## 3 Flutter Critical State and Hopf Bifurcation
### Proof of Hopf Bifurcation  
The Jacobian matrix of the state-space Equation (14) is obtained according to the rule in Equation (17). \[\mathbf{J}_{F(x)}= \begin{bmatrix}0&0&1&0\\ 0&0&0&1\\ \frac{\partial\ddot{u}}{\partial\dot{t}}&\frac{\partial\ddot{u}}{\partial\dot {u}}&\frac{\partial\ddot{t}}{\partial\dot{t}}&\frac{\partial\ddot{u}}{\partial \dot{t}}\\ \frac{\partial\ddot{u}}{\partial\dot{t}}&\frac{\partial\ddot{u}}{\partial\dot {u}}&\frac{\partial\ddot{u}}{\partial\dot{t}}&\frac{\partial\ddot{u}}{\partial \dot{u}}\end{bmatrix}  \tag{17}\]  
Based on the results of the critical values of the flutter, taking the reduced wind speeds \(V_{r}=7\), \(V_{r}=V_{cr}=7.2673\), and \(V_{r}=7.5\), they are substituted into the Jacobian matrix to calculate the corresponding eigenvalues, as shown below:  
\[\lambda\left|{}_{V_{r}=7}\right.= \begin{pmatrix}-0.0543+0.4782i\\ -0.0543-0.4782i\\ -0.0022+1.2001i\\ -0.0022-1.2001i\end{pmatrix} \]  
\[\lambda\left|{}_{V_{r}=7.2673}\right.= \begin{pmatrix}-0.0593+0.4754i\\ -0.0593-0.4754i\\ 0.0000+1.1888i\\ 0.0000-1.1888i\end{pmatrix} \]  
\[\lambda\left|{}_{V_{r}=7.5}\right.= \begin{pmatrix}-0.0638+0.4725i\\ -0.0638-0.4725i\\ 0.0023+1.1785i\\ 0.0023-1.1785i\end{pmatrix} \]  
The obtained eigenvalues from the three sets of results above indicate that the Jacobian matrix consistently has pairs of complex conjugate eigenvalues. Moreover, as the system parameter \(V_{r}\) surpasses the flutter critical value from small to large, the Jacobian matrix exhibits a pair of complex conjugate eigenvalues with a change in the real part from negative to positive. This characteristic satisfies the definition of Hopf bifurcation. Therefore, the aforementioned process demonstrates that the nonlinear flutter system experiences Hopf bifurcation at the flutter critical state. Furthermore, Hopf bifurcation encompasses three types of critical bifurcations. Next, the nonlinear flutter response in the intervals on both sides of the Hopf bifurcation point will be analyzed through calculations to further prove the Hopf bifurcation type. The Runge-Kutta methods will be used to compute the motion response of this nonlinear flutter equation [32]. The flutter equations, as established in this paper, have been formulated into a first-order ordinary differential equation represented as Equation (14). Consequently, for the purpose of solving this equation, the adoption of the fourth-order Runge-Kutta method is appropriate. By carefully selecting an appropriate time step, the temporal response of the flutter can be obtained, thereby facilitating a more in-depth analysis of nonlinear vibrations. * Wind speed: \(U=75\) m/s (beyond the critical state) Initial state of motion: \(\left(h,\dot{h},\alpha,\dot{\alpha}\right)_{1}^{T}=\left(0,0,0.001,0\right)^{T}\) (small perturbation). Taking wind speed \(U=75\) m/s, setting the system vibration frequency \(\omega=1.888\) rad/s, and using the initial conditions for numerical integration \(\left(h,\dot{h},\alpha,\dot{\alpha}\right)_{1}^{T}=\left(0,0,0.001,0\right)^{T}\), the vibration time history response is obtained as shown below. Additionally, the vibration limit cycle is plotted. From Figures 4 and 5, it can be observed that when the wind speed \(U=75\)\(m/s\), the vibration response of the nonlinear flutter system begins to diverge. This indicates that at this point, the focus of the flutter system, i.e., the zero point, becomes unstable. It becomes an unstable focus, and flutter occurs under the influence of a small perturbation \(\alpha\)=0.001 rad. The motion trajectory gradually moves away from this unstable focus. After a certain period of time, the flutter tends towards a stable state known as limit cycle oscillation. Figure 6 represents the phase portraits of Figures 4 and 5 in the velocity and displacement planes, illustrating the occurrence of stable limit cycle oscillation in the system at the current wind speed.