  In this article we investigate the convergence behavior of gathering
protocols with fixed circulant topologies using tools form dynamical systems.
Given a fixed number of mobile entities moving in the Euclidean plane, we model
a gathering protocol as a system of ordinary differential equations whose
equilibria are exactly all possible gathering points. Then, we find necessary
and sufficient conditions for the structure of the underlying interaction graph
such that the protocol is stable and converging, i.e., gathering, in the
distributive computing sense by using tools from dynamical systems. Moreover,
these tools allow for a more fine grained analysis in terms of speed of
convergence in the dynamical systems sense. In fact, we derive a decomposition
of the state space into stable invariant subspaces with different convergence
rates. In particular, this decomposition is identical for every (linear)
circulant gathering protocol, whereas only the convergence rates depend on the
weights in interaction graph itself.

Dynamical Systems