Attraction and Repulsion in Dynamical Systems

Typically, the attraction term dominates at larger distances, while the repulsion is dominant at smaller distances. For and this formula refers to the Lennard–Jones 6-12 potential, used in molecular dynamics models. For it could be viewed as gravitational potential plus an added repulsive component at close distances. In general, this kind of function could be used in simulations related to a variety of problem domains. In [1] such potential functions are applied to model swarming behavior or variations of chase-and-evade algorithms.

Reference

[1] D. M. Bourg and G. Seemann, AI for Game Developers, Sebastopol, CA: O'Reilly Media, 2004 pp. 80–89.