# Singular Phenomenon Analysis of Wind-Driven Circulation System Based on Galerkin Low-Order Model
## 4 Results and Analysis
### Bifurcation Analysis of the System under First-Order Wind Stress  
As the corresponding second-order wind stress intensity parameter \(w_{2}\) is a fixed value, \(w_{1}\) is the bifurcation parameter. First, let \(w_{2}=5\) and focus on the parameter curve \(C_{3}\) shown in Figure 2. Figure 3 is the bifurcation diagram of equilibrium value \(\tilde{A}_{1}\) of the first-order modal amplitude varying with the wind stress intensity parameter \(w_{1}\). In Figure 3, the equilibrium points in the red solid lines \(l_{1}\) and \(l_{2}\) indicate stable focus. The equilibrium points in the blue dashed lines \(P_{1}Q_{1}\) and \(P_{2}Q_{2}\) are unstable. \((w_{1},w_{2})\) focuses. The equilibrium points in the purple-red dashed lines \(Q_{1}S_{1}\) and \(Q_{2}S_{2}\) are unstable nodes. The equilibrium points in the black dashed line \(S_{1}S_{2}\) are saddle points. Correspondingly, the distribution of the real part, \(\kappa_{r}\), of the eigenvalues of the derivative operator in the system at the equilibrium position is also given in Figure 4.  
The two sudden changes occur at \(S_{2}\) and \(S_{1}\) on the \(C_{3}\) curve, respectively. At \(S_{2}\), the equilibrium positions of the system change from one stable focus to one unstable node and one saddle point. However, the equilibrium positions at \(S_{1}\) change from one stable focus, one unstable node, and one saddle point to one stable focus. With the further increase or decrease of \(w_{1}\), the unstable node becomes the unstable focus at \(Q_{1}\) and \(Q_{2}\). It can be seen that an unstable node is experienced during the bifurcation process from the saddle point to the unstable focus. In addition, it changes from the left to the right of point \(P_{1}\), meaning it changes from a stable focus to an unstable focus. So, the system is unstable, and a supercritical Hopf bifurcation occurs at \(P_{1}\). Similarly, a subcritical Hopf bifurcation occurs as the unstable focus changes to the stable focus at \(P_{2}\). Figure 3: Parametric curve corresponding bifurcation diagram. As the bifurcation parameter \(w_{1}\) increases from infinitesimal value, the phase trajectory is initially attracted to the stable branch of \(l_{1}\). As it reaches the equilibrium point at \(P_{1}\), a jump occurs. The phase trajectory jumps over the unstable branch along the green dotted line and is directly attracted to the stable branch \(l_{2}\). Conversely, as the bifurcation parameter \(w_{1}\) decreases from infinity, the phase trajectory at point \(P_{2}\) reaches the stable state attracted to the \(l_{2}\) branch. Then it jumps over the unstable branch and is attracted to the stable branch \(l_{1}\). Therefore, in the middle area of the jump between \(P_{1}\) and \(P_{2}\), there are two possible stable states of the system, which are in the black dashed part. That is, the saddle point on the unstable branch can jump to the stable upper branch \(l_{2}\) or the stable lower \(l_{1}\) branch, meaning the system is in a bistable state. Figure 5 shows the corresponding bifurcation diagrams as the parameter curves \(C_{1}\) and \(C_{2}\) change along the bifurcation set. In Figure 5a, a sudden change occurs at \(w_{1}=0\) with the continuous increase of \(w_{1}\). A stable point is added from the state of no equilibrium point, and then it changes to a stable focus at point \(B\). In Figure 5b, there is no sudden change. Although the equilibrium point is always in a stable state, the type of equilibrium position switches many times as \(w_{1}\) increases continuously. The stable node at the \(B_{1}\) position changes to the stable focus, and the focus of position \(B_{2}\) changes to a stable node. Then it changes to a stable focus at \(B_{3}\), and finally returns to a stable node at the \(B_{4}\) position. Figure 4: The distribution of the real part, \(q\), of the derivative operator of the equilibrium position system. Figure 5: Bifurcation diagram of system Equation (16) when \(w_{2}\) is fixed. (**a**) Along parametric curve \(C_{2}\); (**b**) along parametric curve \(C_{1}\).