## 2 Materials and Methods
### Floquet Exponents  
In this section, the Floquet exponents of the system are obtained, solving the respective equation. In general, this method allows the stability analysis of any periodic orbit; however, the paper focuses on studying \(1T\)-periodic orbits. The idea is to take a periodic solution of the system \(\mathbf{x}^{*}\) and perform a perturbation with a time function \(e^{nt}\). In terms of the unit step function \(\theta(t)\), the system of Equation (6) can be expressed as follows:  
\[\left( \begin{array}{c}\dot{x}_{1}\\ \dot{x}_{2}\end{array} \right) =\left( \begin{array}{cc}-\gamma&0\\ 0&0\end{array} \right)\left( \begin{array}{c}x_{1}\\ x_{2}\end{array} \right)+\left( \begin{array}{cc}0\\ 1\end{array} \right)+\left( \begin{array}{cc}0&1\\ -1&0\end{array} \right)\left( \begin{array}{c}x_{1}\\ x_{2}\end{array} \right)\cdot\theta\left(t-\frac{d}{2}\right)\] \[+\left( \begin{array}{cc}0&-1\\ 1&0\end{array} \right)\left( \begin{array}{c}x_{1}\\ x_{2}\end{array} \right)\cdot\theta\left(t-\left(T-\frac{d}{2}\right)\right). \tag{66}\]  
With  
\[\left( \begin{array}{c}x_{1}\\ x_{2}\end{array} \right)=\left( \begin{array}{c}x_{1}^{*}+e^{nt}a\\ x_{2}^{*}+e^{nt}b\end{array} \right), \tag{67}\]  
a periodic orbit perturbation \(\left(x_{1}^{*}\ x_{2}^{*}\right)^{T}\), where \(a\) and \(b\) are time functions \(t\). Thus, applying this perturbation to the system of Equation (66), the obtained expression is:  
\[\left( \begin{array}{c}\dot{x}_{1}^{*}+\mu e^{nt}a+e^{nt}\dot{a}\\ \dot{x}_{2}^{*}+\mu e^{nt}b+e^{nt}b\end{array} \right) =\left( \begin{array}{cc}-\gamma&0\\ 0&0\end{array} \right)\left( \begin{array}{c}x_{1}^{*}+e^{nt}a\\ x_{2}^{*}+e^{nt}b\end{array} \right)+\left( \begin{array}{c}0\\ 1\end{array} \right)\] \[+\left( \begin{array}{cc}0&1\\ -1&0\end{array} \right)\left( \begin{array}{c}x_{1}^{*}+e^{nt}a\\ x_{2}^{*}+e^{nt}b\end{array} \right)\cdot\theta\left(t-\frac{d}{2}\right)\] \[+\left( \begin{array}{cc}0&-1\\ 1&0\end{array} \right)\left( \begin{array}{c}x_{1}^{*}+e^{nt}a\\ x_{2}^{*}+e^{nt}b\end{array} \right)\cdot\theta\left(t-\left(T-\frac{d}{2}\right) \right).\]  
The duty cycle \(d\) is calculated from Equation (20) and can be expressed as:  
\[d=\frac{(2k_{1}-\gamma Tk_{1}-k_{2}T)x_{1}+(2k_{2}+k_{1}T)x_{2}+(k_{2}T-2k_{1} x_{1 ref}-2k_{2}x_{2 ref})}{k_{1}x_{2}-k_{2}x_{1}}. \tag{68}\]  
Now, \(d=d(x_{1},x_{2})\). Then, after linearizing \(d\) around the point \(X(0)=(x_{1}(0),x_{2}(0))^{T}\) the following expression is obtained:  
\[d=d(x_{1},x_{2})\approx d(x_{1}(0),x_{2}(0))+\left.\frac{\partial d}{\partial x _{1}}\right|_{X(0)}(x_{1}-x_{1}(0))+\left.\frac{\partial d}{\partial x_{2}} \right|_{X(0)}(x_{2}-x_{2}(0)). \tag{69}\]  
After simplifying the expression, the following term is obtained:  
\[\frac{d}{2}(x_{1},x_{2})\approx\zeta_{1}+\zeta_{2}(x_{1}-x_{1}(0))+\zeta_{3}(x_ {2}-x_{2}(0)). \tag{70}\]Therefore,  
\[\theta\bigg{(}t-\frac{d^{*}}{2}\bigg{)}\approx\theta(t-(\tilde{c_{1}}+\tilde{c_{2}} (x_{1}^{*}(0)-x_{1}(0))+\tilde{c_{3}}(x_{2}^{*}(0)-x_{2}(0)))), \tag{71}\]  
and  
\[\theta\bigg{(}t-\frac{d}{2}\bigg{)}\approx\theta(t-(\tilde{c_{1}}+\tilde{c_{2}} (x_{1}^{*}(0)+a(0)-x_{1}(0))+\tilde{c_{3}}(x_{2}^{*}(0)+b(0)-x_{2}(0)))), \tag{72}\]  
where \(d^{*}\) is the duty cycle of the periodic solution \(x^{*}\), and \(d\) is the duty cycle of the perturbation of said periodic orbit. Approximating by first-order Taylor series expansion, then:  
\[\theta\bigg{(}t-\frac{d}{2}\bigg{)}\approx\theta\bigg{(}t-\frac{d^{*}}{2} \bigg{)}+\delta\bigg{(}t-\frac{d^{*}}{2}\bigg{)}(\tilde{c_{2}}a(0)+\tilde{c_{3 }}b(0)), \tag{73}\]  
where \(\tilde{c_{2}}=\frac{-\tilde{c_{2}}}{2}\) and \(\tilde{c_{3}}=\frac{-\tilde{c_{3}}}{2}\). Similarly, by Taylor series expansion for the unit step in \(t-\Big{(}T-\frac{d}{2}\Big{)}\):  
\[\theta\bigg{(}t-\bigg{(}T-\frac{d}{2}\bigg{)}\bigg{)}\approx\theta\bigg{(}t- \bigg{(}T-\frac{d^{*}}{2}\bigg{)}\bigg{)}+\delta\bigg{(}t-\bigg{(}T-\frac{d^{* }}{2}\bigg{)}\bigg{)}(\tilde{c_{2}}a(0)+\tilde{c_{3}}b(0)). \tag{74}\]  
In this way, by substituting in Equation (68), then:  
\[