## 4 Numerical Simulation  
This section demonstrates the feasibility of the proposed theory through a simulation example. It indicates that by employing the devised control protocol, heterogeneous multi-agent systems characterized by cooperative-competitive interactions can attain consensus within a finite time. Consider a hybrid-order heterogeneous multi-agent systems (1) consisting of six agents, and the communication topology is depicted in Figure 1, where 1-3 are first-order agents and 4-6 are second-order agents. Obviously, the weight values of information exchange between agents shown in Figure 1 can divide six agents into two groups:\(\mathcal{V}_{1}=\{1,2,5\}\). \(\mathcal{V}_{2}=\{3,4,6\}\). Moreover, the matrix \(W=\text{diag}(1,1,-1,-1,1,-1)\) can be obtained from Lemma 1, which converts the signed adjacency matrix \(\mathcal{A}\) to matrix \(\overline{\mathcal{A}}\). In addition, we selected agents 1 and 5 with zero in-degree as the pinning nodes and let the pinning gains matrix be \(B=\text{diag}[1,0,0,0,1,0]\). Then, we calculated the parameter \(\mu_{\min}=0.4490\), \(\bar{\mu}_{\min}=0.6378\) according to Theorem 1. The initial position of all agents were \(x_{1}(0)=[11.69;18.37]\), \(x_{2}(0)=[6.22;-18.57]\), \(x_{3}(0)=[13.96;17.35]\), \(x_{4}(0)=[-7.14;10.30]\), \(x_{5}(0)=[9.72;-4.31]\), \(x_{6}(0)=[6.21;-13.15]\). The velocity of the second-order agents were \(v_{4}(0)=[8.24;-18.72]\), \(v_{5}(0)=[-8.92;-18.15]\), \(v_{6}(0)=[-16.11;12.93]\). The nonlinear terms of the six agents were denoted as \(f_{1}(t,x_{1}(t))=\frac{1}{2}\sin(x_{1}(t))\arctan(x_{1}(t)-w_{i}x_{\sigma})\), \(f_{2}(t,x_{2}(t))=\frac{1}{5}\sin(x_{2}(t))\arctan(x_{2}(t)-w_{2}x_{\sigma})\), \(f_{3}(t,x_{3}(t))=-\frac{1}{2}\sin(x_{3}(t))\arctan(x_{3}(t)-w_{3}x_{\sigma})\), \(f_{4}(t,x_{4}(t),v_{4}(t))=\frac{1}{2}\sin(x_{4}(t))\arctan(x_{4}(t))\), \(f_{5}(t,x_{5}(t),v_{5}(t))=-\frac{1}{5}\sin(x_{5}(t))\arctan(v_{5}(t))\), and \(f_{6}(t,x_{6}(t),v_{6}(t))=-\frac{1}{2}\cos(x_{6}(t))\arctan(v_{6}(t))\), respectively. The goal state was \(x_{\delta}=5\). For the control input (5), based on parameter properties and theorem requirements, we chose \(p=3\), \(q=5\), \(k_{1}=33\), \(k_{2}=21\), \(c=10.8\). Correspondingly, the parameters in the event-triggered mechanism were set to \(\gamma_{1i}=\gamma_{2i}=1\). According to above parameters, we obtained through calculation that \(V(0)^{\frac{1}{2}(1-\frac{p}{q})}=3.6808\), \(\mathcal{K}=2.2267\), and the heterogeneous multi-agent systems (1) achieved consensus within the finite time \(T=8.2652\). The data in Table 1 show the variation in finite time \(T\) under different combinations of system parameters \(p\) and \(q\). As the values of \(p\) and \(q\) increase, the finite time \(T\) exhibits certain fluctuations. For example, when \(p=1\) and \(q=3\), \(T=11.7619\); and when \(p=3\) and \(q=5\), \(T=8.2652\), where \(T\) reaches its minimum value. Overall, although \(T\) does not change linearly with \(p\) and \(q\), it tends to increase gradually as \(p\) and \(q\) grow. These data suggest that selecting the parameters \(p\) and \(q\) appropriately can effectively optimize the system's convergence time. In some cases, larger \(p\) and \(q\) do not necessarily result in faster convergence, indicating that a careful analysis of these parameters is required for specific applications. In Figure 2, the position states of the six agents are provided. The position state of agents in group \(\mathcal{V}_{1}\) and the agents in group \(\mathcal{V}_{2}\) converged to \(5\) and \(-5\) before \(t=8\) s, respectively. Figure 3 shows that the velocity of each second-order agents became zero after \(t=8\) s. Obviously, the consensus error in Figure 4 gradually decreased to zero within the finite time \(T\). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(p\) & 1 & 3 & 5 & 7 & 9 & 11 & 13 & 15 \\ \hline \(q\) & 3 & 5 & 7 & 9 & 11 & 13 & 15 & 17 \\ \hline \(T\) & 11.7619 & 8.2652 & 8.4175 & 8.7881 & 9.5253 & 10.4416 & 11.4578 & 12.5351 \\ \hline \hline \end{tabular}
\end{table}