## 2 Stability of the Equilibrium Point
### Stability Analysis by Lyapunov-LaSalle Energy-Based Approach
At this point, we could not use the LaSalle local invariant set theorem as the conditions of the LaSalle local invariant set theorem are not satisfied, and it might also mean that the limit cycle or the equilibrium point is unstable. In order to determine exactly whether the equilibrium point or limit cycle is stable or not, we will need to choose a different Lyapunov function as follows, \(V(x)={x_{1}}^{2}+{x_{2}}^{2}\). If we look at the time derivative of the \(V(\underline{x})\), which is \(\dot{V}(\underline{x})=2({x_{1}}^{2}+{x_{2}}^{2})({x_{1}}^{2}+{x_{2}}^{2}-1)\), when \({x_{1}}^{2}+{x_{2}}^{2}<1\), geometrically it indicates location inside of the circle with a radius 1, then \(\dot{V}(\underline{x})<0\). Based on the [PERSON] global invariant set theorem, we can conclude that \(\dot{V}(\underline{x})=0\) as time goes on, which further indicates that \(\left( \begin{array}{c}x_{1}=0\\ x_{2}=0\end{array} \right)\) or \({x_{1}}^{2}+{x_{2}}^{2}=1\) as time goes on. From here, we can immediately see that the given dynamical system has an equilibrium point \(\left( \begin{array}{c}x_{1}=0\\ x_{2}=0\end{array} \right)\) and a limit cycle \({x_{1}}^{2}+{x_{2}}^{2}=1\). Also, based on the [PERSON] global invariant set theorem, we can conclude that the trajectories either tend to the equilibrium point at \(\underline{x}=\left[ \begin{array}{c}x_{1}\\ x_{2}\end{array} \right]=\left[ \begin{array}{c}0\\ 0\end{array} \right]\) or the limit cycle \({x_{1}}^{2}+{x_{2}}^{2}=1\). However, it is not possible to tend to the limit cycle \({x_{1}}^{2}+{x_{1}}^{2}=1\), as we had the previous assumption that \({x_{1}}^{2}+{x_{2}}^{2}<1\) and therefore, all the trajectories tend to the equilibrium point at \(\underline{x}=\left[ \begin{array}{c}x_{1}\\ x_{2}\end{array} \right]=\left[ \begin{array}{c}0\\ 0\end{array} \right]\), that is to say the equilibrium \(\underline{x}=\left[ \begin{array}{c}x_{1}\\ x_{2}\end{array} \right]=\left[ \begin{array}{c}0\\ 0\end{array} \right]\) is asymptotically stable. In terms of the limit cycle, based on the LaSalle local invariant set theorem and the Lyapunov candidate function we chose \(V(\underline{x})=\left({x_{1}}^{2}+{x_{2}}^{2}-1\right)^{2}\), even though we could not use the LaSalle local invariant set theorem to conclude the result as the conditions of the LaSalle local invariant set theorem are not satisfied. However, we can still use the one of the conditions \(\dot{V}(\underline{x})\) to see how \(V(\underline{x})\) changes as time goes on. Because \(\dot{V}(\underline{x})>0\), it means \(V(\underline{x})\) increases, which indicates that the limit cycle is unstable. We can also use the computer simulation to verify the result. Now, if we plot the Lyapunov candidate function \(V(\underline{x})=\left({x_{1}}^{2}+{x_{2}}^{2}-1\right)^{2}\) as a function of the state \(\underline{x}=\left[ \begin{array}{c}x_{1}\\ x_{2}\end{array} \right]\), as shown in Figure 2, we can see that all the trajectories start nearby the limit cycle will move away from the limit cycle because \(\dot{V}(\underline{x})>0\), as shown in Figure 2c. Please note that the author used the blue arrows in Figure 2c below to indicate how \(V(\underline{x})\) changes as time goes on. The valley in Figure 2a is the limit cycle, and it can be seen from the top view, as indicated in the Figure 2b. Again, according to Figure 2, the equilibrium point is stable, as we can see from Figure 2c by the fact that the trajectories (the blue arrows) are approaching the equilibrium point, not moving away from it. The limit cycle is, however, unstable. As we can see from Figure 2c, the trajectories (the blue arrows) are moving away from the limit cycle, not approaching it. **Case 2**: consider the following dynamical system as an example, and the dynamical equation is given as follows:  
\[\left\{ \begin{array}{l}\dot{x}_{1}=4{x_{1}}^{2}{x_{2}}-{x_{1}}({x_{1}}^{2}+2{x_ {2}}^{2}-9)\\ \dot{x}_{2}=-2{x_{1}}^{3}-{x_{2}}({x_{1}}^{2}+2{x_{2}}^{2}-9)\end{array} \right. \tag{4}\]  
where the state is \(\underset{-}{x}=\left[