## 4 Results
In the above bifurcation process, the strongly excited resonant period-2 and period-4 motion modes in the \(x\) direction become saturated at \(B_{3}\) and \(B_{4}\) points, and the energy spills over into the period-4 and period-8 motion modes in the \(y\) direction as \(\Omega\) decreases, respectively. This typical saturation phenomenon has been discussed in [26, 27]. In short, the 1/2-order subharmonic resonances are aroused by the period-doubling bifurcation instability, and they contain complex hysteresis \begin{table}
\begin{tabular}{c c c c c} \hline \(\Omega\) & **316.3645** & **316.4645** & **316.5645** & **316.6645** \\ \hline \multirow{4}{*}{\(A_{m}\)} & \(0.021+0.987\)i & \(0.019+0.987\)i & \(0.018+0.988\)i & \(0.017+0.988\)i \\  & \(0.021-0.987\)i & \(0.019-0.987\)i & \(0.018-0.988\)i & \(0.017-0.988\)i \\  & \(-0.988+0.009\)i & \(-0.995\) & \(-1.002\) & \(-1.006\) \\  & \(-0.988-0.009\)i & \(-0.980\) & \(-0.974\) & \(-0.969\) \\ \hline \end{tabular}
\end{table} 
Table 2: Period-1 motion Floquet multipliers \(A_{m}\) around \(A_{1}\). \begin{table}
\begin{tabular}{c c c c c} \hline \(\Omega\) & **341.7310** & **341.7316** & **341.7316** & **341.7315** \\ \hline \multirow{4}{*}{\(A_{m}\)} & \(-0.536+0.817\)i & \(-0.535+0.818\)i & \(-0.535+0.818\)i & \(-0.535+0.818\)i \\  & \(-0.536-0.817\)i & \(-0.535-0.818\)i & \(-0.535-0.818\)i & \(-0.535-0.818\)i \\  & \(0.987\) & \(0.998\) & \(1.001\) & \(1.002\) \\  & \(0.968\) & \(0.957\) & \(0.954\) & \(0.953\) \\ \hline \end{tabular}
\end{table} 
Table 3: Period-2 motion Floquet multipliers \(A_{m}\) around \(A_{4}\) (counterclockwise, see Figure 3a). Figure 5: For \(\delta_{0}=1.5\) μm, response characteristics of the VC period-2 motion when \(\Omega=330\) rad/s, (a) orbit (black line) and its Poincare mapping (red dots), (**b**) time series and their frequency spectra. characteristics. Finally, as shown in Figure 6, the above results agree well with the numerical bifurcation diagrams from the local-maxima method [52]. Furthermore, another resonant amplitude is excited around \(\omega_{x0}/N_{b}+\omega_{y0}/N_{b}=400.69\) rad/s (i.e., the resonant range \(R_{5}\) between intervals \(R_{3}\) and \(R_{4}\) presented in Figure 2), and this case of resonance results in strong coupling to the two degrees-of-freedom of the system as shown in Figure 6. In terms of the bifurcation characteristics concerned, as a pair of complex conjugate Floquet multipliers move out of the unit circle (see Table 4), subcritical and supercritical secondary Hopf bifurcations occur at points \(B_{6}\) and \(B_{7}\), respectively, leading quasi-period responses excited at point \(B_{7}\). The response characteristics of the quasi-period motion at \(\Omega=400\) rad/s is presented in Figure 7; as there are incommensurable frequency components, the Poincare mapping of the response orbit tends to a closed point set. Herein, the values of the main incommensurable dimensionless VC frequencies p and q are about 0.5936 and 0.4074, respectively, and p + q = 1, where the value 1 is the dimensionless VC excitation frequency \(\Omega_{\rm VC}=N_{b}\cdot\Omega\). Moreover, for the original time scale, the corresponding main frequency components of the response are \(N_{b}\cdot\Omega\cdot\)p \(\approx\) 1899.52 rad/s and \(N_{b}\cdot\Omega\cdot\)q \(\approx\) 1303.68 rad/s, which are close to the predicted equivalent resonant frequencies \(\omega_{x0}=1908.14\) rad/s and \(\omega_{y0}=1297.36\) rad/s of the two degrees-of-freedom of System (5). This indicates that the case of \(\omega_{x0}+\omega_{y0}=\Omega_{\rm VC}\) combination resonance (i.e., the 1-order summed type of combination resonance) [39; 53] occurs in range \(R_{5}\), and quasi-period resonant responses are excited. In general, the typical combination resonance of multiple degrees-of-freedom systems often causes quasi-period motions [27; 54; 55]. Therefore, it can be considered that the combination resonance is one of the physical mechanisms of the generation of VC quasi-period motions in rolling bearing systems.