Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2013, Article ID 856282, 10 pages http://dx.doi.org/10.1155/2013/856282 Research Article Adaptive Sliding Mode Controller Design for Projective Synchronization of Different Chaotic Systems with Uncertain Terms and External Bounded Disturbances Shijian Cang, 1 Zenghui Wang, 2 and Zengqiang Chen 3 1 Department of Industry Design, Tianjin University of Science and Technology, Tianjin 300457, China 2 Department of Electrical and Mining Engineering, University of South Africa, Florida 1710, South Africa 3 Department of Automation, Nankai University, Tianjin 300071, China Correspondence should be addressed to Shijian Cang; sj.cang@gmail.com and Zenghui Wang; wangzengh@gmail.com Received 22 January 2013; Accepted 1 June 2013 Academic Editor: Constantinos Siettos Copyright © 2013 Shijian Cang et al. Tis 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. Synchronization is very useful in many science and engineering areas. In practical application, it is general that there are unknown parameters, uncertain terms, and bounded external disturbances in the response system. In this paper, an adaptive sliding mode controller is proposed to realize the projective synchronization of two diferent dynamical systems with fully unknown parameters, uncertain terms, and bounded external disturbances. Based on the Lyapunov stability theory, it is proven that the proposed control scheme can make two diferent systems (driving system and response system) be globally asymptotically synchronized. Te adaptive global projective synchronization of the Lorenz system and the L¨ u system is taken as an illustrative example to show the efectiveness of this proposed control method. 1. Introduction Te cooperative behavior of coupled nonlinear oscillators is of interest in connection with a wide variety of diferent phenomena in physics, engineering, biology, and economics. For example, systems of coupled nonlinear oscillators may be used to explain how diferent sectors of the economy adjust their individual commodity cycles relative to one another through the exchange of goods and capital units or via aggregate signals in the form of varying interest rates or raw materials prices. As part of the cooperative behaviors, the synchronization plays very important role in many applica- tions. Synchronization occurs when oscillatory (or repetitive) systems via some kind of interaction adjust their behaviors relative to one another so as to attain a state where they work in unison [1], since the individual oscillators display chaotic dynamics in many cases and it is very important to analyze the synchronization of chaotic systems. Synchronization of two coupled chaotic systems has attracted much attention for both theoretical studies and practical applications since the synchronization of the diferent chaotic systems was observed by Pecora et al. in 1997 [2]. Many investigations have been devoted to syn- chronization due to its potential application in many felds recently, such as security communication [3, 4], information processing [5, 6], and biological systems [7]. Many diferent synchronization strategies of diferent chaotic systems have been developed. Beside the generalized synchronization [8, 9], there are some other types of synchronization which are also very interesting and useful, like phase synchronization [10, 11], antiphase synchronization [12, 13], complete synchro- nization [14, 15], and lag synchronization [16, 17]. In 1999, a new synchronization observed by Mainieri and Rehacek in partially linear chaotic systems is called projective syn- chronization [18]. Te dynamical behavior of projective syn- chronization refers to that two identical or diferent systems which are synchronized up to a constant scaling factor. So far, great progress has been made in the research of projective synchronization among all types of synchroniza- tion because of its adjustable proportionality between the synchronized dynamical states [19, 20]. Xu et al. designed a control scheme to manipulate the scaling factor onto any