How to prove this ODE is stable but not asymptotically stable?

Question

Consider the ODE in polar coordinates: $$ r'=f(r),\theta ' =1 $$ where $$ f(r)=r\sin (1/r^2), r\neq 0, f(0)=0. $$ show that the origin is stable but not asymptotically stable.

Answer 1

Consider the roots of the equation $\sin (1/r^2)=0$. Let $r_2>r_1>0$ be two distinct roots of that equation. Note that we can make these roots arbitrarily small (after all, all points of the form $1/\sqrt{\pi k}$ for $k\in \Bbb N$ are the roots).

Obviously the functions $r(t)=r_1$ and $r(t)=r_2$ both are the solutions of hte differential equation, therefore if the initial data satisfies $r_1<r(0)<r_2$, you obtain that $r_1<r(t)<r_2$. This estimation guarantees that the zero solution is stable, but not asymptotically stable.