We should develop a better understanding of precisely what makes a direction stable or unstable. Schematic diagrams of which directions may be stable and unstable in a close encounter of a moving particle and a fixed particle.
All perturbations happen at the point of closest approach.
If the particle's position is perturbed off the orbit, but still in the plane of the orbit, the particle will diverge from it's original orbit, although not exponentially. Thus an unstable direction at the point of closest approach may be along the line joining the particles. Thus, at the point of closest approach, a stable direction may lie along the orbit. We see that initially the orbits converge, then cross at point , then diverge from one another. It would be relatively easy to compute and output the dot product of the stable and unstable vectors with these special directions during a collision, but an even better understanding could be had by graphical visualization.