## 4 Theoretical Background for Stability Analysis of Limit Cycles
### Approximate Expression of the Monodromy Matrix and Stability Region in the Parametric Space
As can be observed, the matching between the results is remarkable. Figure 9 depicts the Floquet multipliers loci when the parameter \(\kappa_{i}\) is varied. The loci was obtained from the exact and the approximate expression of the monodromy matrix. The loci from both the approximate and the exact expressions of the monodromy matrix shows that the critical value of the feedback gain at which one Floquet multipliers crosses the unit circle is 0.058 which in a perfect agreement with the numerical simulations performed from the switched model and presented in Section 3. For \(0.04<\kappa_{i}<0.058\) (A\({}^{-1}\)), all the Floquet multipliers are inside the unit disk hence indicating that the limit cycle is stable. However, at \(\kappa_{i}\approx 0.058\) A\({}^{-1}\), one Floquet multiplier crosses the unit circle from the point (\(-1\),0) on the complex plane and the system undergoes a flip bifurcation. For \(\kappa_{i}>0.058\), one of the Floquet multiplier is outside the unit disk being real and negative, the period-1 limit cycle becomes unstable. This explains subharmonic oscillations exhibited by the time waveforms and the bifurcation diagrams in Section 3. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\kappa_{i}\) & \(\mathbf{x}(0)\) (Exact) & \(\mathbf{x}(0)\) (Approx) & \(\mathbf{\mu}\) (Exact) & \(\mathbf{\mu}\) (Approx) & **Stability** \\ \hline \multirow{2}{*}{0.04} & \(\left(\begin{matrix}41,3722\\ 390,9831\end{matrix}\right)\) & \(\left(\begin{matrix}41,3793\\ 400,0000\end{matrix}\right)\) & \(\left(\begin{matrix}-0.4500\\ 0.5299\end{matrix}\right)\) & \(\left(\begin{matrix}-0.4500\\ 0.5298\end{matrix}\right)\) & \(\left(\begin{matrix}-0.4500\\ 0.5298\end{matrix}\right)\) & \multirow{2}{*}{stable} \\  & \(\left(\begin{matrix}24,843\\ 390,9837\\ 800,0227\end{matrix}\right)\) & \(\left(\begin{matrix}40,2857\\ 800,0000\end{matrix}\right)\) & \(\left(\begin{matrix}-0.7500\\ 0.5130\end{matrix}\right)\) & \(\left(\begin{matrix}-0.7500\\ 0.5088\end{matrix}\right)\) & \multirow{2}{*}{stable} \\  & \(\left(\begin{matrix}43,8861\\ 800,0240\end{matrix}\right)\) & \(\left(\begin{matrix}A3,9024\\ 800,0000\end{matrix}\right)\) & \(\left(\begin{matrix}-1.0500\\ 0.4951\end{matrix}\right)\) & \(\left(\begin{matrix}-1.0500\\ 0.5011\end{matrix}\right)\) & \multirow{2}{*}{unstable} \\  & \(\left(\begin{matrix}44,659\\ 390,9847\\ 800,0250\end{matrix}\right)\) & \(\left(\begin{matrix}44,689\\ 800,0000\end{matrix}\right)\) & \(\left(\begin{matrix}-1.0500\\ 0.4923\end{matrix}\right)\) & \(\left(\begin{matrix}-1.0500\\ 0.5011\end{matrix}\right)\) & \multirow{2}{*}{unstable} \\  & \(\left(\begin{matrix}44,659\\ 390,9847\\ 800,0250\end{matrix}\right)\) & \(\left(\begin{matrix}44,689\\ 800,0000\end{matrix}\right)\) & \(\left(\begin{matrix}-1.3500\\ 0.4923\end{matrix}\right)\) & \(\left(\begin{matrix}-1.3500\\ 0.4865\end{matrix}\right)\) & \(\left(\begin{matrix}-1.3500\\ 0.4923\end{matrix}\right)\) & \(\left(\begin{matrix}-1.3500\\ 0.4952\end{matrix}\right)\) & \(\left(\begin{matrix}-1.3500\\ 0.4865\end{matrix}\right)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The values of \(\mathbf{x}(0)\), the Floquet multipliers and stability of the corresponding limit cycles obtained with exact and the approximate approaches for \(i_{\text{ref}}=50\) A.  
Figure 9: Floquet multipliers loci from the exact and the approximate expression of the monodromy matrix as the bifurcation parameter \(\kappa_{i}\) is varied for left: \(i_{\text{ref}}=50\) A and right: \(i_{\text{ref}}=80\) A. Now that an approximate expression for the monodromy matrix is available in closed form, the Floquet multipliers can be found analytically.