# Multi-Switching Combination Synchronization of Three Fractional-Order Delayed Systems
## 4 Numerical Examples  
Numerical simulations were carried out to illustrate the above proposed multi-switching combination synchronization scheme. We used the same systems as in [32] with time-delays, which are fractional-order delayed Lorenz, Chen, and Rossler systems, and the numerical simulations were carried out in MATLAB. The fractional-order delayed Lorenz system [37] was considered as the first drive system  
\[ \begin{cases}D^{\alpha}x_{1}=a_{1}(x_{2}-x_{1}),\\ D^{\alpha}x_{2}=c_{1}x_{1}-x_{2}-x_{1}x_{3},\\ D^{\alpha}x_{3}=x_{1}x_{2}-b_{1}x_{3}(t-\tau).\end{cases}  \tag{24}\]  
The system in Equation (24) exhibits a chaotic attractor, as illustrated in Figure 1. The system in Equation (24) can be rewritten as  
\[ \begin{split} D^{\alpha}x(t)&=x(t)+x(t-\tau)+A(x(t),x( t-\tau)),\\ x(t)&=x(0),t\in[-\tau,0],\end{split}  \tag{25}\]  
where  
\[A(x(t),x(t-\tau))=\left( \begin{array}{c}a_{1}x_{2}-(a_{1}+1)x_{1}-x_{1}(t- \tau)\\ c_{1}x_{1}-2x_{2}-x_{1}x_{3}-x_{2}(t-\tau)\\ x_{1}x_{2}-(b_{1}+1)x_{3}(t-\tau)-x_{3}\end{array} \right). \tag{26}\]  
The fractional-order delayed Chen system [14] is the second drive system  
\[ \begin{cases}D^{\alpha}y_{1}=a_{2}(y_{2}-y_{1}),\\ D^{\alpha}y_{2}=(c_{2}-a_{2})y_{1}-y_{1}y_{3}+c_{2}y_{2},\\ D^{\alpha}y_{3}=y_{1}y_{2}-b_{2}y_{3}(t-\tau).\end{cases}  \tag{27}\]  
Figure 1: Chaotic attractor of Lorenz system with \(\alpha=0.95,\ \tau=0.4\): (**a**) \(x_{3}-x_{1}\) plane; (**b**) \(x_{3}-x_{2}\) plane; (**c**) \(x_{2}-x_{1}\) plane; and (**d**) \(x_{3}-x_{1}-x_{2}\) space. The system in Equation (27) displays a chaotic attractor, as shown in Figure 2. We rewrite the system in Equation (27) as  
\[ \begin{split} D^{\alpha}y(t)&=y(t)+y(t-\tau)+B(y(t),y(t -\tau)),\\ y(t)&=y(0),t\in[-\tau,0],\end{split}  \tag{28}\]  
where  
\[B(y(t),y(t-\tau))=\left( \begin{array}{l}a_{2}(y_{2}-y_{1})-y_{1}-y_{1}(t- \tau)\\ (c_{2}-a_{2})y_{1}-y_{1}y_{3}+(c_{2}-1)y_{2}-y_{2}(t-\tau)\\ y_{1}y_{2}-(b_{2}+1)y_{3}(t-\tau)-y_{3}\end{array} \right). \tag{29}\]  
The fractional-order delayed Rossler system is the response system, given by  
\[ \begin{cases}D^{\alpha}z_{1}=-(z_{2}+z_{3})+0.2z_{1}(t-\tau)+U_{1},\\ D^{\alpha}z_{2}=z_{1}+a_{3}z_{2}+U_{2},\\ D^{\alpha}z_{3}=z_{3}(z_{1}-m_{3})+b_{3}+U_{3}.\end{cases}  \tag{30}\]  
where \(U_{1},\ U_{2}\) and \(U_{3}\) are determined afterwards. Without the controllers, the system in Equation (30) exhibits a chaotic attractor, as illustrated in Figure 3. The system in Equation (30) is rewritten as  
\[ \begin{split} D^{\alpha}z(t)&=z(t)+z(t-\tau)+C(z(t), z(t-\tau))+U,\\ z(t)&=z(0),t\in[-\tau,0],\end{split}  \tag{31}\]  
Figure 2: Chaotic attractor of Chen system with \(\alpha=0.95,\ \tau=0.4\): (**a**) \(y_{3}-y_{1}\) plane; (**b**) \(y_{3}-y_{2}\) plane; (**c**) \(y_{2}-y_{1}\) plane; and (**d**) \(y_{3}-y_{1}-y_{2}\) space. where  
\[C(z(t),z(t-\tau))=\left( \begin{array}{c}-(z_{2}+z_{3})-0.8z_{1}(t-\tau)-z_{1}\\ z_{1}+(a_{3}-1)z_{2}-z_{2}(t-\tau)\\ z_{3}(z_{1}-m_{3})+b_{3}-z_{3}-z_{3}(t-\tau)\end{array} \right). \tag{32}\]  
For the systems in Equations (24), (27) and (30), there are eight possible switch combination synchronization cases. When \(k\
eq l\
eq m\), we have \(e_{123},\ e_{231},\ e_{312}\) and \(e_{132},\ e_{213},\ e_{321}\). When \(k=m\
eq l\), we have \(e_{121},\ e_{232},\ e_{313}\) and \(e_{131},\ e_{212},\ e_{323}\). When \(k\
eq l=m\), we have \(e_{122},\ e_{233},\ e_{311}\) and \(e_{133},\ e_{211},\ e_{322}\). When \(k=l\
eq m\), we have \(e_{112},\ e_{223},\ e_{331}\) and \(e_{113},\ e_{221},\ e_{332}\). We randomly pick two cases  
\[ \begin{cases}e_{123}=f_{1}z_{1}-g_{2}x_{2}-h_{3}y_{3},\\ e_{231}=f_{2}z_{2}-g_{3}x_{3}-h_{1}y_{1},\quad\text{\bf case 1}\\ e_{312}=f_{3}z_{3}-g_{1}x_{1}-h_{2}y_{2}.\end{cases}  \tag{33}\]  
and  
\[ \begin{cases}e_{112}=f_{1}z_{1}-g_{1}x_{1}-h_{2}y_{2},\\ e_{223}=f_{2}z_{2}-g_{2}x_{2}-h_{3}y_{3},\quad\text{\bf case 2}\\ e_{331}=f_{3}z_{3}-g_{3}x_{3}-h_{1}y_{1}.\end{cases}  \tag{34}\]  
In the following, we analyze these two cases in detail.