## 4 Theoretical Background for Stability Analysis of Limit Cycles
### Limit Cycles Computation
As stated before, due to interleaving effect, the three switching instants \(0\), \(t_{2}=(D+\frac{1}{3})T\) and \(t_{4}=(D+\frac{2}{3})T\) are given in a fixed pattern and therefore one has  
\[\left.\frac{\partial h_{4}(\mathbf{x},t)}{\partial t}\right|_{t=t_ {4}} = \infty \tag{25}\] \[\left.\frac{\partial h_{2}(\mathbf{x},t)}{\partial t}\right|_{t=t _{2}} = \infty\] (26) \[\left.\frac{\partial h_{0}\mathbf{x},t)}{\partial t}\right|_{t=0} = \infty \tag{27}\]  
This implies that  
\[\mathbf{S}_{010\to 011}=\mathbf{S}_{100\to 110}=\mathbf{S}_{001\to 101}=\mathbf{I} \tag{28}\]  
Therefore, expression (11) for the monodromy matrix becomes  
\[\mathbf{S}_{101\to 100}=\mathbf{I}+\frac{(\mathbf{A}_{101}-\mathbf{A}_{100}) \mathbf{x}(t_{1})+(\mathbf{B}_{101}-\mathbf{B}_{100})}{\mathbf{K}\mathbf{x}( t_{1})+\left.\frac{\partial h_{1}(\mathbf{x},t)}{\partial t}\right|_{t=t_{1}}} \tag{20}\]0  
\[\mathbf{S}_{101\to 100}=\mathbf{I}+\frac{(\mathbf{A}_{101}-\mathbf{A}_{100}) \mathbf{x}(t_{1})+(\mathbf{B}_{101}-\mathbf{B}_{100})}{\mathbf{K}\mathbf{x}( t_{1})+\left.\frac{\partial h_{1}(\mathbf{x},t)}{\partial t}\right|_{t=t_{1}}} \tag{20}\]1 Transition state diagram between the different configurations and their corresponding matrices. To obtain the gradients \(\partial h_{i}(\mathbf{x},t)/\partial t\) for \(i=1,3,5\), the expression of \(\mathbf{x}(0)\) must be obtained in terms of \(t\) for \(t_{5}\leq t<T\). To get this expression, let us write \(\mathbf{x}(t)\) for \(t_{5}\leq t<T\) in terms of \(\mathbf{x}(0)\)  
\[ \begin{array}{rcl}\mathbf{x}(t)&=&\mathbf{\[PERSON]}_{011}(t-t_{4})\mathbf{\[PERSON] }_{010}(t_{4}-t_{3})\mathbf{\[PERSON]}_{110}(t_{3}-t_{2})\mathbf{\[PERSON]}_{100}(t_{2} -t_{1})\mathbf{\[PERSON]}_{101}(t_{1})\mathbf{x}(0)+\\ &\mathbf{\[PERSON]}_{011}(t-t_{4})\Big{(}\mathbf{\[PERSON]}_{010}(t_{4}-t_{3})\Big{(} \mathbf{\[PERSON]}_{110}(t_{3}-t_{2})\big{(}\mathbf{\[PERSON]}_{100}(t_{2}-t_{1}) \mathbf{\[PERSON]}_{101}(t_{1})+\\ &\mathbf{\[PERSON]}_{100}(t_{2}-t_{1})\big{)}+\mathbf{\[PERSON]}_{110}(t_{3}-t_{2}) \Big{)}+\mathbf{\[PERSON]}_{010}(t_{4}-t_{3})\Big{)}+\mathbf{\[PERSON]}_{011}(t-t_{4}) \end{array}  \tag{30}\]  
Therefore, the sampled state \(\mathbf{x}(0)\) can be expressed as follows  
\[ \begin{array}{rcl}\mathbf{x}(0)&=&\eta\Big{(}\mathbf{x}(t)-\mathbf{\Phi}_{0 11}(t-t_{4})\Big{(}\mathbf{\Phi}_{010}(t_{4}-t_{3})\Big{(}\mathbf{\Phi}_{110} (t_{3}-t_{2})(\mathbf{\Psi}_{100}(t_{2}-t_{1})\mathbf{\Psi}_{101}(t_{1})+\\ &\mathbf{\Psi}_{100}(t_{2}-t_{1})\big{)}+\mathbf{\Psi}_{110}(t_{3}-t_{2}) \Big{)}+\mathbf{\Psi}_{010}(t_{4}-t_{3})\Big{)}+\mathbf{\[PERSON]}_{011}(t-t_{4}) \Big{)}\end{array}  \tag{31}\]  
where matrix \(\eta\) is given by  
\[\eta = \Big{(}\mathbf{\[PERSON]}_{011}(t-t_{4})\mathbf{\[PERSON]}_{010}(t_{4}-t_{ 3})\mathbf{\[PERSON]}_{110}(t_{3}-t_{2})\mathbf{\[PERSON]}_{100}(t_{2}-t_{1})\mathbf{ \[PERSON]}_{101}(t_{1})\Big{)}^{-1} \tag{32}\]  
With the expression of \(\mathbf{x}(0)\) in terms \(\mathbf{x}(t)\), the gradient vectors become as follows  
\[\mathbf{K}_{1} = \frac{\partial h_{1}}{\partial\mathbf{x}}=[\kappa_{i}\quad\kappa _{1}\quad 0]\eta \tag{33}\] \[\mathbf{K}_{3} = \frac{\partial h_{3}}{\partial\mathbf{x}}=[\kappa_{i}\quad 0\quad 0]\eta\] (34) \[\mathbf{K}_{5} = \frac{\partial h_{5}}{\partial\mathbf{x}}=[\kappa_{i}\quad 0\quad \kappa_{2}]\eta \tag{35}\]  
Finally, the partial time derivative \(\partial h_{i}(\mathbf{x},t)/\partial t\) for \(i=1,3,5\) are  
\[\frac{\partial h_{1}(\mathbf{x},t)}{\partial t}\bigg{|}_{t=t_{1}} = -1 \tag{36}\] \[\frac{\partial h_{3}(\mathbf{x},t)}{\partial t}\bigg{|}_{t=t_{3}} = -1\] (37) \[\frac{\partial h_{5}(\mathbf{x},t)}{\partial t}\bigg{|}_{t=t_{5}} = -1 \tag{38}\]  
With the previous expressions for all the terms needed for constructing the monodromy matrix, this can be derived and the stability analysis of the limit cycles can be performed. It is worth noting here that the previous theoretical approach can be applied to any kind of limit cycles. These refer to the different periodic orbits under steady state operation.