Research Article Control and Synchronization of Chaotic and Hyperchaotic Lorenz Systems via Extended Backstepping Techniques O. S. Onma, 1 O. I. Olusola, 1 and A. N. Njah 2 1 Nonlinear Dynamics Research Group, Department of Physics, Federal University of Agriculture, PMB 2240, Abeokuta, Nigeria 2 Department of Physics, University of Lagos, Akoka, Lagos, Nigeria Correspondence should be addressed to O. I. Olusola; olasunkanmi2000@gmail.com Received 10 September 2013; Revised 15 November 2013; Accepted 17 November 2013; Published 6 May 2014 Academic Editor: Giovanni P. Galdi Copyright © 2014 O. S. Onma 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. We propose novel controllers for stabilization and tracking of chaotic and hyperchaotic Lorenz systems using extended backstepping techniques. Based on the proposed approach, generalized weighted controllers were designed to control chaotic behaviour as well as to achieve synchronization in chaotic and hyperchaotic Lorenz systems. Te efectiveness and feasibility of the proposed weighted controllers were verifed numerically and showed their robustness against noise. 1. Introduction Chaos theory has found application in many areas of studies; these include mathematics, physics, biology, engineering, economics, and politics [1–3]. One of the most successful applications of chaos theory has been in ecology, where dynamical systems have been used to show how popula- tion growth under density dependence can lead to chaotic dynamics. Chaos theory is also currently being applied to medical studies of epilepsy, specifcally to the prediction of seemingly random seizures by observing initial conditions [4]. Furthermore, a related feld of physics called quantum chaos theory investigates the relationship between chaos and quantum mechanics [5]. In addition, another feld called relativistic chaos [6] has emerged to describe systems that follow the laws of general relativity. Chaotic phenomenon could be benefcial in some appli- cations; however, it is undesirable in many engineering and other physical applications and should therefore be controlled in order to improve the system performance. Chaos control is concerned with using some designed control inputs to mod- ify the characteristics of a parameterized nonlinear system so that the system becomes stable at a chosen position or tracks a desired trajectory [7]. Several techniques have been deviced for chaos control but mostly are for development of two basic approaches: the OGY (Ott, Grebogi, and Yorke) method and Pyragas continuous control. Both methods require a previous determination of the unstable periodic orbits of the chaotic system before the controlling algorithm can be designed. Experimental control of chaos by one or both of these methods has been achieved in a variety of systems, including turbulent fuids, oscillating chemical reactions, magneto-mechanical oscillators, and cardiac tissues [8]. Since the idea of synchronization of chaotic systems was proposed by Pecora and Carroll in 1990, [9], chaos synchronization has received an increasing attention due to its theoretical challenge and its potential applications in secure communications, chemical reactions, biological systems, information science, and plasma technologies [10]. Chaos synchronization involves the coupling or forcing of two systems so that both systems achieve identical dynamics asymptotically with time. A wide variety of approaches have been proposed to achieve chaos synchronization in the cou- pled/forced chaotic systems such as linear state error feedback method [11], time-delay feedback method [12], active control approach [13, 14], impulsive method [15], adaptive control [16, 17], and backstepping approach [18]. Te backstepping approach has been widely applied to achieve synchronization in chaotic and hyperchaotic systems. It has the ability to achieve global stability tracking transient performance for a board class of strict-feedback nonlinear system (see [7] and the references therein). Also the method requires less control Hindawi Publishing Corporation Journal of Nonlinear Dynamics Volume 2014, Article ID 861727, 15 pages http://dx.doi.org/10.1155/2014/861727