# Singular Phenomenon Analysis of Wind-Driven Circulation System Based on Galerkin Low-Order Model
## 4 Results and Analysis
\tag{16}\]  
Following Equation (A10) in Appendix A, the coefficients of the coupling terms can be given as \(\lambda_{1}=1.2973,\lambda_{2}=0.5897\) in Equation (16). Considering that the coefficient of horizontal momentum exchange,\(A_{H}\), is relatively large, it might as well be given as \(Re=20\). Then, the coefficient of the linear term is \(r_{1}=0.4500,r_{2}=0.5045\). For the constant term, \(w_{1}\) and \(w_{2}\) are variable parameters, and the coefficient is taken as \(\sigma_{1,1}=0.3643,\sigma_{2,2}=0.8015\). According to Appendix B, through the analysis of the equilibrium position of Equation (16), it can be concluded that the equilibrium position can be governed by the following equations,  
\[\bar{A}_{1}^{3}+\frac{r_{1}r_{2}-\lambda_{1}\sigma_{2,2}w_{2}}{\lambda_{1} \lambda_{2}}\bar{A}_{1}-\frac{r_{2}\sigma_{1,1}w_{1}}{\lambda_{1}\lambda_{2}}=0 \tag{17}\]  
\[\bar{A}_{2}=\left(\sigma_{2,2}w_{2}-\lambda_{2}\bar{A}_{1}^{2}\right)/r_{2} \tag{18}\]  
Therefore, as parameters \(w_{1}\) and \(w_{2}\) are determined, the equilibrium position \((\bar{A}_{1},\bar{A}_{2})\) can be obtained according to Equation (18). The derivative operator of the system, Equation (16), at the equilibrium position is given by the Jacobian matrix,  
\[\left[ \begin{array}{cc}-r_{1}+\lambda_{1}\bar{A}_{2}&\lambda_{1}\bar{A}_{1 }\\ -2\lambda_{2}\bar{A}_{1}&-r_{2}\end{array} \right] \tag{19}\]  
The eigenvalues, \(\kappa\), of the matrix are given by the following quadratic equation,  
\[\kappa^{2}+(r_{1}+r_{2}-\lambda_{1}\bar{A}_{2})\kappa+2\lambda_{1}\lambda_{2 }\bar{A}_{1}^{2}+r_{2}(r_{1}-\lambda_{1}\bar{A}_{2})=0 \tag{20}\]  
Note that Equation (20) is defined in the complex domain, and the type of equilibrium position depends on the eigenvalues. Since the equilibrium position is governed by the one-dimensional cubic equation, namely Equation (17) with two parameters \((w_{1},w_{2})\), it is clear that such an equation is an extension of the governing equation for the system with a cusp catastrophe. According to the cusp catastrophe theory, Equation (17) gives an equilibrium surface or catastrophe manifold. From Equations (17)-(20), the equilibrium surface shown in Figure 1 and \(\bar{A}_{1}\) changes with the wind stress intensity parameters \(w_{1}\) and \(w_{2}\) can be obtained. The red areas \(H_{1}\) and \(H_{3}\) are stable equilibrium states, and the blue area \(H_{2}\) is the unstable equilibrium state. On the manifold, it shows that the value of \(\bar{A}_{1}\) in the phase point \((w_{1},w_{2},\bar{A}_{1})\) changes steadily as the control parameter changes, and only abruptly changes at some specific positions. According to Appendix B, combined with the one-dimensional cubic equation with one variable, it can be seen that the catastrophe position corresponds to the critical point of the discriminant \(\Delta=0\). Therefore, the condition that the catastrophe point should meet on the parameter surface is  
\[\Delta=\left(-\frac{r_{2}\sigma_{1,1}w_{1}}{2\lambda_{1}\lambda_{2}}\right)^{2}+ \left(\frac{r_{1}r_{2}-\lambda_{1}\sigma_{2,2}w_{2}}{3\lambda_{1}\lambda_{2}} \right)^{3}=0 \tag{21}\]  
Therefore, the catastrophe point can be obtained to form a singular point set. Figure 2 shows its projection onto the control parameter plane \((w_{1},w_{2})\), which is also called a bifurcation set. The curve is a half cubic parabola, which has a cusp at \(O(0,0.4804)\), so it is called a cusp catastrophe. As the phase point moves continuously to the catastrophe point at the upper \(H_{1}\) of the equilibrium surface, it will jump directly over \(H_{2}\) and enter the stable equilibrium surface \(H_{3}\), since the middle \(H_{2}\) of the equilibrium surface is in an unstable state. Figure 1: Equilibrium surface (red for stability, blue for instability). Figure 2: The bifurcation set of system Equation (16) with the variation of control parameter \((w_{1},w_{2})\).