## 2 Rossler Chaotic and RCL-Shunted Josephson Junction System  
### System Description and Transformation  
The Rossler chaotic system has an initial condition \(X_{0}\), which is the drive system in a general form. \[\dot{\mathbf{X}}=\mathbf{A}\mathbf{X}+\mathbf{b}=\left[ \begin{array}{ccc}0&-1&-1\\ 1&a&0\\ 0&0&x_{1}-c\end{array} \right]\left[ \begin{array}{c}x_{1}\\ x_{2}\\ x_{3}\end{array} \right]+\left[ \begin{array}{c}0\\ 0\\ b\end{array} \right] \tag{1}\]  
\(\mathbf{X}=\left[ \begin{array}{cc}x_{1},&x_{2},&x_{3}\end{array} \right]^{T}\) is the state vector. The RCLs-JJ can be expressed by Equation (2) with initial conditions \(Y_{0}=[0\;0\;0]^{T}\). \[\dot{\mathbf{Y}}=\mathbf{B}\mathbf{Y}+\mathbf{b}\psi(y_{1})+\mathbf{U}^{(k)} \tag{2}\]where \(\mathbf{Y}=\left[ \begin{array}{cc}y_{1}&y_{2}&y_{3}\end{array} \right]^{T}\) and  
\[\mathbf{B}=\left[ \begin{array}{ccc}0&1&0\\ 0&-\frac{g(y_{2})}{\beta_{C}}&\frac{1}{\beta_{C}}\\ 0&\frac{1}{\beta_{L}}&-\frac{1}{\beta_{L}}\end{array} \right],\mathbf{b}=\left[ \begin{array}{c}0\\ \frac{1}{\beta_{C}}\\ 0\end{array} \right],\psi(y_{1}(t))=i_{N}-\sin(y_{1}). \tag{3}\]  
where \(g(y_{2})\) is given by  
\[g(y_{2})=\left\{ \begin{array}{c}0.366\text{ as }|y_{2}|>2.9\\ 0.061\text{ as }|y_{2}|\leq 2.9,\end{array} \right. \tag{4}\]  
\(k\) in Equation (2) is the iterative number that uses the iterative learning control law (ILC) as shown in the equation below. \[\mathbf{U}^{(k)}=\left[ \begin{array}{cc}v_{\text{a}}&v_{\text{b}}&v_{\text{c}} \end{array} \right]^{T}+\left[ \begin{array}{cc}u_{1}^{(k)}&u_{2}^{(k)}&u_{3}^ {(k)}\end{array} \right]^{T} \tag{5}\]  
The ILC rule is a sequence of the control input signal for the response system as \(\left\{\mathbf{U}^{(k)}\right\}_{k=1,2,\cdots}\). Equations (1) and (2) are nearly non-identical nonlinear systems, which is seen by their trajectory in Figure 1.  
Figure 1 displays the time response of the state of two distinct systems known as the Rossler and the RCLs-JJ systems with different initial states. The trajectory error between the systems in each state should be enormous in Figure 1. The nonlinear system in Equation (2) should be transferred to the quasi-Rossler system to track the trajectory of the system in Equation (1). Therefore, the active control technique [23; 24] is u  
Figure 1: The time response of the Rossler system and the resistive–capacitive–inductance shunted the Josephson Junction (RCLs-JJ) system. According to the active control technique, the description of the system in Equation (2) and the dynamic transformation between the drive in Equation (1) and the response system in Equation (2) is shown in the equation below. \[\mathbf{z}=\left[ \begin{array}{c}z_{a}\\ z_{b}\\ z_{c}\end{array} \right]=\left[ \begin{array}{c}y_{1}-x_{1}\\ y_{2}-x_{2}\\ y_{3}-x_{3}\end{array} \right] \tag{6}\]  
\[\dot{\mathbf{z}}=\left[ \begin{array}{c}\dot{z}_{a}\\ \dot{z}_{b}\\ \dot{z}_{c}\end{array} \right]=\left[ \begin{array}{c}\dot{y}_{1}-\dot{x}_{1} \\ \dot{y}_{2}-\dot{x}_{2}\\ \dot{y}_{3}-\dot{x}_{3}\end{array} \right] \tag{7}\] \[=\left[ \begin{array}{c}z_{b}+2x_{2}+x_{3}\\ \overline{\beta_{\rm C}}^{-1}g(y_{2})z_{b}+\frac{1}{\beta_{\rm C}}z_{c}+ \frac{1}{\beta_{\rm C}}[i_{N}-\sin(y_{1})]-x_{1}-\Big{(}a+\frac{g(y_{2})}{ \beta_{\rm C}}\Big{)}x_{2}-\frac{x_{3}}{\beta_{\rm C}}\\ \frac{1}{\beta_{\rm L}}z_{b}-\frac{1}{\beta_{\rm L}}z_{c}+\frac{1}{\beta_{\rm L }}(x_{2}-x_{3})-x_{1}x_{3}-cx_{3}-b\end{array} \right]+\left[ \begin{array}{c }v_{\rm a}\\ v_{\rm b}\\ v_{\rm c}\end{array} \right]\]  
where \(\left[ \begin{array}{cc}v_{a}&v_{b}&v_{c}\end{array} \right]^{T}\) is the active control function that eliminates the terms that have no \(z_{i}\) for \(i=a\), \(b\), \(c\). As a result, the active control function can be determined by the equation below. \[\left[ \begin{array}{c}v_{a}\\ v_{b}\\ v_{c}\end{array} \right]=\left[