Exercise 2. Consider the autonomous system $$\left\{\begin{array}{l} u_{1}'=u_{2}\\ u_{2}'=-u_{2}^{3}-u_{1} \end{array}\right.$$ and the function $V(u_{1}, u_{2})$ in $\mathbb{R}^{2}$ given by $$ V(u_{1}, u_{2})=u_{1}^{2}+u_{2}^{2}. $$
Find the linearization of this system at $0$. Can one directly deduce from properties of this linear system the stability/unstability of the trivial solution $u\equiv 0$ of the given system?
Let $u(x)=(u_{1}(x),u_{2}(x))$ be a maximal solution of the given system defined on a maximal interval of existence $I=(a,b)$. Show that $V(u(x))$ is a decreasing function on $I.$
Deduce that the right end point $b$ of $I$ is$+\infty$ and the trivial solution $u\equiv 0$ is Lyapounov stable.
Show that if $u\not\equiv 0$ then $V(u(x))$ is strictly decreasing in $(a,+\infty)$.
Show that $u(x)\rightarrow 0$ as $ x\rightarrow\infty$. Hint: use a function $W(u)=V(u)+\epsilon u_{1}^{3}u_{2}$ with some constant $\epsilon>0$ small enough.
Deduce that all the solutions of the given system are asymptotically stable.
I am able to find what is the linearzation of the system at $0$. But the rest of part(1) and the other parts I am unsure.