TITLE: Determining stability of equilibrium points for a non linear system
QUESTION [6 upvotes]: Given the system:
$$
\left\{ 
\begin{array}{}
\dot x=-x^3y^2 \\ 
\dot y = -2x^2y^3
\end{array}
\right. 
$$
I need to find the equilibrium points and to determine whether the system is stable around them. I'v found $(0,0)$ to be a stable equilibrium point, using the Lyapunov function $V(x)=x^2+y^2$.
The rest of the equilibrium points are $(x_0,0), (0,y_0) \; , \; x_0,y_0 \in \Bbb R$. I'm having trouble with determining wether they are stable or not. Linearization is not useful in this case, and I couldn't find any Lyapunov function.

REPLY [8 votes]: Here's a hint:
Your system is
$$
\frac{d}{dt}
\begin{pmatrix}
x \\ y
\end{pmatrix}
=
x^2 y^2
\begin{pmatrix}
-x \\ -2y
\end{pmatrix}
,
$$
which only differs from the linear system
$$
\frac{d}{dt}
\begin{pmatrix}
x \\ y
\end{pmatrix}
=
\begin{pmatrix}
-x \\ -2y
\end{pmatrix}
$$
by the nonnegative factor $x^2 y^2$.
This means that except for the $x$ and $y$ axes (where $x^2 y^2=0$), the trajectories for your system will be the same as for the linear system, up to a reparametrization of time.
Using this, you should be able to see what happens if you perturb an equilibrium $(x_0,0)$ or $(0,y_0)$ a little (so that you move off the axis).