## 5 Dynamic Characteristics with Bifurcation Diagrams in Sub-Harmonic Regimes  
To understand the nonlinear dynamic behaviors, the bifurcation diagrams should be efficient, specifically under the frequency sweeping conditions based on the time-domain solutions. Figures 8a and 9a show the maximum \(\theta_{rms}\), mean, and minimum values of \(\theta_{f}(t)\) with stable conditions from the HBM. Along with these results, 8b and 9b reflect the bifurcations that occur in the sub-harmonic regimes. To obtain results from the bifurcation phenomena based on the NS, the solutions for the steady state responses were captured by assuming that the transient responses are completely removed after 100 cycles while frequency sweeping was conducted with \(5\times 10^{-3}\) a frequency step such as \(\Delta\omega=5\times 10^{-3}\). Since the HBM for this study is parameterized by including, \(\omega\) as described in Equations (15)-(17), the frequency responses shown in Figures 8a and 9a already include the information of bifurcation areas where unstable conditions are generally observed. Here, to obtain the bifurcation phenomena from the NS, the max, mean, min and rms values for each period are captured repeatedly. In addition, the bifurcations in Figures 8b and 9b show the common dynamic characteristics in terms of \(\theta_{rms}\), max, mean, and min values of \(\theta_{f}(t)\) within the sub-harmonic regimes.  
To investigate the dynamic behaviors in the sub-harmonic regimes in more detail, finer temporal steps of NS were conducted and compared with the HBM solutions, as shown in Figures 10 and 11. For instance, bifurcations from NS were obtained with 300 cycles and \(\Delta\omega=5\times 10^{-4}\). As shown in Figures 10 and 11, nonlinear dynamic behaviors are clearly observed with respect to the stable and unstable conditions in the sub-harmonic areas. The various bifurcation characteristics are recognized with locations (1), (2), (3), and (4) locations, as illustrated in Figure 10. First, the comparisons between the HBM stable regimes and NS solutions are perfectly matched with each other in the areas prior to (1), and the period doubling occurs at location (1), where the unstable conditions start based on the HBM. Second, another period doubling effect is observed in regime (2). Third, more period doubling and complicated behaviors are clearly seen along with the upper and lower branches of solutions between regimes (2) and (3). Finally, the bifurcations still exist between regimes (3) and (4) even though the HBM indicates stable conditions, and finally the solutions from NS comply with the HBM solutions. The maximum, mean, and minimum values of \(\theta_{f}(t)\) also follow the same dynamic characteristics, as shown in Figure 11. Thus, the nonlinear dynamic behaviors can be examined and understood more efficiently by employing both stability and bifurcation analyses.  
Figure 8: Stability conditions from HBM with RMS and its bifurcation diagram at the sub-harmonic region: (**a**) stable and unstable conditions in a frequency response with RMS; (**b**) bifurcation diagram of the sub-harmonic responses. Key:, stable solutions of HBM;, unstable solutions of HBM;, numerical results of the bifurcation.  
Figure 9: Stability conditions from HBM with max, mean and min values, and its bifurcation diagrams at the sub-harmonic region: (**a**) stable and unstable conditions in a frequency response with max, mean and min values; (**b**) bifurcation diagram of the sub-harmonic responses. Key:, stable solutions of HBM;, numerical results of the bifurcation.