Example 1. Investigate the stability of the equilibrium point $x=0, y=0$ of the system by the first approximation

$$
\left\{\begin{array}{l}
\dot{x}=2 x+y-5 y^{2} \\
\dot{y}=3 x^{6}+y+\frac{x^{3}}{2}
\end{array}\left(\dot{x}=\frac{d x}{d t}, \quad \dot{y}=\frac{d y}{d t}\right)\right.
$$
Solution. Systems of the first approximation

$$
\left\{\begin{array}{l}
\dot{x}=2 x+y \\
\dot{y}=3 x+y
\end{array}\right.
$$

Nonlinear terms satisfy the necessary conditions: their order is greater than or equal to two. We form the characteristic equation for system (6):

$$
\left|\begin{array}{cc}
2-\lambda & 1 \\
3 & 1-\lambda
\end{array}\right|=0, \text { or } \quad \lambda^{2}-3 \lambda-1=0
$$

The roots of the characteristic equation (7) $\lambda_{1}=\frac{3+\sqrt{13}}{2}, \lambda_{2}=\frac{3-\sqrt{13}}{2}$ are real and $\lambda_{1}>0$. Therefore, the zero solution $x=0, y=0$ of system (5) is unstable.