Q:

Show that the origin of the following system is stable (non-linear vs linear stability)

$$\dot{x} = -2y -x^3 + x^2 y^2$$
$$\dot{y} = x - x^2 y$$
Jacobian evaluation at $(0, 0)$:
$$\begin{pmatrix}0 & -2\\ 1 & 0\end{pmatrix}$$
Clearly the determinant is greater than zero but the trace is zero. Hence the linearization gives the impression of a center, which is marginally stable.
How do I show the origin is stable?
Related question: If in the case that the linearization of a system at a point is not marginally stable, is the stability of the linear system, the same as that of the non-linear system at the point (stable vs unstable)? 

A:

Near the origin we have
$$
\dot{x} = -2y + O(r^3)\\
\dot{y} = x +O(r^3)
$$
here $r = \sqrt{x^2+y^2}$
The stream plot for the complete system

and for the approximate system

NOTE
Considering the approximated system we have
$$
x\dot x = -2x y\\
2y\dot y = 2x y
$$
adding
$$
\frac{d}{dt}(x^2+2y^2) = 0
$$
or
$$
x^2+2y^2 = C_0
$$
which means that the for the approximated system the origin is a center. 
[![enter image description here][3]][3]

