Available online at www.sciencedirect.com Mathematics and Computers in Simulation 80 (2010) 2286–2296 Synchronization and control of hyperchaotic complex Lorenz system Gamal M. Mahmoud a,∗ , Emad E. Mahmoud b a Department of Mathematics, Faculty of Science, Assiut University, Assiut 71516, Egypt b Department of Mathematics, Faculty of Science, Sohag University, Egypt Received 30 September 2009; accepted 29 March 2010 Available online 13 May 2010 Abstract The aim of this paper is to investigate the phenomenon of projective synchronization (PS) and modified projective synchronization (MPS) of hyperchaotic attractors of hyperchaotic complex Lorenz system which has been introduced recently in our work. The control problem of these attractors is also studied. Our system is a 6-dimensional continuous real autonomous hyperchaotic system. The active control method based on Lyapunov function is used to study PS and MPS of this system. The problem of hyperchaos control is treated by adding the complex periodic forcing. The control performances are verified by calculating Lyapunov exponents. Numerical simulations are implemented to verify the results of these investigations. © 2010 IMACS. Published by Elsevier B.V. All rights reserved. Keywords: Hyperchaotic; Chaos; Control; Synchronization; Complex 1. Introduction In the last thirty years several researchers have focused their attention on the study of hyperchaotic systems with real variables in many important fields in applied nonlinear sciences, e.g., laser physics, secure communications, nonlinear circuits, synchronization, control, neural networks and active wave propagation [1,5,7,11,25,27,28,31]. A hyperchaotic attractor is defined as an attractor with at least two positive Lyapunov exponents. The sum of Lyapunov exponents must be negative to ensure that system is dissipative. The minimal dimension for a continuous hyperchaotic system is 4 [1,2,28]. Synchronization of hyperchaotic systems is a very important nonlinear phenomenon, which has been studied to date on dynamical systems described by real variables [10,27,32]. Hyperchaotic systems have complex dynamical behavior that possess some special feature such as being extremely sensitive to tiny variations of initial conditions and having bounded trajectories in the phase space with at least two positive Lyapunov exponents. 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 output of the drive system asymptotically. Synchronization in chaotic systems has become a direction of intense recent research. In the literature projective synchronization (PS) (or generalized synchronization) and modified projective synchro- nization (MPS) received much attention for chaotic real systems [13,24,29,33,35]. PS is a situation in which the state variables of the drive and response systems synchronize up to a constant scaling factor δ. MPS is defined if the responses ∗ Corresponding author. Tel.: +20 88 2412171; fax: +20 88 2342708. E-mail addresses: gmahmoud@aun.edu.eg, gmahmoud 56@yahoo.com (G.M. Mahmoud), emad eluan@yahoo.com (E.E. Mahmoud). 0378-4754/$36.00 © 2010 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.matcom.2010.03.012