## 4 Simulation Results and Analysis on Chaotic Synchronization and Stability Control
To demonstrate the characteristics of chaos, the controller is first removed after the system is run for 300 s, and then the controller cuts in, thereby validating the performance of the controller. The results show that all the slave systems' trajectories can keep up with the master systems' trajectories, but the steady-state response of ATFSORC is better than those of CMAC and FCMAC. Figure 17j-l show comparisons of the error \(e_{y}\). The steady-state error of CMAC lies between \(-0.002\) and \(0.002\), and that of FCMAC lies between \(-0.0002\) and \(0.0002\); in contrast, the steady-state error of ATFSORC is almost 0. Therefore, the proposed ATFSORC performance is better. The proposed ATFSORC was applied on the stability control of a chaotic system, and it was compared with CMAC and FCMAC to validate the control performance of the proposed ATFSORC. Relevant parameters for simulations are shown in Table 3. At the beginning of the simulation, a set of initial values of controller parameters that allow the system to run smoothly was found by trial and error. These initial values were taken for all the controller parameters for simulation. Simulations contain the following two scenarios:  
1. In the simulation, the controller will first be removed; control will cut in after the chaotic system has run for 400 s; thereby, the feasibility of the controller is validated. 2. Response data of ATFSORC will be controlled with those of CMAC and FCMAC to observe the stabilization time. Figure 16: Comparisons on simulation results of chaotic system synchronization control (\(g=1\)): (**a**) Master system phase diagram for CMAC; (**b**) Master system phase diagram for FCMAC; (**c**) Master system phase diagram for ATFSORC; (**d**) Slave system phase diagram of CMAC; (**e**) Slave system phase diagram for FCMAC; (**f**) Slave system phase diagram for ATFSORC; (**g**) \(x_{1}\), \(x_{2}\) phase diagram for CMAC; (**h**) \(x_{1}\), \(x_{2}\) phase diagram for FCMAC; (**i**) \(x_{1}\), \(x_{2}\) phase diagram for ATFSORC; (**j**) \(y_{1}\), \(y_{2}\) phase diagram for CMAC; (**k**) \(y_{1}\), \(y_{2}\) phase diagram for FCMAC; (**l**) \(y_{1}\), \(y_{2}\) phase diagram for FTORC; (**m**) \(x_{1}\), \(x_{2}\) response for CMAC; (**n**) \(x_{1}\), \(x_{2}\) response for FCMAC; (**o**) \(x_{1}\), \(x_{2}\) response for ATFSORC; (**p**) Error \(e_{x}\) for CMAC; (**q**) Error \(e_{x}\) for FCMAC; (**r**) Error \(e_{x}\) for ATFSORC. Figure 18 shows the stability simulation results of a chaotic system for ATFSORC. In Figure 18a, to observe the characteristics of a chaotic system, the control amount of the chaotic system is set as zero, and the initial condition is set as \((x,y)=(0,\,0)\). Figure 18b \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 & \(\lambda_{2}=1\); \\ Chaotic system parameters & \(\omega=0.7986\); \(A=0.1\); \(b=1\); \(r=10\); \\ Initial point of chaotic system & \((x,y)=(0,\,0)\); \\  & \(\alpha_{ht}=0.98\); \(\beta_{ht}=0.5\); \\ Relevant parameters of ATFSORC & \(m_{j}=0\); \(v_{j}=0.1\); \(k_{1}=1.8\); \\  & \(\beta_{w}=1\times 10^{-5}\); \(\beta_{r}=1\times 10^{-5}\); \(\beta_{m}=8\times 10^{-6}\); \\  & \(\beta_{w}=8\times 10^{-6}\). \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of chaotic system stability simulation. Figure 17: Comparisons on simulation results of chaotic system synchronization control (\(g=0.01\), cut-in control at 300 sec): (**a**) \(x_{1}\), \(x_{2}\) response for CMAC; (**b**) \(x_{1}\), \(x_{2}\) response for FCMAC; (**c**) \(x_{1}\), \(x_{2}\) response for ATFSORC; (**d**) Error \(e_{x}\) for CMAC; (**e**) Error \(e_{x}\) for FCMAC; (**f**) Error \(e_{x}\) for ATFSORC; (**g**) \(y_{1}\), \(y_{2}\) response for CMAC; (**h**) \(y_{1}\), \(y_{2}\) response for FCMAC; (**i**) \(y_{1}\), \(y_{2}\) response for ATFSORC; (**j**) Error \(e_{y}\) for CMAC; (**k**) Error \(e_{y}\) for FCMAC; (**l**) Error \(e_{y}\) for ATFSORC.