## 3 Nonlinear Behavior from Time-Domain Numerical Simulations  
Our concern in this section is to present some graphic results to support and complement the study. The fixed parameter values used in this study are depicted in Table 2. The parameters \(\kappa_{i}\) and \(i_{\text{ref}}\) are varied. A powerful tool to have a wide panoramic view of the different dynamical behavior that the system can have when parameters are varied is a bifurcation diagram. In order to explore the dynamics of the system, bifurcation diagrams for the system are plotted by considering the current feedback gain \(\kappa_{i}\) as bifurcation parameter which is varied within the range \(\kappa_{i}\in(0.04,0.15)\) A\({}^{-1}\) for two different values of \(i_{\text{ref}}=50\) A (25 kW) and \(i_{\text{ref}}=80\) A (64 kW) respectively. The bifurcation diagrams are obtained by sampling the vector of state variables \(\mathbf{x}(t)\) at the switching period thus yielding \(\mathbf{x}[n]\), \(n=0,1\ldots 4000\). The transient samples are eliminated and the last 200 samples are considered as steady-state. The inductor current \(i_{L}[n]\) is plotted as the bifurcation parameter is varied. The results are shown in Figure 5. It can be observed that we get qualitatively similar bifurcation at the onset of instability independently of the value of \(i_{\text{ref}}\). Let us take the bifurcation diagram in the left panel of Figure 5 as an example. From this figure, it is possible to see that the system exhibits a stable periodic behavior for values of \(\kappa_{i}\) less than a critical value \(\kappa_{i,\text{crit}}\approx 0.058\) A\({}^{-1}\). At this critical value of the feedback gain, the system period-1 limit cycle loses its stability and the attractor of the system evolves on a period-2 limit cycle, i.e., the system undergoes a period doubling bifurcation and a period-2 limit cycle emerges. By further increasing the feedback gain \(\kappa_{i}\), the resulting period-2 limit cycle undergoes a period doubling resulting in a period-4 limit cycle. For higher values of \(\kappa_{i}\), the system undergoes more bifurcations phenomena leading eventually to chaotic behavior. Nonlinear behavior manifested by different periodicity parametric windows and chaos can also be observed. In order to check the results obtained from the previous bifurcation diagrams, time-domain simulations have been carried out by using the exact switched circuit-level model implemented in PSIM\(\otimes\) software. The steady-state time-domain waveforms are depicted in Figure 6 for different values of \(\kappa_{i}\). It is worth noting that in the top left panel of Figure 6, the flying capacitor voltages oscillate at the switching frequency \(f_{s}=40\) kHz, the inductor current oscillate at the double of the switching frequency \(2f_{s}=80\) kHz due to interleaving effect and all the duty cycles are the same and constant in steady-state. In the top right panel of Figure 6, the flying capacitor voltages oscillate at one half the switching frequency \(\frac{f_{s}}{2}=20\) kHz, the inductor current oscillate at the switching frequency \(f_{s}=40\) kHz and all the duty cycles are jumping up and down between two different values one of them is saturated to its lower level 0. In the bottom left panel of Figure 6, the flying capacitor voltages oscillate at one fourth the switching frequency \(\frac{f_{s}}{4}=10\) kHz, the inductor current oscillate at one half the switching frequency \(\frac{f_{s}}{2}=20\) kHz and all the duty cycles are jumping between three different values two of them are saturated to their lower level 0. Finally, in the bottom right panel of Figure 6, all the state variables and duty cycles are chaotic and saturated upper and the lower values of the duty cycles are reached. It is also worth to note the increase in the current ripple amplitude after losing the stability of the fundamental period-1 limit cycle which could have a damaging effect on the system efficiency and stresses in the switching devices and which also could shorten their lifetime.