎, 𝑏, 𝑐, 𝑑, and 𝑟 are the real constants. When 𝑎 = 35, 𝑏 = 3, 𝑐 = 12, 𝑑 = 7, and 0 ≤ 𝑟 ≤ 0.085, system (4) is chaotic; when 𝑎 = 35, 𝑏 = 3, 𝑐 = 12, 𝑑 = 7, and 0.085 < 𝑟 ≤ 0.798, system (4) is hyperchaotic; when 𝑎 = 35, 𝑏 = 3, 𝑐 = 12, 𝑑 = 7, and 0.798 < 𝑟 ≤ 0.9, system (4) is periodic [38]. In order to obse (...truncated)