Showing the trivial solution is asymptotically stable Consider the second order ODE $u'' +(u^2-2u+1)u'+u^3-u^5=0$. We'd like to show the $0$ solution is asymptotically stable. Writing it as a first order system and of the form $\vec{u}' = A\vec{u} + F(\vec{u})$ yields
$$
\begin{pmatrix} u \\v \end{pmatrix}' = \begin{pmatrix}0 && 1\\ 0 && -1 \end{pmatrix}\begin{pmatrix} u \\v \end{pmatrix} + \begin{pmatrix} 0 \\-u^2v+2uv-u^3+u^5 \end{pmatrix}.
$$
Unfortunately, the matrix has eigenvalues $0$ and $-1$, and $0$ is not negative, so we can't get asymptotic stability that way. I'm thinking we might want to find a Lyapunov function, but I do not have much experience with this, so any help would be appreciated.
 A: A quite common trick in studying systems of form $\ddot{u} = -\nabla F(u)$ is to multiply both sides by $\dot{u}$:
$$\dot{u} \cdot \ddot{u} = \dot{u} \cdot (-\nabla F(u))$$
$$\dfrac{d}{dt} \left ( \dfrac{\dot{u}^2}{2} \right ) = \dfrac{d}{dt} \left ( -F(u) \right ) $$
$$\dfrac{d}{dt} \left ( \dfrac{\dot{u}^2}{2} + F(u) \right ) = 0 $$
$$\dfrac{\dot{u}^2}{2} + F(u) = \dfrac{\dot{u}^2(0)}{2} + F(u(0)) $$
And in that case the system is conservative and has first integral. It also can't have asymptotically stable and completely unstable equilibria. But your case is not the same: you have terms with $\dot{u}$. Still we can apply the trick to your system:
$$u'' +(u^2-2u+1)u'+u^3-u^5=0$$
$$ u' \cdot u'' +(u^2-2u+1)\cdot (u')^2 + u' \cdot (u^3-u^5)=0 $$
$$ u' \cdot u'' + u' \cdot (u^3-u^5)= -(u^2-2u+1)\cdot (u')^2 $$
$$ \dfrac{d}{dt} \left ( \dfrac{\dot{u}^2}{2} + \dfrac{u^4}{4} - \dfrac{u^6}{6} \right ) = -(u-1)^2 \cdot (u')^2 $$
So what we got here is that derivative of some function w.r.t. time is non-positive. This is almost a setting of Lyapunov theorem. You can check that $\dfrac{v^2}{2} + \dfrac{u^4}{4} - \dfrac{u^6}{6}$ has strict local minimum at $(0, 0)$ so this function could be used in Lyapunov theorem. At least we have that this equilibrium is Lyapunov stable. As far as I understand, the asymptotical stability can be concluded from LaSalle's invariance principle. 
