## 4 Simulation Results and Analysis on Chaotic Synchronization and Stability Control
shows the stable phase diagram for the chaotic system. The controller cuts in at 400 s, and the trajectory of the last 100 s is plotted. It is observed that the proposed ATFSORC can immediately control the chaotic system to a stable state of period one in a short time; the phase plane presents a periodic closed trajectory. Figure 18c,e show the \(x\) response diagram for ATFSORC and the \(y\) response diagram for ATFSORC, respectively. After cut-in control at 400 s, the chaotic system can be stabilized in a short time, and the trajectory presents a periodic variation. Figure 18d,f show error \(e_{x}\) for ATFSORC and error \(e_{y}\) for ATFSORC, respectively. The coordination of the number of association memory layers in Figure 18g indicate that due to the larger transient error when the controller cuts in at 400 s, the number of associative memory layers is increased from the preset one layer to four layers within only 4 s, and then, five layers is fixed after entering the steady state. Figure 18h is a partial enlarged view on the number of association memory layers. Figure 18i is a response diagram of all control amounts for ATFSORC. Figure 19 shows comparisons on the response diagrams of chaotic system stability for CMAC, FCMAC, and ATFSORC, wherein the control amount at the beginning is set as zero with cut-in control after 400 s. Figure 19a-c show the phase diagrams of uncontrolled chaotic systems. The aim is to display the characteristics of a trajectory repeat of the chaotic system. The coordination with the stability phase diagrams from Figure 19d-f shows a clear comparison of the difference before and after control. Figure 19g-i show the \(x\) response diagrams of CMAC, FCMAC, and ATFSORC. Figure 19g shows that CMAC cuts in control at 400 s and the \(x\) response should stabilize the periodic variation at about 500 s. Figure 19h indicates that FCMAC cuts in control at 400 s, and the \(x\) response should be able to stably present periodic variation in a very short time. From Figure 19i, we can see that the ATFCMAC cuts in control at 400 s, and the \(x\) response should be able to stably  
Figure 18: Simulation results of ATFSORC chaotic system stability control: (**a**) Phase diagram for uncontrolled chaotic system; (**b**) Stable phase diagram for ATFSORC; (**c**) \(x\) response for ATFSORC; (**d**) Error \(e_{x}\) for ATFSORC; (**e**) \(y\) response for ATFSORC; (**f**) Error \(e_{y}\) for ATFSORC; (**g**) Number of association memory layers for ATFSORC; (**h**) Number of association memory layers for ATFSORC (partial); (**i**) Control amount for ATFSORC. present the closed trajectory of period one almost immediately. Figure 19j-l show the error \(e_{x}\) for CMAC, FCMAC, and ATFSORC. Figure 19m-o show the \(y\) response diagrams for CMAC, FCMAC, and ATFSORC. Figure 19m shows that CMAC cuts in control at 400 s, and \(y\) response should stabilize the periodic variation at about 500 s. Figure 19n shows that FCMAC cuts in control at 400 s, and the \(y\) response should be able to stably present periodic changes in a very short time. From Figure 19o, we can see that the ATFCMAC cuts in control at 400 s, and the \(y\) response should be able to stably present the closed trajectory of period one almost immediately. Figure 19p-r show the error \(e_{y}\) for CMAC, FCMAC, and ATFSORC. From the simulation results, we know that the proposed ATFSORC possess good control performance and is suitable to be applied on the control and synchronization of chaotic systems. Figure 19: _Cont._