## 4 Results  
The VC 2-Order Superharmonic Resonance, 1/2-Order Subharmonic Resonance and the Case of \(\omega_{x0}+\omega_{y0}\) = \(\Omega_{x0}\) Combination Resonance  
For \(\delta_{0}\) = 1.5 \(\upmu\)m, Figure 2 shows the VC period-1 and period-2 solution branches traced by the HB-AFT method with arc-length continuation. Predicting from Equations (8) and (9), the equivalent resonant frequencies of the two degrees of freedom of System (5) are about \(\omega_{x0}\) = 1908.14 rad/s and \(\omega_{y0}\) = 1297.36 rad/s. Thus, the amplitudes in the \(R_{1}\) and \(R_{2}\) frequency ranges, including \(\omega_{x0}N_{b}\) = 238.52 rad/s and \(\omega_{y0}N_{b}\) = 162.17 rad/s, are the VC primary resonances excited in the \(x\) and \(y\) directions of the system, respectively. These two primary resonances exhibit soft spring and soft-hard spring characteristics in the vertical and horizontal directions, respectively, which agrees with previous theoretical and experimental studies [22; 24]. Moreover, 1/2-order subharmonic resonances in two degrees of freedom are excited in \(R_{3}\) and \(R_{4}\) intervals, where the subharmonic resonant amplitudes are around \(2\cdot\omega_{x0}N_{b}\) = 477.04 rad/s and \(2\cdot\omega_{y0}N_{b}\) = 324.34 rad/s. As shown in Figure 3a, the resonant amplitude in the \(R_{4}\) range is excited by supercritical period-doubling bifurcations at points \(A_{1}\) (see Table 2) and \(A_{2}\), where the leading Floquet multipliers cross the unit circle at -1. The response characteristics of the period-1 and period-2 motions before and after period-doubling transitions at point \(A_{1}\) are presented in Figures 4 and 5, and the response amplitude of \(y(\tau)\) clearly increases after the bifurcation. Then, the leading Floquet multiplier of the stable VC period-2 solution crosses the +1 axis at \(A_{3}\) and \(A_{4}\) (see Table 3), and cyclic fold bifurcation occurs at these two points. This causes significant resonant hysteresis and jumping behaviors in the 1/2-order subharmonic resonance in the horizontal direction. For example, as the control parameter \(\Omega\) increases, the stable period-2 branch suddenly jumps to the stable period-1 branch from point \(A_{4}\) to \(A_{5}\). For the 1/2-order subharmonic resonance as \(\Omega\) around \(2\cdot\omega_{x0}N_{b}\) = 477.04 rad/s (i.e., \(R_{3}\) range), as shown in Figure 3b, supercritical and subcritical period-doubling bifurcations trigger the VC period-2 resonant responses at points \(B_{1}\) and \(B_{2}\), respectively. At this time, the subharmonic resonant frequency-response  
Figure 2: For \(\delta_{0}\) = 1.5 \(\upmu\)m, stable (solid) and unstable (dashed) varying compliance (VC) periodic peak-to-peak frequency–response curves in \(x\) (red line) and \(y\) (black line) directions. curve bends to the left, and this soft spring characteristic is consistent with the hysteresis characteristics of the primary resonance in the vertical direction as \(\Omega\) around \(\omega_{x0}/N_{b}=238.52\) rad/s. Figure 4: For \(\delta_{0}=1.5\)\(\upmu\)m, response characteristics of the VC period-1 motion when \(\Omega=310\) rad/s, (**a**) orbit (black line) and its Poincare mapping (red dots), (**b**) time series and their frequency spectra. Figure 3: For \(\delta_{0}=1.5\)\(\upmu\)m, frequency–response curves of 1/2-order subharmonic resonances of the two degrees-of-freedom of the system, (**a**) in the horizontal and (**b**) vertical directions. Moreover, the stable period-2 solution branch undergoes supercritical period-doubling bifurcation at \(B_{3}\), generating a stable period-4 branch \(B_{3}\)-\(B_{4}\). Then, a period-8 solution branch \(B_{4}\)-\(B_{5}\) is excited by supercritical period-doubling bifurcation at point \(B_{4}\)., where point \(B_{5}\) is the cycle-doubling bifurcation point.