## 4 Numerical Examples
Figures 10 and 11 show the trajectories of synchronization errors based on the ETC strategy and transmission errors of nodes, which prove that all nodes of the systems are synchronized with the distributed controller. It can be seen from Figures 9 and 10 that the proposed ETC strategy saves communication resources and reduces unnecessary transmission of the systems while ensuring the convergence effect. The above simulation results show that as long as the dwell time and attack frequency of DoS attacks satisfy the upper limit constraints calculated in this paper, the complex cyber-physical networks can achieve the synchronization of all nodes with the designed ETC strategy, which also verifies the effectiveness of Theorem 3 to calculate the tolerable attacks._Figure 4: Tracking error trajectories of nodes with event-triggered protocol without DoS attacks. Figure 5: Error trajectories of nodes without DoS attacks. Figure 6: Transmission attempts of follower nodes. Figure 8: DoS signal and switching signal. Figure 7: Switching topologies. Figure 9: Tracking error trajectories of nodes under DoS attacks. **Example 2**. : _In order to verify the superiority of the proposed method, we consider the security control problem solved in [38]. The parameters borrowed from [38] are as follows:_  
\[A=\left[ \begin{array}{cc}0&-1\\ 2&1\end{array} \right],B=\left[ \begin{array}{cc}1&0\\ 0&2\end{array} \right],H=\left[ \begin{array}{ccc}3&-1&-1\\ -1&3&-1\\ -1&-1&3\end{array} \right],\]  
\[\overline{H_{1}}=\left[ \begin{array}{ccc}2&-1&0\\ -1&3&-1\\ 0&-1&2\end{array} \right],\overline{H_{2}}=\left[ \begin{array}{ccc}1&0&0\\ 0&2&-1\\ 0&-1&2\end{array} \right].\]  
_The DoS attack behavior and topologies are the same as [38], as shown in Figures 12 and 13. Other parameters are chosen as follows:_  
\[\varepsilon_{\min}=5.231,\varepsilon_{1}=1,\gamma_{\min}=0.43,q_{1}=4.783,q_{ 2}=6.893,\]  
\[q_{3}=2.785,\alpha=7.981,\beta_{1}=13,\beta_{2}=0.2983,\theta_{1}^{*}=0.5897,\beta_{3}=18.\]  
_By Theorems 1 and 3, one can obtain the controller gains \(K\) and \(F\) as follows:_  
\[K=\left[ \begin{array}{cc}-1.6521&0.7893\\ 0.0672&-2.0972\end{array} \right],F=\left[ \begin{array}{cc}-1.7624&-0.9821\\ 0.0762&-0.9879\end{array} \right].\]  
_and_  
\[F_{a_{\left(\phi,t\right)}}=\frac{N_{a_{\left(10,t\right)}}}{t-t_{0}}\leq \frac{\theta_{1}^{*}}{2\ln(\mu)+(\alpha+\beta_{3})\Lambda_{*}}=0.1087,\tau_{a }>\frac{\alpha+\beta_{3}}{\alpha+\theta_{1}^{*}}=4.1253.\]  
Figure 11: Error trajectories of nodes under DoS attacks. Figure 10: Tracking error trajectories of nodes with event-triggered protocol under DoS attacks. This means that the DoS attacks cannot occur more than 0.1087 s during a unit of time and duration smaller than 4.1253 s. The initial values of the error systems borrowed from [38] is \(e(0)=\left[ \begin{array}{cccc}0.25&0.09&0.09\end{array} \right]^{\mathsf{T}}\). Figure 14 shows the trajectories of synchronization errors based on the ETC strategy. It is easy to see that all nodes of the systems are synchronized under the distributed controller after 110 s compared with 600 s in [38], which verifies the superiority of the results. The above simulation results show that as long as the dwell time and attack frequency of DoS attacks satisfy the upper limit constraints calculated in this paper, the complex cyber-physical networks can achieve the synchronization of all nodes by the designed ETC strategy.