Consider the ODE of damped pendulum (for simplicity I set the mass, the length, and the friction coefficient, to be equal to one):
$$\theta' =\omega $$ $$\omega' =-\sin \theta - \omega$$
The question is from Hirsch's and Smale's ODE book (ch.9 section 3, ex 6b). They ask to discuss the set of trajectories converging to the unstable equilibrium $(\pi,0)$. The difficulty I am facing is this: is it possible that there is an interval $I$ in $\mathbb{R}_+$ such that for all $\omega_0 \in I$, if we start from $(0,\omega_0)$, we converge to $(\pi,0)$ and the angle is always increasing? This seems unintuitive since it would imply that in any neighborhood of $(\pi,0)$, the basin of attraction has positive measure. This is not true for the linearization around $(\pi,0)$, which is a saddle. However, we cannot (at least not directly) use Hartman-Grobman, because a set of positive measure can be homeomorphic to a set of zero measure. These examples are very pathological though. Any help will be appreciated!