## 4 Simulation Results and Analysis on Chaotic Synchronization and Stability Control
We can see that the master-slave system is at an asynchronous stage before 300 s; after approximately 6 s, the controller cuts in, and the slave system can keep up with the master system trajectory. Figure 14c,d show errors \(e_{x}\) and \(e_{y}\), respectively. Looking coordinately with the number of association memory layers and their partial enlarged views in Figure 14e,f, it is found that because the errors at the transient state are larger, the number of layers is added from the preset one layer to six layers only within 0.25 s; after 0.5 s, since the error is smaller, the number of layers is reduced by one layer. Figure 14g shows the response diagram of all the control amount for ATFSORC, wherein adding the improved compensation controller output \(u_{c}\) on the TSK fuzzy self-organizing recurrent cerebellar model articulation controller output \(u_{\text{ITSORC}}\) will be equal to the control amount \(u\) in Equation (29). Figure 15 compares the response diagrams of chaotic system synchronization for ATFSORC, FCMAC, and CMAC, with a coupling strength of g = 0.01 and an initial condition set as (\(x_{1}\), \(x_{2}\), \(y_{1}\), \(y_{2}\)) = (1, 0, 1, 0). Figure 15a-c show comparisons of response diagrams for the master systems; Figure 15d-f show comparisons of response diagrams for the slave systems. The comparisons show 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 is more obvious than that of ATFSORC. Figure 15g-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 line strip from CMAC is thicker, and its transient time is longer; FCMAC is second, and ATFSORC is the best. Figure 15j-l show comparisons of the \(y_{1}\) 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 comparisons show that the line strip of CMAC is thicker and the transient time is longer, FCMAC is second, and ATFSORC is the best. Figure 12: Simulation results of ATFSORC chaotic system synchronization control (\(g\) = 0.01): (**a**) Master system phase diagram for ATFSORC; (**b**) Slave system phase diagram for ATFSORC; (**c**) \(x_{1}\), \(x_{2}\) phase diagram for ATFSORC; (**d**) \(y_{1}\), \(y_{2}\) phase diagram for ATFSORC; (**e**) \(x_{1}\), \(x_{2}\) response for ATFSORC; (**f**) Error \(e_{y}\) for ATFSORC; (**g**) \(y_{1}\), \(y_{2}\) response for ATFSORC; (**h**) Error \(e_{y}\) for ATFSORC; (**i**) Number of association memory layers for ATFSORC; (**j**) Number of association memory layers for ATFSORC (partial); (**k**) Control amount for ATFSORC. Figure 13: Simulation results of ATFSORC chaotic system synchronization control (\(g=1\)): (**a**) Master system phase diagram for ATFSORC; (**b**) Slave system phase diagram for ATFSORC; (**c**) \(x_{1}\), \(x_{2}\) phase diagram for ATFSORC; (**d**) \(y_{1}\), \(y_{2}\) phase diagram for ATFSORC; (**e**) \(x_{1}\), \(x_{2}\) response for ATFSORC; (**f**) Error \(e_{x}\) for ATFSORC; (**g**) \(y_{1}\), \(y_{2}\) response for ATFSORC; (**h**) Error \(e_{y}\) for ATFSORC; (**i**) Number of association memory layers for ATFSORC; (**j**) Number of association memory layers for ATFSORC (partial); (**k**) Control amount for ATFSORC. Figure 16 compares the response diagrams of chaotic system synchronization for AFSORC, FCMAC, and CMAC, with a coupling strength of g = 1 and the initial condition set as (\(x_{1}\), \(x_{2}\), \(y_{1}\), \(y_{2}\)) = (1, 0, 1, 0). Figure 16a-c show comparisons of the master systems' response diagrams. Figure 16d-f show comparisons of the slave systems' response diagrams.