# Multi-Switching Combination Synchronization of Three Fractional-Order Delayed Systems
## 4 Numerical Examples
: _Supposing that \(f_{i}\
eq 0\), \(g_{i}=0\) and \(h_{i}=0\)\((i=1,2,3)\), the system in Equation (30) can be stabilized to its equilibrium \(O(0,0,0)\) with the following controllers_  
\[ \begin{cases}U_{1}=\dfrac{1}{f_{1}}\{(k_{1}-1)(f_{1}z_{1})-f_{1}(-(z_{2}+z_{3 })-0.8z_{1}(t-\tau)-z_{1})\},\\ U_{2}=\dfrac{1}{f_{2}}\{(k_{2}-1)(f_{2}z_{2})-f_{2}(z_{1}+(a_{3}-1)z_{2}-z_{2}( t-\tau))\},\\ U_{3}=\dfrac{1}{f_{3}}\{(k_{3}-1)(f_{3}z_{3})-f_{3}(z_{3}(z_{1}-m_{3})+b_{3}-z_{ 3}-z_{3}(t-\tau))\}.\end{cases}  \tag{41}\]  
**Case 2**  
From the systems in Equations (24), (27) and (30), we have  
\[ \begin{cases}D^{\alpha}e_{112}=f_{1}D^{\alpha}z_{1}-g_{1}D^{\alpha}x_{1}-h_{ 2}D^{\alpha}y_{2},\\ D^{\alpha}e_{223}=f_{2}D^{\alpha}z_{2}-g_{2}D^{\alpha}x_{2}-h_{3}D^{\alpha}y_ {3},\\ D^{\alpha}e_{331}=f_{3}D^{\alpha}z_{3}-g_{3}D^{\alpha}x_{3}-h_{1}D^{\alpha}y_ {1}.\end{cases}  \tag{42}\]  
such that  
\[ \begin{cases}\lim\limits_{t\rightarrow+\infty}\parallel f_{1}z_{1}-g_{1}x_{ 1}-h_{2}y_{2}\parallel=0,\\ \lim\limits_{t\rightarrow+\infty}\parallel f_{2}z_{2}-g_{2}x_{2}-h_{3}y_{3} \parallel=0,\\ \lim\limits_{t\rightarrow+\infty}\parallel f_{3}z_{3}-g_{3}x_{3}-h_{1}y_{1} \parallel=0.\end{cases}  \tag{43}\]Substituting the systems in Equations (24), (27), and (30) into the system in Equation (42) yields:  
\[ \begin{cases}D^{\alpha}e_{112}=f_{1}(-(z_{2}+z_{3})+0.2z_{1}(t-\tau)+U_{1})-g_{1} (a_{1}(x_{2}-x_{1}))\\ \qquad-h_{2}((c_{2}-a_{2})y_{1}-y_{1}y_{3}+c_{2}y_{2}),\\ D^{\alpha}e_{223}=f_{2}(z_{1}+a_{3}z_{2}+U_{2})-g_{2}(c_{1}x_{1}-x_{2}-x_{1}x_{ 3})\\ \qquad-h_{3}(y_{1}y_{2}-b_{2}y_{3}(t-\tau)),\\ D^{\alpha}e_{331}=f_{3}(z_{3}(z_{1}-m_{3})+b_{3}+U_{3})-g_{3}(x_{1}x_{2}-b_{1}x _{3}(t-\tau))\\ \qquad-h_{1}(a_{2}(y_{2}-y_{1})).\end{cases}  \tag{44}\]  
Here, we have the following similar results. **Theorem 3**. : _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_  
\[ \begin{cases} U_{1}=\dfrac{1}{f_{1}}\{(k_{1}-1)(f_{1}z_{1}-g_{1}x_{1}-h_{2}y_{2 })+g_{1}(a_{1}x_{2}-(a_{1}+1)x_{1}-x_{1}(t-\tau))\\ \qquad+h_{2}((c_{2}-a_{2})y_{1}-y_{1}y_{3}+(c_{2}-1)y_{2}-y_{2}(t-\tau))-f_{1} (-(z_{2}+z_{3})-0.8z_{1}(t-\tau)-z_{1})\},\\ U_{2}=\dfrac{1}{f_{2}}\{(k_{2}-1)(f_{2}z_{2}-g_{2}x_{2}-h_{3}y_{3})+g_{2}(c_{1} x_{1}-2x_{2}-x_{1}x_{3}-x_{2}(t-\tau))\\ \qquad+h_{3}(y_{1}y_{2}-(b_{2}+1)y_{3}(t-\tau)-y_{3})-f_{2}(z_{1}+(a_{3}-1)z_{ 2}-z_{2}(t-\tau))\},\\ U_{3}=\dfrac{1}{f_{3}}\{(k_{3}-1)(f_{3}z_{3}-g_{3}x_{3}-h_{1}y_{1})+g_{3}(x_{ 1}x_{2}-(b_{1}+1)x_{3}(t-\tau)-x_{3})\\ \qquad+h_{1}(a_{2}(y_{2}-y_{1})-y_{1}-y_{1}(t-\tau))-f_{3}(z_{3}(z_{1}-m_{3}) +b_{3}-z_{3}-z_{3}(t-\tau))\}.\end{cases}  \tag{45}\]  
**Corollary 4**. : _(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_  
\[ \begin{cases} U_{1}=\dfrac{1}{f_{1}}\{(k_{1}-1)(f_{1}z_{1}-h_{2}y_{2})+h_{2}((c _{2}-a_{2})y_{1}-y_{1}y_{3}+(c_{2}-1)y_{2}-y_{2}(t-\tau))\\ \qquad-f_{1}(-(z_{2}+z_{3})-0.8z_{1}(t-\tau)-z_{1})\},\\ U_{2}=\dfrac{1}{f_{2}}\{(k_{2}-1)(f_{2}z_{2}-h_{3}y_{3})+h_{3}(y_{1}y_{2}-(b_ {2}+1)y_{3}(t-\tau)-y_{3})\\ \qquad-f_{2}(z_{1}+(a_{3}-1)z_{2}-z_{2}(t-\tau))\},\\ U_{3}=\dfrac{1}{f_{3}}\{(k_{3}-1)(f_{3}z_{3}-h_{1}y_{1})+h_{1}(a_{2}(y_{2}-y_ {1})-y_{1}-y_{1}(t-\tau))\\ \qquad-f_{3}(z_{3}(z_{1}-m_{3})+b_{3}-z_{3}-z_{3}(t-\tau))\}.\end{cases}  \tag{46}\]  
_(ii) Supposing that \(f_{i}\
eq 0\), \(g_{i}\
eq 0\) and \(h_{i}=0\)\((i=1,2,3)\), hybrid projective synchronization between the systems in Equations (24) and (30) can be achieved with the following controllers_  
\[