# Proof Strategy for a Dynamical System of Points on the Plane

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!

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This is a simplified and slightly tweaked version of a Boids type setup (en.wikipedia.org/wiki/Boids)? – Sharkos Apr 11 '13 at 0:41
@Sharkos Yes, I can see the link to Boids, and I am in fact studying swarming systems. The main difference is that in my case the points only have a binary sensor, whereas in Boids they know the distance to and velocities of the ones around them. This is part of my active research in the field. – MGA Apr 11 '13 at 0:44
@Amzoti I can give you the equations for the dynamical system representation if you like: $\dot{x}=R\omega \cos{x}$, $\dot{y}=R\omega \sin{x}$, $\dot{\theta} = \omega$, where $R$ and $\omega$ depend on whether the LOS makes an intersection or not. If it does not, $R$ is positive, $\omega$ is negative. If it does, $R=0$, $\omega$ is positive. The equation for whether an intersection occurs or not seems difficult to write down in closed form, but it is definitely an OR combination of a function on each point $j\neq i$. – MGA Apr 11 '13 at 0:54
@Amzoti I wouldn't see it that way. $\omega$ is a fixed parameter (2 actually, one for each possible state). I would probably see it that for each point $i$, $(\dot{x}_i,\dot{y}_i,\dot{\theta}_i)=f(X,Y,\theta_i)$, where $X$ and $Y$ are the positions of all the points and $f$ computes the activation for point $i$ given the positions of the other points (note that you only need your own orientation to compute your activation). So it's a homogeneous system of ODEs, the problem is with computing the activations. – MGA Apr 11 '13 at 1:35
The response on non-intersection is not clear to me. What happens to $\theta$ if there is no intersection ? Does it satisy $\dot\theta=-\omega$ ? – nonlinearism Apr 11 '13 at 2:54