# Robust Adaptive Synchronization of a Class of Uncertain Chaotic Systems with Unknown Time-Delay
## 1 Introduction  
The dynamic behaviors of a chaotic system looks like a random behavior. But in reality, the behavior of chaotic systems follows a natural order. On the other hand, the dynamic behavior of these chaotic systems is entirely dependent on the initial conditions, so that with the slightest change in these conditions, the behavior of these systems undergoes extreme changes [1]. Due to the properties of chaotic systems (quasi-random dynamic behavior and drastic dependence on initial conditions), it isdifficult to control these systems [2]. Accordingly, this challenge has become a fascinating topic for researchers in various fields [3]. One of the significant issues in the field of control of chaotic systems is the synchronization of such systems in the presence of various disturbances and uncertainties so that, by designing a control strategy, the dynamic behavior of the slave system follows the dynamics of the master system [4; 5; 6]. This issue was first introduced by [PERSON] and [PERSON] [7]. Owing to the highly substantial applications of synchronization such as secure communications, chemical processes, biological systems, and information processing, numerous control approaches have been submitted to synchronize chaotic systems. Among these techniques the active control method [7], adaptive control strategy [8; 9], adaptive observer base control strategy [10], sliding mode control method [11], predictive control [12], Linear feedback control [13], back-stepping control [14], robust control method [15], etc. can be mentioned. In many real and engineering systems comprising biological, physical, electrical, chemical, and communication systems, various uncertainties and time delays are unavoidable and create a lot of changes in the dynamic behavior of these systems and increase the complexity of the systems models. In these systems, achieving the desired behavior is hard and, therefore, the issues of stability and control of these systems have turned out to be a crucially important and a considerable topic. Accordingly, researchers have developed a variety of control strategies for synchronizing systems with uncertainty [16; 17; 18] and systems with a time delay [19; 20; 21; 22; 23]. In [16], fuzzy-neural network control for synchronization of uncertain chaotic systems is presented. In [20], an iterative learning controller to synchronize two non-linear systems with free time delay and couple free is proposed. An adaptive fuzzy controller design for synchronizing a class of uncertain non-linear systems is proposed in [17]. In [22], an adaptive intelligent controller is introduced to control uncertain systems with time delay. An observer-based fuzzy adaptive output feedback controller [21] is designed to control the stochastic system with time delay. In [19], the design of a synchronization controller based on robust adaptive neural networks for a class of time-delay uncertain chaotic systems is yielded. Adaptive synchronization has been proposed for a class of time-delay uncertain chaotic systems via fuzzy fractional-order neural networks [23]. Due to the increasing complexity of time-delayed systems, the synchronization of time-delayed chaotic systems with the application approach in secure telecommunications has also received great attention from researchers in this field. For example, finite-time synchronization of non-identical chaotic systems with multiple time-varying delays and bounded disturbances was investigated in [24]. [PERSON] et al. [25] adopted an improved synchronization to synchronize time-delayed chaotic Lur'e systems using sampled-data control. Also, in [26], [PERSON] et al. proposed a novel synchronization criterion of chaotic Lur'e systems with time delays using sampled-data control. A control strategy for heterogeneous uncertain chaotic systems with a time delay [27] is proposed in which a robust framework for synchronization error estimation is presented.