## 4 Controller Design  
We adopted the following time delay complex Chen system as the drive system \(L1\) at the sending end and response system \(L2\) at the receiving end:  
\[L1:\ \ \left\{\begin{array}{rcl}\dot{y}_{1}&=&a_{1}(y_{2}-y_{1}),\\ \dot{y}_{2}&=&(a_{2}-a_{1})y_{1}+a_{2}y_{2}-y_{1}y_{3}(t-\tau),\\ \dot{y}_{3}&=&-a_{3}y_{3}(t-\tau)+(1/2)(\ddot{y}_{1}y_{2}+y_{1}\ddot{y}_{2}), \end{array}\right. \tag{9}\]\[L2:\ \left\{\begin{array}{rcl}\dot{x}_{1}&=&a_{1}(x_{2}-x_{1})+v_{1}+jv_{2},\\ \dot{x}_{2}&=&(a_{2}-a_{1})x_{1}+a_{2}x_{2}-x_{1}x_{3}(t-\tau)+v_{3}+jv_{4},\\ \dot{x}_{3}&=&-a_{3}x_{3}(t-\tau)+(1/2)(\dot{x}_{1}x_{2}+y_{1}\dot{x}_{2})+v_{5 },\end{array}\right. \tag{10}\]  
where \(y_{1}=u_{1}+ju_{2}\) and \(y_{2}=u_{3}+ju_{4}\) are the complex state variables of the drive system \(L1\), \(x_{1}=u_{1}^{\prime}+ju_{2}^{\prime}\), and \(x_{2}=u_{3}^{\prime}+ju_{4}^{\prime}\) are the complex state variables of the response system \(L2\), and the real variables are \(y_{3}=u_{5}\) and \(x_{3}=u_{5}^{\prime}\). The controller is \(\mathbf{v}=diag\{v_{1}+jv_{2},v_{3}+jv_{4},v_{5}\}\). Here, \(\tau=1\) s and the system is chaotic, as shown as Figure 1.  
The system parameters (\(a_{1}=35,\ a_{2}=23,\ a_{3}=1,\tau=1\) s) and initial conditions (\(\mathbf{u}(0)=(1,2,3,4,1)^{\mathrm{T}}\), \(\mathbf{u}^{\prime}(0)=(1,2,3,4,1)^{\mathrm{T}}\)) are certified by the sender and receiver. We employed controller (6), with \(k_{1}=k_{2}=k_{3}=k_{4}=k_{5}=-100\). Therefore, the controller was designed as:  
\[\left\{\begin{array}{rl}v_{1}^{\prime}&=s_{1}^{\prime}-f_{1}^{\prime}(x)+k_{ 1}e_{1}^{\prime}\\ &=s_{1}-a_{1}(u_{3}^{\prime}(t-\tau)-u_{1}^{\prime})-k_{1}(u_{1}^{\prime}-h_{1 }*u_{1}+h_{2}*u_{2})\\ v_{1}^{\prime}&=s_{1}^{\prime}-f_{1}^{\prime}(x)+k_{2}e_{1}^{\prime}\\ &=s_{2}-a_{1}(u_{4}^{\prime}-u_{2}^{\prime})-k_{2}(u_{2}^{\prime}-h_{1}*u_{2}- h_{2}*u_{1})\\ v_{2}^{\prime}&=s_{2}^{\prime}-f_{2}^{\prime}(x)+k_{3}e_{2}^{\prime}\\ &=s_{3}-(a_{2}-a_{1})u_{1}^{\prime}-a_{2}u_{3}^{\prime}(t-\tau)-u_{1}^{\prime }u_{5}^{\prime}\\ &-k_{3}(u_{3}^{\prime}(t-\tau)-h_{3}*u_{3}+h_{4}*u_{4})\\ v_{2}^{\prime}&=s_{2}^{i}-f_{2}^{i}(x)+k_{4}e_{2}^{i}\\ &=s_{4}-(a_{2}-a_{1})u_{2}^{\prime}-a_{2}u_{4}^{\prime}-u_{2}^{\prime}u_{5}^{ \prime}\\ &-k_{4}(u_{4}^{\prime}-h_{3}*u_{4}-h_{3}*u_{4})\\ v_{3}&=s_{3}-f_{3}(x)+k_{5}e_{5}\\ &=s_{5}+a_{3}u_{3}^{\prime}(t-\tau)-(u^{\prime}1*u_{3}^{\prime}(t-\tau)+u_{2}^ {\prime}*u_{4}^{\prime})\\ &-k_{5}(u_{5}^{\prime}-h_{5}*u_{5})\end{array}\right. \tag{11}\]  
First, we chose \(h_{1}=h_{2}=j,h_{3}=-1\), which means that \(e_{1}=(u_{1}^{\prime}+u_{2})+j(u_{2}^{\prime}-u_{1})=0\), \(e_{2}=(u_{3}^{\prime}(t-\tau)+u_{4})+j(u_{4}^{\prime}-u_{3})=0\), and \(e_{3}=u_{5}^{\prime}+u_{5}=0\). In fact, \(x_{1},x_{2}\) and \(y_{1},y_{2}\) realize CCS, while \(x_{3}\) and \(y_{3}\) realize anti-synchronization. The attractor projections of this synchronization for the time delay complex Chen system are shown in Figure 7.  
Figure 7: Attractor projections of special synchronization for time delay complex Chen system. Red attractor, system \(L1\); green attractor, system \(L2\). (**a**) \(u_{2}-u_{5}\), (**b**) \(u_{1}-u_{4}\), (**c**) \(u_{1}-u_{3}-u_{5}\), and (**d**) \(u_{3}-u_{4}-u_{5}\).