## 4 Simulation Results and Analysis on Chaotic Synchronization and Stability Control
The comparisons indicate that although all the slave systems can catch up with the trajectories of the master systems, the trajectory jittering phenomena of CMAC and FCMAC at the transient state are more obvious than that of ATFSORC. Figure 16g-i show comparisons of the \(x_{1}\) and \(x_{2}\) phase diagrams. Under the ideal circumstance, when the steady-state error is 0, i.e., \(x_{1}\) is equal to \(x_{2}\), the phase diagram should present a strip of a straight line. The comparisons show that the CMAC line strip is thicker and the transient time is longer, FCMAC is second, and ATFSORC is the best. Figure 16j-l show comparisons of \(y_{1}\) and \(y_{2}\) phase diagrams. Under the ideal circumstance, when the steady-state error is 0, i.e., \(y_{1}\) is equal to \(y_{2}\), the phase diagram should present a strip of a straight line. The comparison shows that the transient time for CMAC is longer, FCMAC is second, and ATFSORC is the best. Figure 16m-o show comparisons of the \(x_{1}\) and \(x_{2}\) response diagram. Coordinating with the comparisons of error e\({}_{x}\) from Figure 16p-r, we can see that although all the slave systems can catch up with the trajectories of the master systems, the steady-state error of CMAC lies between \(-\)0.002 and 0.002, and the steady-state error of FCMAC lies between \(-\)0.0002 and 0.0002, whereas the steady-state error of ATFSORC is almost 0. Therefore, the proposed ATFSORC error is smaller and the convergence velocity is faster. Figure 14: Simulation results of ATFSORC chaotic system synchronization control (g = 0.01, Cut-in control at 300 sec): (**a**) \(x_{1}\), \(x_{2}\) response for ATFSORC; (**b**) error \(e_{x}\) for ATFSORC; (**c**) \(y_{1}\), \(y_{2}\) response diagram for ATFSORC; (**d**) Error \(e_{y}\) for ATFSORC; (**e**) Number of association memory layers for ATFSORC; (**f**) Number of association memory layers for ATFSORC (partial); (**g**) Control amount for ATFSORC. Figure 15: Comparisons on simulation results of chaotic system synchronization control (\(g=0.01\)): (**a**) Master system phase diagram for cerebellar model articulation controller (CMAC); (**b**) Master system phase diagram for fuzzy CMAC (FCMAC); (**c**) Master system phase diagram for ATFSORC; (**d**) Slave system phase diagram for 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 ATFSORC. Figure 16: _Cont._  
Figure 17 compares the response diagrams of chaotic system synchronization for ATFSORC, FCMAC, and CMAC, with a coupling strength of \(g=0.01\) and \(g=1\). Moreover, the chaotic system is first allowed to run for 300 s before the controller cuts in, and the initial conditions are set as (\(x_{1}\), \(x_{2}\), \(y_{1}\), \(y_{2}\)) = (1, 0, 1, 0). Figure 17a-c show comparisons of \(x_{1}\) and \(x_{2}\) response diagrams. To demonstrate the characteristics of chaos, the controller is first removed after the system is run for 300 s; 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 17d-f show comparisons of error \(e_{x}\). The steady-state errors lie between \(-0.002\) and \(0.002\) for CMAC and between \(-0.0002\) and \(0.0002\) for FCMAC, whereas the steady-state error of ATFSORC is almost 0; therefore, the proposed ATFSORC performance is better. Figure 17g-i show a comparison of \(y_{1}\) and \(y_{2}\) response diagrams.