## 4 Simulation Results and Analysis on Chaotic Synchronization and Stability Control  
In this study, the proposed ATFSORC was applied on the synchronization control of chaotic systems, and it was compared with CMAC and fuzzy CMAC (FCMAC) to validate the control performance of the proposed ATFSORC. Relevant parameters for simulation are shown in Table 2. At the beginning of the simulation, a trial-and-error method was used to find a set of initial values of controller parameters that can enable the system to run smoothly. After that, all the controller parameters were subjected to simulation using these initial values. The simulation contains the following two scenarios:  
1. By adjusting the coupling strength \(g\) of the gap junction in Equations (28) and (29) to 0.01 and 1 as well as by changing the cut-in control time point of ATFSORC to 300 s after the chaotic system running, the simulation would validate the feasibility of the controller proposed in this study. 2. By comparing the response data of ATFSORC with those of CMAC and FCMAC, the controller performance was determined via root mean square errors and average errors. Figures 10 and 11 show the uncontrolled chaotic system response diagrams for coupling strengths \(g=0.01\) and \(g=1\), respectively. To observe the unpredictable behavior of the chaotic system, the control amount \(u\) for the chaotic system in Equation (29) was set to zero, and the initial condition was set to \((x_{1},\,x_{2},\,y_{1},\,y_{2})=(1,\,0,\,1,\,0)\). Figures 10a and 11a show the phase diagrams of the master system. The phase diagrams show that the master system indeed operates at a chaotic state. Figures 10b and 11b show the phase diagrams of the slave system, which has different initial starting points from those of the master system; thus, the diagrams demonstrate the characteriztics of chaos and have phase plane trajectories different from those of the master system. Figure 10c shows a phase diagram constructed by \(x_{1}\) and \(x_{2}\). As shown very clearly from the diagram, when the primary and secondary systems are asynchronous, the entire phase plane trajectory is disorganized and in disorder. Figure 11c shows a phase diagram constructed by \(x_{1}\) and \(x_{2}\). As shown very clearly from the diagram, when the primary and secondary systems are asynchronous, the entire phase plane is slightly disorganized; however, due to the large coupling strength, the \(x\) coefficients of the master-slave system will gradually synchronize. Figure 10d shows a phase diagram constructed by \(y_{1}\) and \(y_{2}\). Similarly, we can clearly discover that when the control amount is zero, the asynchronous state is presented. Figure 11d shows the phase diagram constructed of \(y_{1}\) and \(y_{2}\). We can similarly discover that due to the large coupling strength, the \(y\) coefficient for the master-slave system will gradually synchronize. \begin{table}
\begin{tabular}{l l} \hline \hline
**Relevant Parameters of Simulation** & **Parameter Setting Values** \\ \hline System sampling time & 0.01 s; \\ Signed distance parameter & \(\lambda_{1}=1\); \\ Chaotic system parameters & \(\omega=0.7986\); \(A=0.1\); \(b=1\); \(r=10\); \(f=0.1271\); \\ Initial point of chaotic system & \((x_{1},\,x_{2},\,y_{1},\,y_{2})=(1,\,0,\,1,\,0)\); \\  & \(\alpha_{th}=0.98\) (\(g=0.01\)); \\  & \(\alpha_{th}=0.98\) (\(g=1\)); \\  & \(\alpha_{th}=0.98\) (\(g=0.01\), cut in at 300 s); \\  & \(\beta_{th}=0.4\) (\(g=0.01\)); \\ Relevant parameters of ATFSORC & \(\beta_{th}=0.4\) (\(g=1\)); \\  & \(\beta_{th}=0.35\) (\(g=0.01\), cut in at 300 s); \\  & \(m_{j}=0\); \(v_{j}=0.1\); \(k_{1}=1.8\); \\  & \(\beta_{w1}=1\times 10^{-4}\); \(\beta_{r1}=1\times 10^{-5}\); \\  & \(\beta_{m1}=8\times 10^{-6}\); \(\beta_{r1}=8\times 10^{-6}\). \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Simulation parameters for the synchronization control of chaotic systems.