*3.1. Floquet Theory on the Stability*

3.1.1. Simulations Results in MATLAB/SIMULINK Software

The monodromy matrix is explained in Equation (23). Using a code developed in the MATLAB software, we find the eigenvalues noted as *λ* of this monodromy matrix. These eigenvalues are solutions of equation det(*P* − *λI*) = 0. Figure 5 is the representation in the complex plane of the Floquet multipliers when the amplitude *VM* varies from 0.0 to 3, for three values of the cut frequency *fc* = *fs* <sup>2</sup> , *fc* = *fs*, *fc* = 2 *fs* with *fs* = 50*kHz* and for two values of the output voltage (*E* = 60 *V*and*E*= 48 *V*). It should be noted that the period-1 orbit will lose stability and bifurcate into period-2 (sub-harmonic oscillation) if the system has the phenomenon of smooth period-doubling when a control parameter varies. Moreover, at this value of the control parameter, period-1 is destroyed and period-2 is created. Indeed, one of the Floquet multipliers is approximately equal to 1, which allows us to say that at this value of the amplitude *VM* the period-1 orbit destabilizes and gives way to a sub-harmonic oscillation of period 2. In Figure 5, the followings remarks can be noticed:

**Remark 1.** *We find that for several values of the gain VM, the moduli of the Floquet multipliers are located in the circle of unit radius synonymous with the stability of the period-1 orbit. In addition, for other ranks of the parameter VM, the Floquet multipliers are located outside the circle of unit radius, which is synonymous with destabilization of the periodic orbit of period-1.*

**Remark 2.** *We observe that when for <sup>E</sup>* <sup>=</sup> <sup>60</sup> *V, <sup>E</sup>* <sup>=</sup> <sup>48</sup> *V, fc is increasing, i.e., fc* = *fs* 2 < (*fc* = *fs*) < (*fc* = 2 *fs*)*, the critical values VM (allowing us to set the boundary between period-1 oscillations and subharmonic oscillations) evolve in a decreasing manner, i.e., VM fs* > *VMfs* >

2 *VM*<sup>2</sup> *fs . We can conclude that the stability zone increases with the cut-off frequency, this conclusion justifies once again the bifurcation diagrams in Figure 6.*
