# Robust Adaptive Synchronization of a Class of Uncertain Chaotic Systems with Unknown Time-Delay
## 4 Simulation Example and Results  
In this section, applying the adaptive control strategy proposed in the previous section, the robust synchronization of two Jerk chaotic systems with uncertainty and bounded external disturbances as well as unknown time delays is simulated and evaluated. The state space equations of the Jerk system are described as follows [30]:  
\[ \begin{split}\dot{x}_{1}(t)&=x_{2}(t),\\ \dot{x}_{2}(t)&=x_{3}(t),\\ \dot{x}_{3}(t)&=-\varepsilon_{1}x_{1}(t)-x_{2}(t)- \varepsilon_{2}x_{3}(t)-f(x_{1}(t)),\end{split}  \tag{24}\]  
where, \(\varepsilon_{1}\) and \(\varepsilon_{2}\) are system parameters and \(f(. )\) is a piecewise linear function that is defined as follows:  
\[f(x_{1}(t),t)=\frac{1}{2}(v_{0}-v_{1})\Big{[}|x_{1}(t)+1|-\big{|}x_{1}(t)-1| \Big{]}+v_{1}x_{1}(t).\]  
where, \(v_{0},\ v_{1}\) are constant values. Accordingly, the dynamic equations of Jerk chaotic master and slave systems based on Equation (24) in the presence of external disturbances, non-linear uncertainties and unknown time delays are presented in the following form:  
\[ \begin{split}\dot{x}_{1}(t)&=x_{2}(t),\\ \dot{x}_{2}(t)&=x_{3}(t),\\ \dot{x}_{3}(t)&=-\varepsilon_{1}x_{1}(t)-x_{2}(t)- \varepsilon_{2}x_{3}(t)-f(x_{1}(t-\tau_{1}),t)+\Delta f(x(t),t)+d_{1}(t),\end{split}  \tag{25}\]  
and the slave system is provided with a robust adaptive control strategy in the following form:  
\[ \begin{split}\dot{y}_{1}(t)&=y_{2}(t),\\ \dot{y}_{2}(t)&=y_{3}(t)+u_{0}(t),\\ \dot{y}_{3}(t)&=-\varepsilon_{1}y_{1}(t)-y_{2}(t)- \varepsilon_{2}y_{3}(t)-g(y_{1}(t-\tau_{2}),t)+\Delta g(y(t),t)+d_{2}(t)+u(t). \end{split}  \tag{26}\]In the simulation process, the disturbances and boundary uncertainties of the system that satisfy the conditions of Assumption 1 are regarded as:  
\[d_{1}(t)=(\sin(3t)+(\cos(2t))^{2}),\;\;d_{2}(t)=(0.2 sin(t)+\sin(\pi t)), \tag{27}\]  
\[\Delta f(x(t),t)=0.1 sin(x_{1}(t)+x_{2}(t)-5x_{3}(t)),\;\Delta g(y(t),t)=0.3 sin(y _{1}(t)+y_{2}(t)-y_{3}(t)). \tag{28}\]  
Additionally, piecewise linear functions with unknown time delays in the master and slave systems are illustrated as follows:  
\[f(x_{1}(t-\tau_{1}),t)=\frac{1}{2}(v_{0}-v_{1})\left[\left|x_{1}(t-\tau_{1})+1 \right|-\left|x_{1}(t-\tau_{1})-1\right|\right]+v_{1}x_{1}(t-\tau_{1}), \tag{29}\]  
\[g(y_{1}(t-\tau_{2}),t)=\frac{1}{2}(v_{0}-v_{1})\left[\left|y_{1}(t-\tau_{2})+1 \right|-\left|y_{1}(t-\tau_{2})-1\right|\right]+v_{1}y_{1}(t-\tau_{2}). \tag{30}\]  
Established on Equations (25)-(30), the dynamic equations of synchronization error are defined as follows:  
\[ \begin{array}{c}\dot{e}_{1}(t)=e_{2}(t),\\ \dot{e}_{2}(t)=e_{3}(t)+u_{0}(t),\\ \dot{e}_{3}(t)=-\varepsilon_{1}e_{1}(t)-e_{2}(t)-\varepsilon_{2}e_{3}(t)-g(y_ {1}(t-\tau_{2}),t)+f(x_{1}(t-\tau_{1}),t)\\ +\Delta g(y(t),t)-\Delta f(x(t),t)+d_{2}(t)-d_{1}(t)+u(t).\end{array}  \tag{31}\]  
By describing \(u_{0}(t)=k_{1}e_{1}(t)+k_{2}e_{2}(t)-e_{3}(t)\), the LTI segment of the synchronization error system (31) is defined as follows:  
\[ \begin{array}{c}\dot{e}_{1}(t)=e_{2}(t),\\ \dot{e}_{2}(t)=k_{1}e_{1}(t)+k_{2}e_{2}(t).\end{array}  \tag{32}\]  
Therefore \(k_{1},\;k_{2}\) should be determined in such a way that according to Theorem 1 the error dynamics \(e_{1}(t),\;e_{2}(t)\) in system (32) are stable and \(u(t)\) based on a robust adaptive controller designed based on the adaptive rules in Theorem 2 are considered in form (11). Accordingly, in order to evaluate the performance of the adaptive control strategy proposed for robust synchronization of systems (25) and (26), the initial conditions of the master and slave systems are equal to \([-0.5,\;0.5,\;0.87]\) and \([2,\;-1,\;-2]\), respectively, and are considered, the controller parameters \(u_{0}(t)\) are equal to \(k_{1}=-1,\;k_{2}=-2\) and in the controller (11), \(k_{0}=+1\). Moreover, to improve the controller's performance in detracting the output chattering rate, instead of the \(sign(\cdot)\) function, the \(tanh(\cdot)\) function is exploited in the controller structure (11).