Consider an autonomous vector field on the plane having a hyperbolic fixed point with a homoclinic orbit connecting the hyperbolic fixed point. Can a trajectory starting at any point on the homoclinic orbit reach the hyperbolic fixed point in a finite time?
I believe that by the definition of a homoclinic orbit that it cannot be reached in a finite time as $\phi(t)$ is a homoclinic orbit if $\psi(t) \rightarrow x_0$ as $t\rightarrow \pm\infty$ where $x'=f(x) $ at $x=x_0$. Can anyone confirm this?