# Stability of an equilibrium

From a Center-Manifold reduction I get the following system: $$\begin{pmatrix}\dot x \\\dot y\end{pmatrix}=\begin{pmatrix}-y(2x^2-2xy+y^2)\\x\end{pmatrix}$$ The aim is to analyze the stability of the origin. Using Matlab I had the impression that the origin is unstable. My idea was to introduce the time-reverse system $$\begin{pmatrix}\dot x \\\dot y\end{pmatrix}=\begin{pmatrix}y(2x^2-2xy+y^2)\\-x\end{pmatrix}$$ which appears to be asymptotically stable (Matlab). Unfortunately I didn't manage to find a Lyapunov-function and as far as I know it's the only possibility to proof stability of a non-hyperbolic equilibrium. Any advice or a different approach is welcome :D