CYBERNETICS AND PHYSICS, VOL. 2, NO. 1, 2013 , 31–36 MIXED TRACKING AND PROJECTIVE SYNCHRONIZATION OF 5D HYPERCHAOTIC SYSTEM USING ACTIVE CONTROL Kayode Ojo Department of Physics University of Lagos, Akoka Nigeria kaystephe@yahoo.com Samuel Ogunjo Condensed Matter and Statistical Physics Group Department of Physics Federal University of Technology Akure, Nigeria stogunjo@fut.edu.ng Oluwafemi Williams Dept. of Physics FUT, Akure Nigeria Abstract This paper examines mixed tracking control and hy- brid synchronization of two identical 5-D hyperchaotic Lorenz systems via active control technique. The de- signed control functions for the mixed tracking enable each of the system state variables to stabilize at differ- ent chosen positions as well as control each state vari- ables of the system to track different desired smooth function of time. Also, the active control technique is used to design control functions which achieve projec- tive synchronization between the slave state variables and the master state variables. We also show that the coupling strength is inversely proportional to the syn- chronization time. Numerical simulations are carried out to validate the effectiveness of the analytical tech- nique. Key words Hyperchaos, synchronization, tracking, active control. 1 Introduction Since the discovery of the first chaotic system by Lorentz in 1963 many new chaotic systems have been successively developed [Chen and Ueta, 1999; Chua and Lin, 1990; Qi et al, 2005; R¨ ossler, 1976]. Chaos has gradually moved from simply being a scientific curiosity to a promising subject with practical sig- nificance and applications in different fields such as communication[Mengue and Essimbi, 2012], biologi- cal systems[Shi, 2012], economics and other fields. During the beginning of the last decade, one of the most fascinating discoveries that transformed research in the field of nonlinear dynamics and chaos theory is the fact that two or more chaotic systems evolving from different initial conditions can be made to synchronize, either by coupling the systems (locally or globally) or by forcing them. Synchronization means that the state of a response system eventually approaches that of a driving system. This was first demonstrated by Pecora and Carroll [Pecora and Carroll, 1990]. Unrelenting research in chaotic systems has given rise to different types of synchronization including complete synchro- nization (CS) [Pecora and Carroll, 1990], generalized synchronization (GS) [Kacarev and Parlitz, 1996], pro- jective synchronization (PS) [Mainieri and Rehacek, 1999], function projective synchronization (FPS)[An and Chen, 2008] amongst others. Chaotic behaviours could be beneficial feature in some cases, but can be undesirable in some engineer- ing, biological and other physical applications; and therefore it is often desired that chaos should be con- trolled, so as to improve the system performance. Thus, it is of considerable interest and potential utility, to devise control techniques capable of forcing a system to maintain a desired dynamical behaviour even when intrinsically chaotic. The control of chaos and bifur- cation is concerned with using some designed control input(s) to modify the characteristics of a parameter- ized nonlinear system. There might be need for dif- ferent components of a chaotic system will be required to follow different trajectories when controlled, there- fore, the need for mixed tracking or control. A number of methods such as OGY closed-loop feedback method [Ott, Grebogi and Yorke, 1990], active control [Bai and Lonngren, 1997], active backstepping [ Zhang, Ma, Li, and Zou, 2005] and recursive active control [Vincent, Laoye, and Odunaike, 2009] exist for the control of chaos in systems. Chaos control is considered as a spe- cial case of chaos synchronization. Despite the numer- ous advantages of mixed tracking, no research work has been done in this regard to the best of our understand- ing. The active control method introduced by [Bai and Lonngren, 1997] is efficient technique for the synchro- nization of chaotic systems because it can be used to