Synchronization of uncertain chaotic systems via backstepping approach Samuel Bowong a, * , F.M. Moukam Kakmeni b a INRIA Lorraine, Project CONGE, I.S.G.M.P. B^ at A, Ile de Saulcy, 57045 Metz Cedex 01, France b Laboratoire de M ecanique, D epartement de Physique, Facult e des Sciences, Universit e de Yaound e I, B.P. 812 Yaound e, Cameroon Accepted 8 December 2003 Abstract In this paper, adaptive synchronization of two uncertain chaotic systems is presented using adaptive backstepping approach. The master system is any smooth nonlinear chaotic system, while the slave system is a nonlinear chaotic system in the feedback form. Global stability and exponential synchronization between the master and slave systems can be achieved. The proposed approach offers a systematic design procedure for adaptive synchronization of a large class of continuous-time chaotic systems in the chaos research literature. Computer simulations are provided to verify the operation of the designed synchronization scheme. Ó 2004 Elsevier Ltd. All rights reserved. 1. Introduction Motivated by potential applications in physics, electrical engineering, communication theory and many other fields, the synchronization of chaotic systems has received an increasing interest [1–8]. It has been shown that two chaotic systems exhibit identical oscillations, i.e., synchronization error tends to zero for all t P 0 [1]. But despite the amount of theoretical and experimental results already obtained, chaos synchronization seems difficult task, over all if we think that: (i) due to sensitive dependence of chaos on initial conditions, it is almost impossible to reduce the same starting conditions, (ii) in matching exactly the master and slave systems, even infinitesimal parametric variations of any model will eventually result in divergence of orbits starting nearby each other, and (iii) parametric differences between chaotic systems (for instance, due to inaccuracy design or time variations) yield different attractors. To avoid these problems, some strategies have been reported (see [3,4,7,9,11–14], and references therein). In particular, several authors have reported adaptively estimation techniques. These techniques present an acceptable performance and allow synchro- nization, although the parameters are not known [9,12,14] or they are time-varying [3,4,7,11,13]. But the only drawback of these strategies is that the structure of parameters for a given model must be known. Although the structure of the parameters can be known in some cases, it would be desirable to have a scheme to achieve synchronization even if slave oscillator has little prior knowledge about the master system. This necessity of robustness can be required in some systems (for instance, the multimode laser, animal gait or oscillatory neural systems). In this paper, an algorithm for the adaptive chaos synchronization is developed. In particular, the synchronization problem is interpreted as a stabilization one. The goal is to stabilize, at the origin, the discrepancy between the drive and response systems. Discrepancy is defined as the dynamical differences between the drive and response systems and include: (1) model mismatches, which means that the model of the drive system might not be the same as that of the * Corresponding author. Address: B.P. 12639 Yaound e, Cameroon. Tel.: +237-996-41-64/785-5788; fax: +237-231-9584. E-mail addresses: sbowong@uycdc.uninet.cm (S. Bowong), fmoukam@uycdc.uninet.cm (F.M. Moukam Kakmeni). 0960-0779/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2003.12.084 Chaos, Solitons and Fractals 21 (2004) 999–1011 www.elsevier.com/locate/chaos