## 4 Simulation Results and Analysis on Chaotic Synchronization and Stability Control
Figures 12 and 13 show the response diagrams of chaotic system synchronization for ATFSORC with coupling strengths of \(g=0.01\) and \(g=1\), respectively; initial conditions were set as (\(x_{1}\), \(x_{2}\), \(y_{1}\), \(y_{2}\)) = (1, 0, 1, 0). Figures 12a and 13a show the phase diagrams of the master system, starting from the coordinate at (1, 1) and performing a chaotic counterclockwise trajectory motion. Figures 12b and 13b show the phase diagram of the slave system, starting from the coordinates at (0, 0); it can be seen that after inserting the controller, the slave system can rapidly catch up with the trajectory of the master system. Figures 12c and 13c show the phase diagrams for \(x_{1}\) and \(x_{2}\). The observation shows that due to the addition of ATFSORC, the \(x\) parameters for the master-slave system can be synchronized in a short period of time, demonstrating excellent control performance. Figure 11: Uncontrolled coupled nonlinear chaotic system (\(g=1\)): (**a**) Master system phase diagram; (**b**) Slave system phase diagram; (**c**) \(x_{1}\), \(x_{2}\) phase diagram; (**d**) \(y_{1}\), \(y_{2}\) phase diagram. Figure 10: Uncontrolled coupled nonlinear chaotic system (\(g=0.01\)): (**a**) Master system phase diagram; (**b**) Slave system phase diagram; (**c**) \(x_{1}\), \(x_{2}\) phase diagram; (**d**) \(y_{1}\), \(y_{2}\) phase diagram. Figures 12d and 13d show the phase diagrams for \(y_{1}\) and \(y_{2}\); we can observe that the slave system can be rapidly synchronized with the master system. Figures 12e and 12f show the response diagram and errors for \(x_{1}\) and \(x_{2}\), respectively. The observation implies that the master-slave system can be fully synchronized in about 9 s, and the error after the steady state is almost approaching zero. Figure 13e,f are the response diagram and errors for \(x_{1}\) and \(x_{2}\), respectively. The observation shows that the master-slave system can be fully synchronized in about 9 s, and the error after the steady state is almost approaching zero. Figure 12g,h show the response diagram and errors for \(y_{1}\) and \(y_{2}\). In the same manner, it can be observed that the synchronization can be complete in about 9 s, and the steady-state error approaches zero. Looking coordinately with the number of associate memory layers in Figure 12i,j, it is found that because the transient error is larger, the number of layers is increased from the preset one layer to six layers only within 0.3 s, the number of layers is increased layer by layer, and the number of association memory layers at the steady state is fixed as 7 layers. Figure 13g,h show the response diagram and errors for \(y_{1}\) and \(y_{2}\). In the same manner, it can be observed that the synchronization can be completed in about 9 s, and the steady-state error approaches zero. Looking coordinately with the number of associate memory layers in Figure 13i,j, it can be found that because the transient error is larger, the number of layers increases from the preset one layer to six layers only within 0.2 s, and the number of association memory layers at steady state is fixed as seven layers. Figures 12k and 13k show the response diagrams of all control amounts of 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 14 shows the response diagrams of the chaotic system for ATFSORC, wherein the coupling strength \(g=0.01\) and \(g=1\); 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 14a,b show the \(x_{1}\), \(x_{2}\) response and \(y_{1}\), \(y_{2}\) response for ATFSORC, respectively.