Finding a Lyapunov function for a given system I need to find a Lyapunov function for $(0,0)$ in the system:
\begin{cases} x' = -2x^4 + y \\ y' = -2x - 2y^6 \end{cases}
Graph built using this tool showed that there should be stability but not asymptotic stability. I'd like to prove that fact by means of Lyapunov function but cannot find the appropriate one among the most common ones such as $Ax^{2n}+By^{2m}$, $Ax^2+By^2+Cxy$, etc.
Please, give me some hint about how to proceed with the search of the suitable function or even the desired function itself it you know the easy way of finding it in this case.
Thanks in advance

Dynamical system
$$
%
\begin{align}
%
  \dot{x} &= -2x^{4} + y \\
%
  \dot{y} &= -2x - 2y^{6} \\
%
\end{align}
%
$$
There are two critical points: the origin $(0,0)$ and 
$$\left\{-\frac{1}{2^{6/23}},2 \left(\frac{1}{2^{6/23}}\right)^4\right\} =\left\{ -\frac{1}{2^{6/23}}, \frac{1}{2^{1/23}} \right\} \approx \{-0.834585,0.970313\}$$
Phase portrait
The phase portrait below displays the nullcline $\dot{x}=0$ with a red, dashed line and $\dot{x}=0$ in purple.

Stability
The polar transformations
$$
x = r \cos \theta, \qquad y = r \sin \theta
$$
lead to 
$$
 r^{2} = x^{2} + y^{2} 
$$
which implies
$$
 \dot{r} = \frac{x\dot{x}+y\dot{y}}{\sqrt{x^2+y^2}} = \frac{-2 x^5-x y-2 y^7}{\sqrt{x^2+y^2}},
$$
plotted below.

The problem is that the sign of $\dot{r}$ changes twice in every neighborhood of the origin.
