## 2 Materials and Methods
### Generalization of the Poincare Map  
Next, a sufficient condition for the existence of \(nT\)-periodic orbits is determined. The idea is to successfully apply Equation (26) and use the multiplication theorem for Jacobian determinants [27].  
For convenience, the initial condition \(\mathbf{x}_{1}\) and the final condition \(\mathbf{x}_{n}\) are selected. The duty cycle in the interval \([nT,(n+1)T]\) is denoted by \(d_{n}\). In addition, \(\Sigma\) is the state space of the system (which is the same at each switching instant of the converter).  
If \(\Sigma\xrightarrow{P_{n-1}}\Sigma\) corresponds to the Poincare application given by Equation (26), then the following expression is defined:  
\[\Sigma\xrightarrow{P_{1}}\Sigma\xrightarrow{P_{2}}\Sigma\cdots\Sigma \xrightarrow{P_{n-1}}\Sigma, \tag{55}\]  
and the Poincare application \(P\) is defined such as \(P(\mathbf{x}_{1})=\mathbf{x}_{n}\). Then, it follows that:  
\[P=P_{n-1}\circ\cdots\circ P_{2}\circ P_{1}. \tag{56}\]  
First, from Equation (26) is determined that:  
\[\mathbf{x}_{2} =\phi_{1}\bigg{(}\frac{d_{1}}{2}\bigg{)}\phi_{2}(T-d_{1})\phi_{1 }\bigg{(}\frac{d_{1}}{2}\bigg{)}\cdot\mathbf{x}_{1}+\phi_{1}\bigg{(}\frac{d_{1 }}{2}\bigg{)}\phi_{2}(T-d_{1})\psi_{1}\bigg{(}\frac{d_{1}}{2}\bigg{)}\] \[+\phi_{1}\bigg{(}\frac{d_{1}}{2}\bigg{)}\psi_{2}(T-d_{1})+\psi_{1 }\bigg{(}\frac{d_{1}}{2}\bigg{)}. \tag{57}\]  
Now, applying the equality of Equation (56), it presents \(n\geq 2\):  
\[\mathbf{x}_{n}=\left[\prod_{i=1}^{n-1}\phi_{1}\bigg{(}\frac{d_{i}}{2}\bigg{)} \phi_{2}(T-d_{i})\phi_{1}\bigg{(}\frac{d_{i}}{2}\bigg{)}\right]\mathbf{x}_{1} +\mathbf{H}_{n}, \tag{58}\]  
where the matrix \(\mathbf{H}_{n}\) is defined recursively, as follows:  
\[\mathbf{H}_{1} =\phi_{1}\bigg{(}\frac{d_{1}}{2}\bigg{)}\phi_{2}(T-d_{1})\psi_{1 }\bigg{(}\frac{d_{1}}{2}\bigg{)}+\phi_{1}\bigg{(}\frac{d_{1}}{2}\bigg{)}\psi_{ 2}(T-d_{1})+\psi_{1}\bigg{(}\frac{d_{1}}{2}\bigg{)}, \tag{59}\]  
\[\mathbf{H}_{n} =\phi_{1}\bigg{(}\frac{d_{n}}{2}\bigg{)}\phi_{2}(T-d_{n})\phi_{1} \bigg{(}\frac{d_{n}}{2}\bigg{)}\cdot\mathbf{H}_{n-1}+\phi_{1}\bigg{(}\frac{d_ {n}}{2}\bigg{)}\phi_{2}(T-d_{n})\psi_{1}\bigg{(}\frac{d_{n}}{2}\bigg{)} \tag{60}\] \[+\phi_{1}\bigg{(}\frac{d_{n}}{2}\bigg{)}\psi_{2}(T-d_{n})+\psi_{1 }\bigg{(}\frac{d_{n}}{2}\bigg{)}.\]  
By considering \(\mathbf{x}_{n}=\mathbf{x}_{1}\), the following condition for the existence of \(nT\)-periodic orbits is obtained:  
\[\mathbf{x}_{1}=(\mathbf{I}_{2}-\phi(\mathbf{d}))^{-1}\cdot\mathbf{H}_{n}, \tag{61}\]  
where  
\[\phi(\mathbf{d})=\prod_{i=1}^{n-1}\phi_{1}\bigg{(}\frac{d_{i}}{2}\bigg{)}\phi_{ 2}(T-d_{i})\phi_{1}\bigg{(}\frac{d_{i}}{2}\bigg{)}. \tag{62}\]  
If the spectral radius of the matrix \(\phi_{1}(d)\) is less than \(1\), Equation (61) can be applied to guarantee that the system has \(nT\)-periodic orbits.  
The stability of the \(nT\)-periodic orbits is carried out by considering Equations (47) and (56). Then, from Equation (56):  
\[\mathbf{JP}=\mathbf{JP}_{n-1}\cdots\mathbf{JP}_{2}\cdot\mathbf{JP}_{1}. \tag{63}\]Now, from Equation (47), the new expression presented in Equation (64) is obtained:  
\[\mathbf{JP}_{j}=\frac{\partial\mathbf{P}_{j}}{\partial\mathbf{x}_{j}}+\frac{ \partial\mathbf{P}_{j}}{\partial d_{j}}\cdot\frac{\partial d_{j}}{\partial \mathbf{x}_{j}}. \tag{64}\]  
Finally, using Equations (63) and (64), and the matrix multiplication by blocks, the new expression obtained is:  
\[\mathbf{JP}=\prod_{i=1}^{n-1}\left(\frac{\partial\mathbf{P}_{n-i}}{\partial \mathbf{x}_{n-i}}\quad\frac{\partial\mathbf{P}_{n-i}}{\partial d_{n-i}}\right) \left(\begin{array}{c}\mathbf{I}_{2}\\ \frac{\partial d_{n-i}}{\partial\mathbf{x}_{n-i}}\end{array}\right). \tag{65}\]  
Therefore, the eigenvalues of the Jacobian matrix given in Equation (65) are calculated to analyze the stability of the \(nT\) periodic orbits.