# Multi-Switching Combination Synchronization of Three Fractional-Order Delayed Systems
## 4 Numerical Examples
**Case 1**  
From the systems in Equations (24), (27) and (30), we have the error dynamical system  
\[ \begin{cases}D^{\alpha}e_{123}=f_{1}D^{\alpha}z_{1}-g_{2}D^{\alpha}x_{2}-h_{3}D^ {\alpha}y_{3},\\ D^{\alpha}e_{231}=f_{2}D^{\alpha}z_{2}-g_{3}D^{\alpha}x_{3}-h_{1}D^{\alpha}y_{1 },\\ D^{\alpha}e_{312}=f_{3}D^{\alpha}z_{3}-g_{1}D^{\alpha}x_{1}-h_{2}D^{\alpha}y_{2 }.\end{cases}  \tag{35}\]  
such that  
\[ \begin{cases}\lim_{t\to+\infty}\parallel f_{1}z_{1}-g_{2}x_{2}-h_{3}y_{3} \parallel=0,\\ \lim_{t\to+\infty}\parallel f_{2}z_{2}-g_{3}x_{3}-h_{1}y_{1} \parallel=0,\\ \lim_{t\to+\infty}\parallel f_{3}z_{3}-g_{1}x_{1}-h_{2}y_{2} \parallel=0.\end{cases}  \tag{36}\]  
Substituting the systems in Equations (24), (27) and (30) into the system in Equation (35) yields  
\[ \begin{cases}D^{\alpha}e_{123}=f_{1}(-(z_{2}+z_{3})+0.2z_{1}(t-\tau)+U_{1})-g_ {2}(c_{1}x_{1}-x_{2}-x_{1}x_{3})\\ \phantom{D^{\alpha}e_{123}=}-h_{3}(y_{1}y_{2}-b_{2}y_{3}(t-\tau)),\\ D^{\alpha}e_{231}=f_{2}(z_{1}+a_{3}z_{2}+U_{2})-g_{3}(x_{1}x_{2}-b_{1}x_{3}(t- \tau))\\ \phantom{D^{\alpha}e_{231}=}-h_{1}(a_{2}(y_{2}-y_{1})),\\ D^{\alpha}e_{312}=f_{3}(z_{3}(z_{1}-m_{3})+b_{3}+U_{3})-g_{1}(a_{1}(x_{2}-x_{ 1}))\\ \phantom{D^{\alpha}e_{312}=}-h_{2}((c_{2}-a_{2})y_{1}-y_{1}y_{3}+c_{2}y_{2}). \end{cases}  \tag{37}\]  
Here, we obtain the following results. **Theorem 2**. : _Multi-switching combination synchronization between the systems in Equations (24) and (27) and the system in Equation (30) can be achieved with the following controllers_  
\[ \begin{cases}\mathcal{U}_{1}=\dfrac{1}{f_{1}}\{(k_{1}-1)(f_{1}z_{1}-g_{2}x_{2} -h_{3}y_{3})+g_{2}(c_{1}x_{1}-2x_{2}-x_{1}x_{3}-x_{2}(t-\tau))\\ \phantom{D^{\alpha}e_{123}=}+h_{3}(y_{1}y_{2}-(b_{2}+1)y_{3}(t-\tau)-y_{3})-f_ {1}(-(z_{2}+z_{3})-0.8z_{1}(t-\tau)-z_{1})\},\\ \mathcal{U}_{2}=\dfrac{1}{f_{2}}\{(k_{2}-1)(f_{2}z_{2}-g_{3}x_{3}-h_{1}y_{1})+ g_{3}(x_{1}x_{2}-(b_{1}+1)x_{3}(t-\tau)-x_{3})\\ \phantom{D^{\alpha}e_{231}=}+h_{1}(a_{2}(y_{2}-y_{1})-y_{1}-y_{1}(t-\tau))-f_ {2}(z_{1}+(a_{3}-1)z_{2}-z_{2}(t-\tau))\},\\ \mathcal{U}_{3}=\dfrac{1}{f_{3}}\{(k_{3}-1)(f_{3}z_{3}-g_{1}x_{1}-h_{2}y_{2})+ g_{1}(a_{1}x_{2}-(a_{1}+1)x_{1}-x_{1}(t-\tau))\\ \phantom{D^{\alpha}e_{231}=}+h_{2}((c_{2}-a_{2})y_{1}-y_{1}y_{3}+(c_{2}-1)y_{2 }-y_{2}(t-\tau))\\ \phantom{D^{\alpha}e_{231}=}-f_{3}(z_{3}(z_{1}-m_{3})+b_{3}-z_{3}-z_{3}(t- \tau))\}.\end{cases}  \tag{38}\]  
Supposing \(F\
eq 0\) and \(G=0\) or \(H=0\), we have the following results. **Corollary 2**. : _(i) Supposing that \(f_{i}\
eq 0\), \(g_{i}=0\) and \(h_{i}\
eq 0\)\((i=1,2,3)\), multi-switching hybrid projective synchronization between the systems in Equations (27) and (30) can be achieved with the following controllers_  
\[ \begin{cases}U_{1}=\dfrac{1}{f_{1}}\{(k_{1}-1)(f_{1}z_{1}-h_{3}y_{ 3})+h_{3}(y_{1}y_{2}-(b_{2}+1)y_{3}(t-\tau)-y_{3})\\ \quad-f_{1}(-(z_{2}+z_{3})-0.8z_{1}(t-\tau)-z_{1})\},\\ U_{2}=\dfrac{1}{f_{2}}\{(k_{2}-1)(f_{2}z_{2}-h_{1}y_{1})+h_{1}(a_{2}(y_{2}-y_{ 1})-y_{1}-y_{1}(t-\tau))\\ \quad-f_{2}(z_{1}+(a_{3}-1)z_{2}-z_{2}(t-\tau))\},\\ U_{3}=\dfrac{1}{f_{3}}\{(k_{3}-1)(f_{3}z_{3}-h_{2}y_{2})+h_{2}((c_{2}-a_{2})y_ {1}-y_{1}y_{3}+(c_{2}-1)y_{2}-y_{2}(t-\tau))\\ \quad-f_{3}(z_{3}(z_{1}-m_{3})+b_{3}-z_{3}-z_{3}(t-\tau))\}.\end{cases}  \tag{39}\]  
_(ii) Similarly, supposing that \(f_{i}\
eq 0\), \(g_{i}\
eq 0\) and \(h_{i}=0\)\((i=1,2,3)\), multi-switching hybrid projective synchronization between the systems in Equations (24) and (30) can be achieved with the following controllers_  
\[ \begin{cases}U_{1}=\dfrac{1}{f_{1}}\{(k_{1}-1)(f_{1}z_{1}-g_{2}x_{ 2})+g_{2}(c_{1}x_{1}-2x_{2}-x_{1}x_{3}-x_{2}(t-\tau))\\ \quad-f_{1}(-(z_{2}+z_{3})-0.8z_{1}(t-\tau)-z_{1})\},\\ U_{2}=\dfrac{1}{f_{2}}\{(k_{2}-1)(f_{2}z_{2}-g_{3}x_{3})+g_{3}(x_{ 1}x_{2}-(b_{1}+1)x_{3}(t-\tau)-x_{3})\\ \quad-f_{2}(z_{1}+(a_{3}-1)z_{2}-z_{2}(t-\tau))\},\\ U_{3}=\dfrac{1}{f_{3}}\{(k_{3}-1)(f_{3}z_{3}-g_{1}x_{1})+g_{1}(a_{ 1}x_{2}-(a_{1}+1)x_{1}-x_{1}(t-\tau))\\ \quad-f_{3}(z_{3}(z_{1}-m_{3})+b_{3}-z_{3}-z_{3}(t-\tau))\}.\end{cases}  \tag{40}\]  
**Corollary 3**.