## 4 Analysis of Nonlinear Bifurcation and Limit Cycle Oscillation  
Using the Runge-Kutta method discussed in the previous section, the LCO amplitude at a specific wind speed can be obtained through integration. Therefore, by selecting wind speed values within a certain range and substituting them into the reference case of the nonlinear motion system, we can integrate them to obtain a set of correspondences between wind speed and the stable LCO amplitude. This data can be used to plot the bifurcation curve of the flutter, as shown in Figure 10, which illustrates the relationship between wind speed and the stable LCO amplitude of the torsional degree of freedom for this particular operating condition. In this context, the abscissa U signifies the actual wind speed, while the ordinate A denotes the maximum amplitude of the limit cycle for the torsional degree of freedom. At the flutter critical state \(U_{cr}=57.3374\) m/s, the nonlinear flutter system undergoes a Hopf bifurcation, and a stable LCO bifurcation curve appears on the right side of the bifurcation point. The LCO amplitude rises with the increasing wind speed. From the graph, it can be observed that after the wind speed exceeds \(U=77.7\) m/s, the trend of the limit cycle amplitude shows a steep increase, and stable limit cycle vibration cannot be obtained at higher wind speeds. Therefore, for this operating condition of the nonlinear flutter system, vibration divergence occurs when the wind speed exceeds \(U=77.7\) m/s. Further research can be conducted on the bifurcation curve in Figure 10, which may reveal the presence of unstable LCO within the range of wind speeds \(U<77.7\) m/s, with amplitudes larger than the stable limit cycle. This can be investigated by setting larger initial conditions and performing integration calculations. * Wind Speed: Wind speed \(U=75\) m/s (beyond the critical state); Initial state of motion: \(\left(h,\dot{h},\alpha,\dot{\alpha}\right)_{3}^{T}=(0,0,0.45,0)^{T}\). By selecting a wind speed of \(U=75\) m/s and initial conditions of \(\left(h,\dot{h},\alpha,\dot{\alpha}\right)_{3}^{T}=(0,0,0.45,0)^{T}\), the time response of the flutter system vibration can be computed. As shown in Figures 11 and 12, under the given initial conditions, the vibration amplitude of the system initially decays and eventually stabilizes into a LCO vibration response after a period of time. Moreover, this stable LCO vibration response is consistent with the stable limit cycle response shown in Figures 4 and 5. The phase portraits of these two figures, representing the limit cycles, are plotted in Figure 13.
* Wind Speed: Wind speed \(U=75\) m/s (beyond the critical state); Initial state of motion: \(\left(h,\dot{h},\alpha,\dot{\alpha}\right)_{4}^{T}=(0,0,0.452,0)^{T}\). Figure 10: Bifurcation curves for the relationship between stable LCO amplitudes and wind speed. The vibration response of the system shows a slow increase in amplitude and max speed in both DOFs, indicating a gradual divergence of the oscillations. The vibration phase diagram is shown in Figure 16. A comparison with the motion phase diagram in Figure 13, which had the initial conditions \(\left(h,\dot{h},\alpha,\dot{\alpha}\right)_{3}^{T}=(0,0,0.45,0)^{T}\) under the same wind speed, reveals the presence of an unstable limit cycle oscillation within the range of torsional amplitudes between 0.45 and 0.452. The vibration phase trajectories on either side of this LCO will gradually move away from it. From the analysis above, at a wind speed of \(U=75\) m/s, the origin point of this nonlinear flutter system is an unstable focus. Additionally, there are two LCOs in the phase diagram. The smaller-amplitude LCO is stable, and vibration starting from the unstable focus will eventually converge to this stable LCO. The larger amplitude limit cycle is unstable one. Vibrations within the range between the unstable and stable LCOs will gradually approach the stable one, while vibrations with amplitudes greater than the unstable LCO will diverge.