# Local asymptotical stability for an ODE

Consider a system: \begin{align*} \frac{d}{dt} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \begin{pmatrix} x_2 \\ a x_1 + b x_1^2 \end{pmatrix} \end{align*} with $a < 0$ and $b\ne0$. My question is whether the equilibrium point $x=0$ is a locally asymptotically stable equilibrium point. I tried some Lyapunov functions, for example: $$V = -\frac{a}{2} x_1^2 + \frac{1}{2} x_2^2 - \frac{b}{3}x_1^3$$ give $$L_f V = \frac{\partial V}{\partial x}f = 0,$$ hence the Lyapunov-stability, but not asymptotic stability.

The fact that $dV/dt = 0$ says that the trajectories in the $x_1,x_2$ plane lie on the curves $V = \text{constant}$. The origin is a strict local minimum of $V$, so near the origin these form closed curves around the origin.
From another point of view, the lack of asymptotic stability is the result of the invariance of the system under the symmetry $x_1 \to x_1$, $x_2 \to -x_2$, $t \to -t$. If such a system is started at, say, $x_1 = p, x_2 = 0, t = 0$, and reaches $x_2 = 0$ again at $x_1 = q, t = T$, then by symmetry we will have $(x_1(T+t),x_2(T+t)) = (x_1(T-t),-x_2(T-t))$, so that at $t=2T$ it comes back to $x_1 = p, x_2 = 0$, and will continue in a closed orbit of period $2T$.