## 3 Stability of Limit Cycle
### Stability of Limit Cycle by Poincare-Bendixson (PB) Theorem  
It is noted that the PB theorem is mainly used to prove whether there exists a limit cycle instead of proving the stability of the limit cycle, and that the PB theorem is only valid in two-dimensional systems. By using the PB theorem, we are able to determine whether there exists a limit cycle [73], and once there exists a limit cycle, we then need to go back to use the invariant set theorem to show if the limit cycle is stable or not. The Poincare-Bendixson theorem is described as follows [61]:  
\"If we have the following conditions satisfied: (1) if we can find a closed and bounded region, (2) and there is no equilibrium point in the region, (3) and there exists a trapped trajectory that lies in the region from the beginning and stays in the region for all future times, then the trajectory is either a closed trajectory itself or it spirals toward a closed trajectory as time goes on\".  
The standard way to find this region is to find a closed region so that the vector field points into the region on its boundary [61; 74; 75; 76]. Consider the following system as an example:  
\[\left\{\begin{array}{l}\dot{x}=y+\frac{1}{1+x}-3\\ \dot{y}=-x^{2}y-0.05y+0.6\end{array}\right. \tag{20}\]  
The question is that can we find a region such that there exists a limit cycle in this region. We can plot the system when \(\dot{x}=0\) and \(\dot{y}=0\), as shown in Figure 10(a). The red and black lines divide the entire region into four different small regions, as shown in Figure 10(b). In region 1, we know that \(\dot{x}>0\) and \(\dot{y}<0\), which indicates that \(x\) increases and \(y\) decreases, so the vector field in the region 1 points toward the southeast. In region 2, \(\dot{x}<0\) and \(\dot{y}<0\), which indicates that \(x\) decreases and \(y\) decreases as well, so the vector field in region 2 points southwest. In region 3, \(\dot{x}<0\) and \(\dot{y}>0\), which indicates that \(x\) decreases and \(y\) increases, so the vector field in region 3 point towards northwest. In region 4, \(\dot{x}>0\) and \(\dot{y}>0\), which indicates that \(x\) increases and \(y\) increases as well, so the vector field in region 4 points towards northeast. So, overall the vector field in the entire field is in a clockwise direction. By observing the vector field in the entire region, we know that there might be a limit cycle as the vector field forms a cycle. To further verify if indeed there exists a limit cycle, we can construct a closed region, i.e., the region between the right trapezoid and the small circle, as shown in Figure 10(b). In order to see if there exists a limit cycle in that region, we need to look at the vector field, and as we already knew that the vector field indeed points into the right trapezoid region on its boundary, all we know to see next is whether the cycle is a repelling cycle or not, i.e., if the vector field in the cycle points into the right trapezoid region or not. We can calculate the Jacobian matrix at the equilibrium point, i.e., the interaction point between and the red line and black line, as follows:  
\[J=\begin{bmatrix}-\frac{1}{(x+1)^{2}}&1\\ -2 xy&-x^{2}-0.05\end{bmatrix} \tag{21}\]  
At the equilibrium point \(\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}0.45\\ 2.31\end{array}\right]\), the trace of the Jacobian matrix is smaller than 0, which indicates that the equilibrium point is a saddle point. This means that the vector field surrounding the circle does not point into the right trapezoid region and therefore, based on the PB theorem, we cannot conclude that there exists a limit cycle.