## 4 Comparison among Different Approaches  
Comparisons among the approaches to analyze the stability of the equilibrium point of two-dimensional nonlinear dynamical systems, i.e., Lyapunov-LaSalle energy-based approach, graphical phase plane approach, and linearizing-around-an-equilibrium-point approach, are presented in the Table 1. The advantages and disadvantages of each method are illustrated. Similarly, the comparisons among the approaches to analyze the stability of the limit cycle of two-dimensional nonlinear dynamical system, i.e., Poincare-Bendixson theorem and LaSalle local invariant set theorem, are also presented in Table 1. \begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \multicolumn{2}{p{113.8 pt}}{**Approaches for Stability**} & \multicolumn{1}{c}{**Abvantages**} & \multicolumn{1}{c}{**Disadvantages**} \\ \multicolumn{2}{p{113.8 pt}}{**Equilibrium Point**} & & \\ \hline \multirow{6}{*}{
\begin{tabular}{} \end{tabular} } & 1. This analytical method can give quantitative result based on the Lyapunov and LaSalle theorems; & 1. It is difficult to choose the right Lyapunov candidate function, if one did not manage finding one, it does not mean the equilibrium point being analyzed is unstable, it could be stable or it could be unstable; & 1. This method has limitations, it is not applicable to all dynamical systems. It can only give us the correct dynamics information fast near the equilibrium point \\ \cline{2-3}  & 1. It is a visualization method, which can give the conclusion without being getting into calculation; & 1. This method most of the time is limited to the two-dimensional systems; & 1. This method most of the time is limited to the two-dimensional systems; & 2. As it is visualization method, it cannot give us the quantitative result, only the qualitative result; & 3. This method can only be applied to autonomous systems, not non-autonomous systems \\ \cline{2-3}  & & & \\ \cline{2-3}  & & & \\ \cline{2-3}  & & & \\ \cline{2-3}  & & & \\ \cline{2-3}  & & & \\ \hline \end{tabular}
\end{table} 
Table 1: Comparison among different approaches for stability analysis of an equilibrium point and limit cycle. Figure 11: Plot of the system when \(\dot{x}=0\) and \(\dot{y}=0\). From the above comparison, we can see that the Lyapunov-LaSalle energy-based analytical method can give quantitative result about the stability of an equilibrium point, and the Lyapunov-LaSalle energy-based approach is applicable to all dynamical systems [77; 78; 79; 80; 81; 82]. However, in order to use this method, we need to choose the right Lyapunov function, and usually it is difficult to choose the right Lyapunov candidate function. If one did not manage finding one, it does not mean the equilibrium point being analyzed is unstable, it could be stable or it could be unstable. Meanwhile, this method involves with large calculations when we calculate the time derivative of the Lyapunov candidate function [83; 84; 85; 86; 87; 88; 89]. The graphical phase plane approach is a better choice to analyze the stability of an equilibrium point if quantitative result is not needed [90; 91; 92], and this approach is a visualization-based method, which can give the conclusion without being getting into calculation, and it is faster than the Lyapunov method when determining the stability of an equilibrium point. However, this method most of the time is limited to the two-dimensional systems and can only be applied to autonomous systems, not non-autonomous systems. Regarding the linearizing-around-an equilibrium-point approach, this involves a very simple calculation because the nonlinear system has been linearized, and this method can give us the qualitatively correct dynamics information quickly near the equilibrium point. Use with care is needed when using this method, as it is not applicable to all dynamical systems. It can only give us the correct dynamics information quickly near the equilibrium point when the equilibrium point is saddle, node, or focus/spiral.