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
2026-03-27 23:48:55.1774655335
Stability of an equilibrium
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