Let $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ be a two-dimensional dynamical system with a fixed point at $\mathbf{x}=\mathbf{0}$. Define a Lyapunov function $V(\mathbf{x})$ and explain what it means for $\mathbf{x}=\mathbf{0}$ to be Lyapunov stable.

For the system

$$\begin{aligned} &\dot{x}=-x-2 y+x^{3} \\ &\dot{y}=-y+x+\frac{1}{2} y^{3}+x^{2} y \end{aligned}$$

determine the values of $C$ for which $V=x^{2}+C y^{2}$ is a Lyapunov function in a sufficiently small neighbourhood of the origin.

For the case $C=2$, find $V_{1}$ and $V_{2}$ such that $V(\mathbf{x})<V_{1}$ at $t=0$ implies that $V \rightarrow 0$ as $t \rightarrow \infty$ and $V(\mathbf{x})>V_{2}$ at $t=0$ implies that $V \rightarrow \infty$ as $t \rightarrow \infty$