# Multi-Switching Combination Synchronization of Three Fractional-Order Delayed Systems
## 4 Numerical Examples
\begin{cases} U_{1}=\dfrac{1}{f_{1}}\{(k_{1}-1)(f_{1}z_{1}-g_{1}x_{1})+g_{1}(a_{ 1}x_{2}-(a_{1}+1)x_{1}-x_{1}(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}-g_{2}x_{2})+g_{2}(c_{1}x_{1}-2x_{2} -x_{1}x_{3}-x_{2}(t-\tau))\\ \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}-g_{3}x_{3})+g_{3}(x_{1}x_{2}-(b_ {1}+1)x_{3}(t-\tau)-x_{3})\\ \qquad-f_{3}(z_{3}(z_{1}-m_{3})+b_{3}-z_{3}-z_{3}(t-\tau))\}.\end{cases}  \tag{47}\]  
**Corollary 5**. : _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_  
\[ \begin{cases} U_{1}=\frac{1}{f_{1}}\{(k_{1}-1)(f_{1}z_{1})-f_{1}(-(z_{2}+z_{3})-0.8 z_{1}(t-\tau)-z_{1})\},\\ U_{2}=\frac{1}{f_{2}}\{(k_{2}-1)(f_{2}z_{2})-f_{2}(z_{1}+(a_{3}-1)z_{2}-z_{2}(t- \tau))\},\\ U_{3}=\frac{1}{f_{3}}\{(k_{3}-1)(f_{3}z_{3})-f_{3}(z_{3}(z_{1}-m_{3})+b_{3}-z_ {3}-z_{3}(t-\tau))\}.\end{cases}  \tag{48}\]  
The system parameters are given as \(a_{1}=10\), \(b_{1}=\frac{8}{3}\), \(c_{1}=28\), \(a_{2}=35\), \(b_{2}=3\), \(c_{2}=28\), \(a_{3}=0.4\), \(b_{3}=0.2\), \(m_{3}=10\), thus the systems in Equations (24), (27) and (30) exhibit chaotic behaviors, respectively. We assume \(f_{1}=f_{2}=f_{3}=1\), \(g_{1}=g_{2}=g_{3}=1\) and \(h_{1}=h_{2}=h_{3}=1\), and the initial values are \((x_{1}(0),x_{2}(0),x_{3}(0))=(-20,2,3),(y_{1}(0),y_{2}(0),y_{3}(0))=(7,4.04,20)\) and \((z_{1}(0),z_{2}(0),z_{3}(0))=(1,2,40)\), respectively. Multi-switching combination synchronization between the systems in Equations (24), (27) and (30) can be realized with \(K=diag\{-10,-10,-10\}\). Figure 4 illustrates synchronization errors \(e_{123}\), \(e_{231}\), \(e_{312}\). Figure 5 shows synchronization states \(x_{2}+y_{3}\) vs. \(z_{1}\), \(x_{3}+y_{1}\) vs. \(z_{2}\) and \(x_{1}+y_{2}\) vs. \(z_{3}\) of the drive systems in Equations (24) and (27) and the response system in Equation (30). Figure 6 displays synchronization errors \(e_{112}\), \(e_{223}\), and \(e_{331}\). Figure 7 illustrates synchronization states \(x_{1}+y_{2}\) vs. \(z_{1}\), \(x_{2}+y_{3}\) vs. \(z_{2}\) and \(x_{3}+y_{1}\) vs. \(z_{3}\) of the drive systems in Equations (24) and (27) and the response system in Equation (30). In Figures 4-7, we can see that the multi-switching combination synchronization errors converge to zero, i.e., the multi-switching combination synchronizations for Cases 1 and 2 are achieved, respectively. The feedback gain matrix \(K\) is an important factor to affect the convergence of the error systems. With the increase of the absolute value of \(k_{i}\), the convergence time will be shortened. Thus, we carried out one more simulation with \(K=diag\{-40,-40,-40\}\). Figures 8 and 9 illustrate the synchronization errors for Cases 1 and 2, respectively. By comparing Figures 4 and 8, as well as Figures 6 and 9, it is easy to see that convergence times are shortened obviously. Figure 5: Responses for states between the systems in Equations (24) and (27) and the system in Equation (30): (**a**) \(x_{2}+y_{3}\) vs. \(z_{1}\); (**b**) \(x_{3}+y_{1}\) vs. \(z_{2}\); and (**c**) \(x_{1}+y_{2}\) vs. \(z_{3}\). Figure 6: Synchronization errors among the systems in Equations (24), (27) and (30) with \(k_{1}=k_{2}=k_{3}=-10\): (**a**) \(e_{112}\); (**b**) \(e_{223}\); and (**c**) \(e_{331}\). Figure 8: Synchronization errors among the systems in Equations (24), (27) and (30) with \(k_{1}=k_{2}=k_{3}=-40\): (**a**) \(e_{123}\); (**b**) \(e_{231}\); and (**c**) \(e_{312}\). Figure 7: Responses for states between the systems in Equations (24) and (27) and the system in Equation (30): (**a**) \(x_{1}+y_{2}\) vs. \(z_{1}\); (**b**) \(x_{2}+y_{3}\) vs. \(z_{2}\); and (**c**) \(x_{3}+y_{1}\) vs. \(z_{3}\).