## 2 Materials and Methods
### System Discretization  
In a continuous dynamic system, discretization is accomplished by selecting an appropriate hyperplane at which the trajectories intersect at an angle other than zero (transversality condition) [22; 23]. Thus, the Poincare map is defined as in Equation (21). The term \(P(x)\) is the intersection of the orbit passing through \(x\) with the hyperplane \(\Sigma\).  
\[\begin{array}{rl}P:\Sigma&\rightarrow\Sigma\\ x&\mapsto P(x).\end{array} \tag{21}\]  
Note that, if \(L_{0}\) is a limit cycle, then the orbit through \(x_{0}\) always intersects \(\Sigma\) at \(x_{0}\). Thus, a \(1T\)-periodic orbit corresponds to a single point in \(\Sigma\); a \(2T\)-periodic orbit will correspond to 2 points in \(\Sigma\); and so on. This is because our system is \(T\)-periodically forced. A sample of the system is taken each period to build the Poincare application used to study the dynamics of the Boost converter, concatenating the solutions in each of the intervals \([nT,nT+\frac{d_{n}}{2}]\), \([nT+\frac{d_{n}}{2},(n+1)T-\frac{d_{n}}{2}]\), \([(n+1)T-\frac{d_{n}}{2},(n+1)T]\), where \(d_{n}\) is the duty cycle at a point \(n\), following the methodology presented in [16]. Equation (13) is used to achieve this, as follows:  
If \(\mathbf{x}(nT)\) is the state of the system at the time \(nT\); then, at the end of the interval \([nT,nT+\frac{d_{n}}{2}]\):  
\[\mathbf{x}_{1}\bigg{(}nT+\frac{d_{n}}{2}\bigg{)}=\phi_{1}\bigg{(}\frac{d_{n}}{2 }\bigg{)}\mathbf{x}(nT)+\psi_{1}\bigg{(}\frac{d_{n}}{2}\bigg{)}. \tag{22}\]  
Now, at the end of the second stretch \([nT+\frac{d_{n}}{2},(n+1)T-\frac{d_{n}}{2}]\):  
\[\mathbf{x}_{2}\bigg{(}(n+1)T-\frac{d_{n}}{2}\bigg{)}=\phi_{2}(T-d_{n})\mathbf{ x}_{1}\bigg{(}nT+\frac{d_{n}}{2}\bigg{)}+\psi_{2}(T-d_{n}). \tag{23}\]  
Consequently, at the end of the second section:  
\[\mathbf{x}_{2}\bigg{(}(n+1)T-\frac{d_{n}}{2}\bigg{)}=\phi_{2}(T-d_{n})\phi_{1 }\bigg{(}\frac{d_{n}}{2}\bigg{)}\mathbf{x}(nT) \tag{24}\]  
Now, at the end of the third trace \([(n+1)T-\frac{d_{n}}{2},(n+1)T]\):  
\[\mathbf{x}_{3}((n+1)T)=\phi_{1}\bigg{(}\frac{d_{n}}{2}\bigg{)}\mathbf{x}_{2} \bigg{(}(n+1)T-\frac{d_{n}}{2}\bigg{)}+\psi_{1}\bigg{(}\frac{d_{n}}{2}\bigg{)}. \tag{25}\]  
Therefore, at the end of the third section:  
\[\mathbf{x}_{3}((n+1)T) =\phi_{1}\bigg{(}\frac{d_{n}}{2}\bigg{)}\phi_{2}(T-d_{n})\phi_{1} \bigg{(}\frac{d_{n}}{2}\bigg{)}\mathbf{x}(nT)\] \[+\phi_{1}\bigg{(}\frac{d_{n}}{2}\bigg{)}\phi_{2}(T-d_{n})\psi_{1} \bigg{(}\frac{d_{n}}{2}\bigg{)}\] \[+\phi_{1}\bigg{(}\frac{d_{n}}{2}\bigg{)}\psi_{2}(T-d_{n})+\psi_{1 }\bigg{(}\frac{d_{n}}{2}\bigg{)}. \tag{26}\]  
The Poincare map \(P\) used in this research is that corresponding to  
\[P(\mathbf{x}(nT))=\mathbf{x}_{3}((n+1)T).\]  
The term \(\mathbf{x}_{3}((n+1)T)\) is given by Equation (26). Note that in this case, no duty cycle saturation is assumed; however, \(d_{n}\in(0,T)\) is assumed. When the duty cycle saturates, then:  
1. If \(d_{n}=0\), the Poincare map corresponds to \[P(\mathbf{x}(nT))=\phi_{2}(T)\mathbf{x}(nT)+\psi_{2}(T).\] (27)
2. If \(d_{n}=T\), the Poincare map corresponds to \[P(\mathbf{x}(nT))=\phi_{1}(T)\mathbf{x}(nT)+\psi_{1}(T).\] (28)  
Note that the Poincare map \(P\) is a function of the state of the system at the instant \(nT\) and of the duty cycle in the interval \([nT,(n+1)T]\), that is, \(P=P(\mathbf{x}(nT),d_{n})\). The latter must be considered when analyzing the stability of periodic orbits.