# Singular Phenomenon Analysis of Wind-Driven Circulation System Based on Galerkin Low-Order Model
## 4 Results and Analysis
### Bifurcation Analysis of the System under Second-Order Wind Stress  
As the circulation system has only second-order wind stress, the corresponding intensity parameter \(w_{1}\) of the first-order wind stress is a fixed value, and \(w_{2}\) is the bifurcation parameter. First, let \(w_{1}=1\) and focus on the parameter curve \(L_{3}\) in Figure 2. Figure 6 is a bifurcation diagram of the value \(\bar{A}_{1}\) of the first-order modal amplitude equilibrium point changing with the wind stress intensity parameter \(w_{2}\). In Figure 6, the equilibrium points in the purple solid line \(l_{1}\) are stable nodes. The equilibrium points in the red solid lines \(l_{2}\) and \(l_{4}\) are stable focuses. However, the equilibrium points in the black dashed line \(l_{3}\) are saddle points, the equilibrium points in the purple-red dashed line \(TN\) are unstable nodes, and the equilibrium points in the blue dashed line \(NK\) are unstable focuses. The sudden change in the system only occurs at point \(T\) on the parameter curve \(L_{3}\). In this situation, an unstable node and a saddle point are added from the stable focus state on branch \(l_{2}\). With the further increase in the parameter \(w_{2}\), the unstable node becomes an unstable focus at \(N\). Finally, the subcritical Hopf bifurcation occurs in the lower part of the equilibrium curve at \(K\) from an unstable focus to a stable focus, which is consistent with the sudden change along the parameter curve \(C_{3}\). At \(M\), the transition from a stable node to a stable focus is consistent with switching the stable point as changing along the parameter curves \(C_{1}\) and \(C_{2}\). In fact, as the parameter \(w_{2}\) increases, the phase trajectory of the system is always attracted to the stable node branch \(l_{1}\) or the stable focus branch \(l_{2}\) before point \(K\). However, the phase trajectory of the system may be attracted to the stable focus branch \(l_{2}\) or \(l_{4}\) after point \(K\). In this situation, the system has two possible stable states, i.e., in a bistable state. Figure 7 shows the bifurcation diagrams as the parameter curves \(L_{1}\) and \(L_{2}\) change along the bifurcation set. For the parametric curve \(L_{1}\), the bifurcation behavior is similar to the variation along the parametric curve \(L_{3}\), as shown in Figure 7a. The stable branch is continuous in the lower branch after bifurcation, while an unstable state appears in the upper branch. From Figure 7b, it can be seen that a sudden change occurs at point \(O\) as the parameter curve \(L_{2}\) changes. Then, the number and type of equilibrium points before and after point \(O\) change topologically, which is a fork bifurcation. In addition, it is clear that the bifurcation occurring along the parameter curves \(L_{1}\) and \(L_{3}\) is an imperfection bifurcation, and its prototype is a fork bifurcation that occurs as \(L_{2}\) changes. The transition from stable nodes to stable focus also occurs at \(P_{1}\) and \(P_{2}\), which is the same as the change along curve \(C_{2}\) with abrupt changes at the cusp. Figure 6: The bifurcation diagram corresponding to the parameter curve \(L_{3}\). Overall, as the value of the system parameters \((w_{1},w_{2})\) is distributed in the \(D_{1}\) area on the corresponding bifurcation set (Figure 2), system Equation (16) has only one equilibrium state and is a stable node, \(P_{SN}\), as shown in Figure 8a, or stable focus, \(P_{SF}\), as shown in Figure 8b.