## 3 Results and Discussion
### Influence of Amplitude Excitation  
To determine the influence of the excitation amplitude on the system dynamics and nonlinear resonance, the excitation frequency \(\omega\) was fixed, and then magnet responses were computed versus the bifurcation parameter \(A\). Two fixed frequency values were applied: \(\omega=60\) rad/s (blue line) and \(\omega=120\) rad/s (black line). The two displacements were analyzed for each magnet (\(MAX\) and \(MIN\)). This means that the magnet oscillates between both lines. It is similar to the envelope of time series. For a certain range of parameters, the lines intersect, which means that more than one solution exists.  
Figure 9a shows the bifurcation diagram for the top magnet. Figure 9b,c presents the bifurcation diagrams for the bottom magnet and induced current, respectively.  
Figure 8: Numerical magnets’ oscillation of the top and the bottom magnets and induced current for a frequency \(\omega=60\) rad/s and excitation amplitude of \(A=0.6\)\(g\) (**a**), and time responses in the first and second resonances (**b**) for an excitation amplitude of \(A=1\,g\).  
All bifurcation diagrams correspond to each other and provide similar results. The bifurcation branches computed begin from the equilibrium point, which causes an increase in \(A\). An analysis of the results obtained for \(\omega\) = 60 rad/s shows four possible scenarios: (1) one stable solution, if the amplitude ranges from \(A\) = 0 m/s\({}^{2}\) to the \(FB2\) point, and from the \(NS2\) point (\(A\) = 12.5 m/s\({}^{2}\)) to \(A\) = 20 m/s\({}^{2}\); (2) two stable and one unstable solution between the bifurcation points \(FB2\) and \(NS1\); (3) two unstable and one stable solution between \(NS1\) and \(FB1\); and (4) a small region with only one unstable solution between \(FB1\) and \(NS2\) (pink region). The unstable periodic time courses in this region are shown in Figure 9. The coexistence of the stable solutions (branches) results from the fold bifurcation points that change stability from stable to unstable (\(FB1\)) and from unstable to stable (\(FB2\)) during turning, and the coexistence of the solutions depends on the initial conditions.  
For a frequency of \(\omega\) = 120 rad/s (black line), only two bifurcation scenarios are observed. The first is one stable branch between 0 and \(FB2\) and between \(FB1\) and 20 rad/s. The second is two stable and one unstable solution occurring between the \(FB2\) and \(FB1\) bifurcation points. It should be noted that in the case with two stable solutions, one of them has a higher energy output. For example, for a frequency of \(\omega\) = 120 rad/s and an amplitude of \(A\) = 10 m/s\({}^{2}\), the recovered energy from the bottom branch is about 0.016 W and that from the top branch is 0.064 W. The behavior of the two frequencies is different in the studied cases. In the first resonance, unstable solutions caused by NS bifurcations appear in the same branch. The unstable solution appears between the \(NS1\) and \(NS2\) bifurcation points.  
Figure 9: Influence of the amplitude of excitation \(A\) on the bifurcation scenario for the top magnet (**a**), bottom magnet (**b**), and induced current (**c**) for two fixed excitation frequencies \(\omega\) = 60 rad/s (blue line) and \(\omega\) = 120 rad/s (black line). The pink region means a small region with only one unstable solution. The top line means \(MAX\), while the bottom line means \(MIN\) values of the magnet displacements.