Fuzzy model-based adaptive synchronization of time-delayed chaotic systems Nastaran Vasegh a, * , Vahid Johari Majd b,1 a Faculty of Electrical Engineering, K.N. Toosi University of Technology, P.O. Box 16315-1355, Tehran, Iran b Electrical Engineering Department, Tarbiat Modares University, P.O. Box 14115-143, Tehran, Iran Accepted 10 September 2007 Communicated by Prof. L. Marek-Crnjac Abstract In this paper, fuzzy model-based synchronization of a class of first order chaotic systems described by delayed-dif- ferential equations is addressed. To design the fuzzy controller, the chaotic system is modeled by Takagi–Sugeno fuzzy system considering the properties of the nonlinear part of the system. Assuming that the parameters of the chaotic sys- tem are unknown, an adaptive law is derived to estimate these unknown parameters, and the stability of error dynamics is guaranteed by Lyapunov theory. Numerical examples are given to demonstrate the validity of the proposed adaptive synchronization approach. Ó 2007 Elsevier Ltd. All rights reserved. 1. Introduction Chaos synchronization is an important topic in nonlinear systems. Generally, in the synchronization phenomenon, the trajectories of two different systems (drive and response systems) become identical in spite of having different initial conditions and/or different structures. Since the work of Pecora and Carroll [10], there have been many publications on synchronization of chaotic systems ([1] and the references therein). Moreover, it has been widely explored in a variety of fields including physical, chemical and ecological processes as well as secure communications [2]. Various synchronization schemes such as backstepping design [13], active control [5], nonlinear control [6], adaptive control [3], and adaptive fuzzy control [17] have been successfully employed for chaos synchronization. In most liter- ature on synchronization, however, low-dimensional systems are considered such as Lorenz and Duffing systems [3,5,6,8,13]. On the other hand, it is well known that low-dimensional chaotic signals can be easily detected, so they are not suitable for applications such as secure communications [11]. Time-delayed chaotic systems are infinite-dimen- sional and have finite high-dimensional hyper-chaotic attractors, which can yield low detectability and high safety for secure communication [12]. 0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.09.030 * Corresponding author. Tel.: +98 21 44213993. E-mail addresses: vasegh@eetd.kntu.ac.ir (N. Vasegh), majd@modares.ac.ir (V.J. Majd). 1 Tel.: +98 21 8801 1001x3353; fax: +98 21 8800 6544. Available online at www.sciencedirect.com Chaos, Solitons and Fractals 40 (2009) 1484–1492 www.elsevier.com/locate/chaos