## 4 Analysis of Nonlinear Bifurcation and Limit Cycle Oscillation
In summary, the Runge-Kutta integration method can be used to solve flutter systems with nonlinear aerodynamic self-excitation and obtain bifurcating LCOs beyond the critical state. The stable LCO amplitude at a specific wind speed can be obtained through the steady-state time response, while the unstable LCO requires the following calculation steps: select a wind speed \(U\) and an initial condition \(\left(0,0,\alpha_{0},0\right)^{T}\), calculate the time response of the nonlinear system vibrations, and continuously change the initial amplitude \(\alpha_{0}\). When  
Figure 14: Continuous vibration of bridge flutter in vertical bending DOF at wind speed of 75 m/s and initial state of motion \(\left(h,\dot{h},\alpha,\dot{\alpha}\right)_{4}^{T}=(0,0,0.452,0)^{T}\). Figure 15: Continuous vibration of bridge flutter in torsional DOF at wind speed of 75 m/s and initial state of motion \(\left(h,\dot{h},\alpha,\dot{\alpha}\right)_{4}^{T}=(0,0,0.452,0)^{T}\). the response above this value converges to the stable limit cycle and the response above this value diverges, it can be concluded that there exists an unstable limit cycle at the wind speed \(U\), with an amplitude of \(\alpha_{0}\). By substituting the parameter values of the reference case into the flutter Equations (4) and (5), the stable and unstable limit cycles corresponding to the wind speed can be obtained. Figure 17 shows the relationship between the limit cycle amplitudes and wind speed. Taking into account the parameter settings for the other three operating conditions in Table 2, the relationship between the LCO amplitude and wind speed after the supercritical Hopf bifurcation is calculated using the aforementioned method. The results are plotted in Figures 18 and 19. As shown in the results of the LCO bifurcation curves, the flutter systems of all four operating conditions undergo supercritical Hopf bifurcation at the same critical value, leading to stable limit cycle oscillations on the side above the critical value. The stable LCO amplitude rises with the increasing wind speed. Comparing the curves of the reference case and case A, the two curves nearly overlap, but the stable LCO amplitude of case A is much higher than the corresponding one. The stable LCO amplitude rises with the increasing wind speed. Figure 16: Phase trajectory diagram of LCO in vertical and torsional DOF at wind speed of 75 m/s and initial state of motion \(\left(h,\dot{h},\alpha,\dot{\alpha}\right)_{4}^{T}=\left(0,0,0.452,0\right)^{T}\). Figure 17: Bifurcation curves for the relationship between LCO amplitudes and wind speed. A is relatively smaller than that of the reference case, and the unstable LCO amplitude is larger than that of the reference case. This indicates that applying stronger nonlinear damping on the torsional degree of freedom has a limited dissipating effect on limit cycle oscillations under supercritical flutter. Furthermore, comparing the curves of case A and case B, it can be observed that to achieve the same LCO amplitude, case B needs higher wind speeds. This suggests that considering the nonlinear stiffness of the bridge structure reduces the stable LCO amplitude and increases the unstable LCO amplitude under supercritical flutter, indicating a higher stability of the bridge compared to a system without considering structural nonlinear stiffness. Finally, in case C, by adding strong nonlinear damping on the vertical bending degree of freedom, the corresponding wind speed for the same LCO amplitude is the highest, demonstrating strong energy dissipation and suppression capabilities for supercritical flutter. In conclusion, by considering various nonlinear factors, the variation relationship between the limit cycle amplitude and wind speed for the four operating conditions has been analyzed. Among them, considering structural nonlinear stiffness and cubic damping on the vertical bending degree has the most significant influence on the LCO amplitude under supercritical flutter.