I'm trying to prove that the equilibirum $(0,0)$ of the second order equation $$\frac{d^{2}x}{dt^{2}}+\frac{dx}{dt}+x+x^{3}=0$$ is globally asymptotically stable.
So far, by writing it as $$\frac{dx}{dt}=y$$ $$\frac{dy}{dt}=-y-x-x^{3}$$ I have seen that the origin is locally an antracting focus due to Hartman's theorem. But now I don't know how to proceed. Any ideas?
Thanks!
Oh, I've just realised that if I rewrite the equation as the motion of a particle under a potential field V(x), I have $$\frac{d^{2}x}{dt^{2}}=-V^{'}(x)-bx$$ and it is easy to check that $$E(x,y)=\frac{y^{2}}{2}+V(x)$$ is a Lyapunov function. Since the only subset in $Q=\{(x,y)\in\mathbb{R}^{2};\frac{dE}{dt}(x,y)=0\}$ that is forward invariant is the origin, then $(0,0)$ is globally asymptotically stable!