## 2 Materials and Methods
### Floquet Exponents
\begin{split}\bigg{(}\begin{array}{c}\dot{x}_{1}^{*}+\mu e^{ \mu t}a+e^{\mu t}\dot{a}\\ \dot{x}_{2}^{*}+\mu e^{\mu t}b+e^{\mu t}\dot{b}\end{array}\bigg{)}&=\bigg{(} \begin{array}{cc}-\gamma&0\\ 0&0\end{array}\bigg{)}\bigg{(}\begin{array}{cc}x_{1}^{*}+e^{\mu t}a\\ x_{2}^{*}+e^{\mu t}b\end{array}\bigg{)}+\bigg{(}\begin{array}{cc}0\\ 1\end{array}\bigg{)}\\ &+\bigg{(}\begin{array}{cc}0&1\\ -1&0\end{array}\bigg{)}\bigg{(}\begin{array}{cc}x_{1}^{*}+e^{\mu t}a\\ x_{2}^{*}+e^{\mu t}b\end{array}\bigg{)}\cdot\bigg{[}\theta\bigg{(}t-\frac{d ^{*}}{2}\bigg{)}\\ &+\delta\bigg{(}t-\frac{d^{*}}{2}\bigg{)}(\tilde{c_{2}}a(0)+\tilde{c_{3}}b(0)) \bigg{]}\\ &+\bigg{(}\begin{array}{cc}0&-1\\ 1&0\end{array}\bigg{)}\bigg{(}\begin{array}{cc}x_{1}^{*}+e^{\mu t}a\\ x_{2}^{*}+e^{\mu t}b\end{array}\bigg{)}\cdot\bigg{[}\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))\bigg{]}.\end{split}  \tag{75}\]  
As \(\big{(}x_{1}^{*},x_{2}^{*}\big{)}^{T}\) is a solution of the system, then from the last equation it follows that:  
\[ \begin{split}\bigg{(}\begin{array}{c}\mu e^{\mu t}a+e^{\mu t} \dot{a}\\ \mu e^{\mu t}b+e^{\mu t}\dot{b}\end{array}\bigg{)}&=\bigg{(}\begin{array}{ cc}-\gamma&0\\ 0&0\end{array}\bigg{)}\bigg{(}\begin{array}{c}e^{\mu t}a\\ e^{\mu t}b\end{array}\bigg{)}\\ &+\bigg{(}\begin{array}{cc}0&1\\ -1&0\end{array}\bigg{)}\bigg{(}\begin{array}{cc}x_{1}^{*}\\ x_{2}^{*}\end{array}\bigg{)}\delta\bigg{(}t-\frac{d^{*}}{2}\bigg{)}(\hat{c_{2}}a (0)+\hat{c_{3}}b(0))\\ &+\bigg{(}\begin{array}{cc}0&1\\ -1&0\end{array}\bigg{)}\bigg{(}\begin{array}{cc}e^{\mu t}a\\ e^{\mu t}b\end{array}\bigg{)}\bigg{[}\theta\bigg{(}t-\frac{d^{*}}{2}\bigg{)}+ \delta\bigg{(}t-\frac{d^{*}}{2}\bigg{)}(\hat{c_{2}}a(0)+\hat{c_{3}}b(0))\\ &+\bigg{(}\begin{array}{cc}0&-1\\ 1&0\end{array}\bigg{)}\bigg{(}\begin{array}{cc}x_{1}^{*}\\ x_{2}^{*}\end{array}\bigg{)}\delta\bigg{(}t-\bigg{(}T-\frac{d^{*}}{2}\bigg{)} \bigg{)}(\hat{c_{2}}a(0)+\hat{c_{3}}b(0))\\ &+\bigg{(}\begin{array}{cc}0&-1\\ 1&0\end{array}\bigg{)}\bigg{(}\begin{array}{cc}e^{\mu t}a\\ e^{\mu t}b\end{array}\bigg{)}\cdot\bigg{[}\theta\bigg{(}t-\bigg{(}T-\frac{d^{*}}{2} \bigg{)}\bigg{)}\\ &+\delta\bigg{(}t-\bigg{(}T-\frac{d^{*}}{2}\bigg{)}\bigg{)}(\hat{c_{2}}a(0)+ \hat{c_{3}}b(0))\bigg{]}.\end{split}  \tag{76}\]  
As the delta function:  
\(\delta\Big{(}t-\frac{d^{*}}{2}\Big{)}\) only contribute to:\(t=\frac{d^{*}}{2}\), and the matrices  
\[\left( \begin{array}{c}e^{\mu t}a\\ e^{\mu t}b\end{array} \right)\delta\left(t-\frac{d^{*}}{2}\right)(\zeta_{2}a(0)+ \zeta_{3}b(0))\approx\left( \begin{array}{c}0\\ 0\end{array} \right), \tag{77}\]  
\[\left( \begin{array}{c}e^{\mu t}a\\ e^{\mu t}b\end{array} \right)\delta\left(t-\left(T-\frac{d^{*}}{2}\right) \right)(\zeta_{2}a(0)+\zeta_{3}b(0))\approx\left( \begin{array}{c}0\\ 0\end{array} \right), \tag{78}\]  
then the perturbed system can be written as:  
\[\left( \begin{array}{c}\mu a+\dot{a}\\ \mu b+b\end{array} \right) =\left( \begin{array}{cc}-\gamma&0\\ 0&0\end{array} \right)\left( \begin{array}{c}a\\ b\end{array} \right)+\mathbf{A}\delta\left(t-\frac{d^{*}}{2}\right)\left( \begin{array}{c}a(0)\\ b(0)\end{array} \right) \tag{79}\] \[+\left( \begin{array}{cc}0&1\\ -1&0\end{array} \right)\left( \begin{array}{c}a\\ b\end{array} \right)\theta\left(t-\frac{d^{*}}{2}\right)\] \[+\mathbf{B}\delta\left(t-\left(T-\frac{d^{*}}{2}\right)\right) \left( \begin{array}{c}a(0)\\ b(0)\end{array} \right)\] \[+\left( \begin{array}{cc}0&-1\\ 1&0\end{array} \right)\left( \begin{array}{c}a\\ b\end{array} \right)\theta\left(t-\left(T-\frac{d^{*}}{2}\right)\right).\]  
where  
\[\mathbf{A}=e^{-\mu\left(\frac{C}{2}\right)}\left( \begin{array}{cc}0&1\\ -1&0\end{array} \right)\left( \begin{array}{c}x_{1}^{*}\left(\frac{C}{2} \right)\\ x_{2}^{*}\left(\frac{C}{2}\right)\end{array} \right)\left( \begin{array}{cc} \zeta_{2}&\zeta_{3}\end{array} \right), \tag{80}\]  
and  
\[\mathbf{B}=e^{-\mu\left(T-\frac{d^{*}}{2}\right)}\left( \begin{array}{cc}0&-1 \\ 1&0\end{array} \right)\left(