## 3 Results and Discussion
### Exhibiting Simulation Results and Discussion  
Figure 2 displays two distinct phase portraits of the two systems known as the Rossler and RCLs-JJ systems with different initial states. The chaotic behavior of the Rossler system for each phase portrait is displayed in Figure 2a-c. The phase portrait behaviors of the RCLs-JJ system is shown in Figure 2d-f. The chaotic behavior of the RCLs-JJ system is clear only in Figure 2e and the others are not chaos. The challenge is how to find the controller approach in the minimum trajectory error. The first stage overcomes the non-identical trajectory between the two systems and employs the active control to change the RCLs-JJ system into a quasi-Rossler system from Equation (6) to Equation (13). The phases of two systems after utilizing the active control technique are exhibited in Figure 3a-c. The new phase portraits of the RCLs-JJ system are almost completely different from the original phase portraits and are now closer to the Rossler system. Therefore, the new system is called a quasi-Rossler system. The time response of each component of the two systems indicated in Figure 3d-f in the paths of Rossler and RCLs-JJ systems, respectively, are distant from each other. Figure 2: Original phase portrait of the Rossoler system with \([ \begin{array}{cccc}x_{10}&x_{20}&x_{30}\end{array} ]^{T}=[ \begin{array}{cccc}0. 2&0.4&0.1\end{array} ]^{T}\) in the left column, (**a**) the Rossoler system \(x_{1}\), \(x_{2}\), (**b**) the Rossoler system \(x_{2}\), \(x_{3}\), and (**c**) the Rossoler system \(x_{1}\), \(x_{3}\), and the original phase portrait of the RCL-shunted Josephson Junction system with an initial condition at the original in the right column, (**d**) the RCLs-JJ system \(y_{1}\), \(y_{2}\), (**e**) the RCLs-JJ system \(y_{2}\), \(y_{3}\), and (**f**) the RCLs-JJ system \(y_{1}\), \(y_{3}\). Figure 4 demonstrates the tracking error between the Rossler and the quasi-Rossler systems, which transfers from the RCLs-JJ system. The vibration of the tracking error has many large amplitudes in the second (\(y_{2}\)-\(x_{2}\)) and third (\(y_{3}\)-\(x_{3}\)) components at the specific moment. Figure 5 demonstrates the phase portrait of the \(x_{1}\) (\(y_{1}\)) and \(x_{2}\) (\(y_{2}\)) by utilizing the ILC update law to track the trajectory. The two trajectories almost overlap, which validates the tracking error phenomena in Figure 6.  
The tracking errors oscillation in Figure 6 are between \(0.1408\) and \(-0.1959\) for the first tracking error, between \(0.1434\) and \(-0.4217\) for the second, and between \(0.4784\) and \(-0.8344\) for the third tracking error, respectively. Figure 3: After applying the active control procedure to (**a**–**c**), the phase portrait of the Rossler (\(x_{1}\)-blue) and RCLs-JJJ system (\(y_{1}\)-red) and (**c**–**f**) time response of the system is stated in the Rossler (\(x_{1}\)-blue) system and the RCLs-JJJ system (\(y_{1}\)-black). (**a**) Phase portrait \(x_{1}\) (\(y_{1}\)), \(x_{2}\) (\(y_{2}\)); (**b**) Phase portrait \(x_{2}\) (\(y_{2}\)), \(x_{3}\) (\(y_{3}\)); (**c**) Phase portrait \(x_{1}\) (\(y_{1}\)), \(x_{3}\) (\(y_{3}\)); (**d**)Time response of state \(x_{1}\)-blue, \(y_{1}\)-black; (**e**) Time response of state \(x_{2}\)-blue, \(y_{2}\)-black; (**f**) Time response of state \(x_{3}\)-blue, \(y_{3}\)-black. Figure 4: The tracking error between the Rossler and RCLs-JJ systems through an active control procedure. Figure 5: The phase portrait of \(x_{1}\) (\(y_{1}\)) and \(x_{2}\) (\(y_{2}\)) using the iterative learning control update law. Figure 7 exhibits the tracking error after using the ILC law. The third ingredient varies the most when comparing to subfigures in Figure 1. The lager vibration will always occur at a particular moment \(t_{\rm i}\) because the tracking error between two systems became larger at same moments.