A ug 1 99 8 Marginal hyperchaos synchronization with a single driving variable

The seminal papers by Pecora and Carrol (PC) [1] and Ott, Grebogi and Yorke (OGY) [2] in 1990 have induced avalanche of research works in the field of chaos control. Chaos synchronization in dynamical systems is one of methods of controling chaos, see, e.g. [1-8] and references therein.The interest to chaos synchronization in part is due to the application of this phenomenen in secure communications, in modeling of brain activity and recognition processes,etc [1-8]. Also it should be mentioned that this method of chaos control may result in improved performance of chaotic systems [1-8]. According to PC [1] synchronization of two systems occurs when the trajectories of one of the systems will converge to the same values as the other and they will remain in step with each other. For the chaotic systems synchronization is performed by the linking of chaotic systems with a common signal or signals (the so-called drivers): suppose that we have a chaotic dynamical system of three or more state variables. In the above mentioned way of chaos control one or some of these state variables can be used as an input to drive a subsystem consisting of remaining state variables and which is a replica of part of the original system.In [1] it has been shown that if the real parts of the Lyapunov exponents for the subsystem (below: sub-Lyapunov exponents) are negative then the subsystem synchronizes to the chaotic evolution of original system. If the largest sub-Lyapunov exponent is not negative, then one can use the nonreplica approach to chaos synchronization [9]. Within the nonreplica approach to chaos synchronization one can try to perform chaos synchronization between the original chaotic system and nonreplica response system with control terms vanishable upon synchronization.To be more specific, one can try to make negative the real parts of the conditional Lyapunov exponents of the nonreplica response system. As it has been shown in [9] from the application viewpoint nonreplica approach has some advantages over the replica one. Recently in [10] it has been indicated that for more secure communication purposes the use of hy-perchaos is more reliable. Quite naturally in the light of this result the investigation of hyperchaos synchronization is of paramount importance. According to Pyragas for hyperchaos synchronization at least two drive variables are needed [11]. Recently this idea was challenged in [12] in the sense that instead of several driving variables one can try to …