I have a rather simple-looking system which exhibits a particular behaviour in simulation, and I would now like to attempt to prove this formally. The problem is, I don't really know where to start, and would very much appreciate some suggestions on different ways that I could try out.
System Description
I will try to describe the system as concisely as I can (but I'm happy to add details as comments on request). There are $n$ points on a plane, which are initially randomly distributed. Each point has a position, $(x,y)$, and an orientation, $\theta$, and projects a line of sight (LOS) along this orientation.
Every point moves in one of two ways, depending on whether its LOS intersects at least one circle of radius $r$ centered around another point ($r$ being a fixed parameter). If there is an intersection, the point turns clockwise on the spot (i.e. $\dot{\theta}=\omega$, $\omega$ being a fixed parameter). Otherwise, it moves backwards (i.e. with the LOS pointing opposite the direction of motion) along a circular trajectory of radius $R$ ($R$ being a fixed parameter).
Emergent Behaviour
For the right values of $R$ and $\omega$ (which I have found through simulation), the points will aggregate into one 'cluster' (one can define a metric, e.g., by saying that every point is at a distance of at most $d$ from at least one other point). I want to prove that this aggregation happens from any given set of initial positions and orientations.
What I Have Tried So Far
Two approaches come to my mind. One can either attack the problem using geometry, or represent it in state space and try to analyse the resultant system of ODEs.
Using geometry, I have made the assumption that only one point moves at a time (this would not be a complete proof, but is acceptable for me). I have rather easily proven that one moving point will approach one static point, but I have no idea how to generalize this.
A dynamical systems approach seems hard because of the severe non-linearities in the input function. The $n$-body gravitational problem is unsolvable, and to my mind, its dynamics seem simpler than this!