  In this article we investigate a family of nonlinear evolutions of polygons
in the plane called the $\beta$-polygon flow and obtain some results analogous
to results for the smooth curve shortening flow: (1) any planar polygon shrinks
to a point and (2) a regular polygon with five or more vertices is
asymptotically stable in the sense that nearby polygons shrink to points that
rescale to a regular polygon. In dimension four we show that the shape of a
square is locally stable under perturbations along a hypersurface of all
possible perturbations. Furthermore, we are able to show that under a lower
bound on angles there exists a rescaled sequence extracted from the evolution
that converges to a limiting polygon that is a self-similar solution of the
flow. The last result uses a monotonicity formula analogous to Huisken's for
the curve shortening flow.
