# Robust Adaptive Synchronization of a Class of Uncertain Chaotic Systems with Unknown Time-Delay
## 2 Preliminaries and Problem Formulation  
In this paper, a class of uncertain chaotic systems with unknown time delay is considered so that the general form of demonstrating dynamic equations to its companion form in the format of master and slave systems is defined as follows:  
\[\left\{ \begin{array}{c}\dot{x}_{i}=x_{i+1},\quad 1\leq i\leq n-1\\ \dot{x}_{n}=f(x(t-\tau_{1}),t)+\Delta f(x(t),t)+d_{1}(t),\end{array} \right. \tag{1}\]  
and the slave system is written in the following form:  
\[\left\{ \begin{array}{c}\dot{y}_{i}=y_{i+1},\quad 1\leq i\leq n-1\\ \dot{y}_{n}=g(y(t-\tau_{2}),t)+\Delta g(y(t),t)+d_{2}(t)+u(t).\end{array} \right. \tag{2}\]  
So that \(x(t),\ y(t)\in R^{n}\) denote the dynamic states of the master and slave systems, \(f(x(t-\tau_{1}),t),\ g(y(t-\tau_{2}),t)\in R\) nonlinear functions with unknown time delays with delays \(\tau_{1},\ \tau_{2}\) and \(\Delta f(x(t),t),\ \Delta g(x(t),t)\) express nonlinear bounded uncertainties in the slave and master systems. Also, \(d_{1}(t),\ d_{2}(t)\) indicate the external disturbances applied to the master and slave systems, and \(u(t)\) the control law applied to the slave system. The differential equations expressed in the forms correspond to a number of known chaotic systems such as: Van der Pol oscillator systems, Duffing's oscillator, [PERSON]'s system, [PERSON]'s system, etc. [29]. **Definition 1**. : _The master and the slave systems defined in forms (1) and (2) have robust synchronization if for all conditions governing the system, including external disturbance, uncertainties and unknown time-delays, for each initial condition, the following condition must be met:_  
\[\lim_{t\rightarrow\infty}\|y_{i}(t)-x_{i}(t)\|=lim_{t\rightarrow\infty}\|e_{ i}(t)\|=0,\ \ i=1,\ldots,n. \tag{3}\]  
Therefore, \(e_{i}(t)\) describes the synchronization error of the master and the slave systems. Accordingly, the differential equations of the synchronization error dynamics for uncertain master and slave chaotic systems with unknown time-delay express in forms (1) and (2) are defined as:  
\[\left\{ \begin{array}{c}\dot{e}_{i}=e_{i+1},\\ \dot{e}_{n}=g(y(t-\tau_{2}),t)+\Delta g(x(t),t)+d_{2}(t)-(f(x(t-\tau_{1}),t)+ \Delta f(x(t),t)+d_{1}(t))+u(t).\end{array} \right. \tag{4}\]  
Therefore, in this paper, we seek to design a robust adaptive controller to perform robust synchronization of chaotic systems (1) and (2) in the presence of external disturbances, bounded non-linear uncertainties and existing unknown time-delays in accordance with Definition 1. Put another way, the proposed designed controller can operate despite the existing conditions in the master and slave systems in such a way that the state dynamics of the slave system in a finite time is in accordance with the behavior of the dynamics of the master system. Also, the adaptation of state dynamics under the assumed conditions remain stable and robust until the synchronization error tends to zero. **Assumption 1**. : _Uncertain external disturbances \(d_{1}(t),\ d_{2}(t)\) and uncertain bounded non-linear uncertainties \(\Delta f(x(t),t)\) and \(\Delta g(x(t),t)\) in master and slave systems (1) and (2) satisfy the following conditions:_  
\[\|\Delta f(x(t),t)\|\leq\alpha_{1}h_{1}(x),\ \|\Delta g(y(t),t)\|\leq\alpha_{2}h_{2 }(y),\ \|d_{1}(t)\|\leq\gamma_{1}\ \text{and}\ \|d_{2}(t)\|\leq\gamma_{2}. \tag{5}\]  
_Thus \(\|\cdot\|\) signifies the norm \(l_{1}\) and \(\alpha_{1},\alpha_{2},\ \gamma_{1},\gamma_{2}\) are the values of the unknown real positive constant. Also, \(h_{1}(\cdot),\,h_{2}(\cdot)\) are generally known functions._  
**Assumption 2**.